THE UN I V E R S ITY OF MI C H I G A N COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report INTEGRALS OF THE CALCULUS OF VARIATIONS Arthur W. J, Stoddart ORA Project 05304 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NO. GP-57 WASHINGTON, D. C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR April 1964

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1964.

PREFACE This thesis is dedicated to Professor Lamberto Cesari Professor Do Bo Sawyer My parents My wife I must express special gratitude to Professor Cesari for his patient, yet inspiring supervision of my work over the past two and a half years. Part I of this thesis was carried out under partial support of NSF research grant GP-57 at The University of Michigano I also wish to thank my committee) and particularly Professor Go W. Hedstrom, the second reader, ii

TABLE OF CONTENTS Page ABSTRACT v 1. INTRODUCTION 1 Part I 2. THE BC-INTEGRAL 6 2.1 Introduction 6 2.2 Quasi Additive Functions 8 2.3 The Weierstrass-Type Integral ff(T,I) as a BCIntegral 11 2.4 Induced Measures 13 2.5 Representation of BC-Integrals 19 3. THE LEBESGUE-STIELTJES INTEGRAL AS A BC-INTEGRAL 20 3.1 The Interval Function r 20 3.2 Comparison with Previous Results 25 4. THE BEND OF A CURVE AS A BC-INTEGRAL 28 4.1 The Bend of a Curve 28 4.2 Continuous Light Curves with Finite Bend 29 4.3 General Light Curves 33 4.4 Angle Swept Out by Direction 38 5. GENERALIZED WEIERSTRASS-TYPE INTEGRALS Jf(, i) AS BCINTEGRALS 42 5.1 A Lemma 43 5.2 Existence of the Integral ff(5,i) 44 5.3 Transformation of the Integral ff(, ) 49 5.4 The Integral ff(C,i) as a Lebesgue-Stieltjes Integral 52 5.5 The Conditions (5) and (Z) 55 6. INVARIANCE PROPERTIES OF INTEGRALS ff(,~) 58 6.1 Relations R Between Interval Functions 58 6.2 Invariance of Integrals ff(l,1) Under Relations R 60 6.3 Substitution of the Invariance of V in the Relations 65 6.4 Properties of the Relations R 67 6.5 Parametric Curve Integrals 69 6.6 Parametric Surface Integrals 71 iii

TABLE OF CONTENTS (Concluded) Page 7. ROTATIONAL PROPERTIES OF INTEGRALS ff(, ) 76 7oi Approximative Rotational Relations 76 7.2 Relation Between Integrals 77 7.3 Substitution of Special Relations 79 8. SEMICONTINUITY OF INTEGRALS 81 8.1 The Topology T 81 802 The First Semicontinuity Theorem 83 8.3 The Second Semicontinuity Theorem 88 8 4 Convexity Conditions 92 8.5 The Homogeneous Case 95 9. SEMICONTINUITY IN PARTICULAR CASES 97 9,1 Parametric Curve Integrals ff(X,X')dE 97 9.2 Parametric Surface Integrals ff(X,J)du dv 100 9o3 Non-Parametric Integrals:f(w,X,grad X) d. 104 9,4 Curve Integrals Involving Higher Derivatives 107 Part II 10o THE SHAPE OF LEVEL SURFACES OF HARMONIC FUNCTIONS IN THREE DIMENSIONS 117 10.1 Introduction 117 10o2 Star-Shaped Regions 118 1.0,3 Convex Regions 121 10.4 A Counter Example 123 BIBLIOGRAPHY 125 iv

ABSTRACT The general purpose of this thesis is to study-in an abstract and unified. formulation-properties of the integrals of the calculus of variations which are usually discussed separately in a number of particular situations (parametric and non-parametric curves, surfaces, varieties, with differential elements of orders one, two, etc.). An abstract form of the integrals of the calculus of variations has been given by Cesari in two recent papers, where Burkill-type or BC-integrals of vector-valued set functions relative to a mesh function are treated in a very general setting. Cesari introduced a condition of "quasi additivity" on the set function, that is sufficient for the existence of the corresponding BCintegral. In particular, the formulation includes Weierstrass-type integrals Jf(T,O) over a Euclidean variety T with a quasi additive set function 0 of bounded variation, and therefore the Weierstrass integrals of the calculus of variations for curves and surfaces studied by Tonelli, Bouligand, Menger, and Pauc. Under suitable conditions, the integral Jf(T,O) can be expressed both as a BC-integral and as a Lebesgue-Stieltjes integral with respect to a measure induced by i. In this thesis, a modification of the Weierstrass-type integral is made to allow a more convenient expression of the integrals of the calculus of variations. Specifically, Cesari's results for the integral Jf(T,O) are extended to an integral of the form ff(S,O), where now 5 is a set function with appropriate properties. The particular properties considered in this thesis are invariance, behavior under rotation, and semicontinuity. Here invariance means that the integrals corresponding to two systems (o,), (',n') of set functions have the same value. We introduce relations between the systems (~,), (1',d') that ensure invariance. Frechet invariance for parametric curve and surface integrals is framed under these invariance theorems. Concerning behavior under rotation, the integrals are proved invariant under rotations in Euclidean space provided such rotation leads to pairs (Gq), (1',T') of set functions related in an appropriate sense. Some very general theorems of lower semicontinuity of the integrals ff(S,O) in the Lebesgue-Stieltjes form are proved under suitable convexity conditions on f. The semicontinuity is relative to a topology appropriate to the formulation. These general theorems are then shown to contain as corollaries the particular lower semicontinuity theorems of Tonelli and Turner for parametric curves, of Cesari and Turner for parametric surfaces, of Tonelli for non-parametric curves, and of Cinquini for parametric curves in E3 depending on differential elements of orders two, or three. v

In addition, further functionals are considered in Cesari's formulation. The Lebesgue-Stieltjes integral is shown to be a BC-integral of a quasi additive set function relative to the standard mesh function. The bend or total curvature of a curve is expressed as a BC-integral of various set functions relative to appropriate mesh functions. The last part of the thesis concerns the shape of level surfaces of harmonic functions in three dimensions. In terms of the corresponding regions of higher potential, or "regions of potential," the results can be summarized as follows. If two regions of potential are convex, then every intermediate region of potential is convex. If two regions of potential are star-shaped relative to some point, then every intermediate region of potential is similarly star-shaped. On the other hand, we prove by an example that if two regions of potential are merely simply connected, the intermediate regions of potential need not be simply connected. vi

I. INTRODUCTION The techniques used in the direct method of the calculus of variations show an underlying similarity and unity which has long been noted by many authors such as Bouligand,2 Tonelli,26 and Menger.l7 These similarities can be seen in the so far parallel but quite separated discussions of the integrals of the calculus of variations for parametric and non-parametric curves in Em, for parametric and non-parametric surfaces in Em, for the same integrals depending on differential elements of higher orders, and so on. Our objective in this thesis has been to give a unified discussion of the main properties of the integrals of the calculus of variations in the frame of an axiomatic treatment of the same integrals. We shall discuss essentially the properties of semicontinuity, invariance with respect to representation, and invariance with respect to orthogonal linear transformations in Em. A major step in this direction was made by Cesari6'7 who introduced the concept of quasi additive set function i with respect to a given mesh function 6, and developed an axiomatic treatment of the corresponding integral B = f$ for set functions 0 which are quasi additive and of bounded variation. We shall call fA a Burkill-Cesari integral, or BCintegral. This integral includes both the usual Burkill-type integrals, and the apparently unrelated parametric Weierstrass-type integrals 1

2 f f(T,~) relative to a mapping T: A + Em and a set function 0 which is quasi additive and of bounded variation with respect to a mesh function 6. Indeed, as Cesari proved in Ref. 6 under general assumptions, the set function 0 = f(T,O) is again quasi additive and of bounded variation with respect to the same mesh function 6, and hence f f(T,O) can be defined as a BC-integral f f(T,T) = f 0. Under a convenient system of axioms, the BC-integral can be represented as a Lebesgue-Stieltjes integral, in particular f f(T,I) = (A) ff(T,9)dt in a convenient measure space (Aj, l) ). The line integrals and surface integrals of the calculus of variations in parametric form are included in the axiomatic treatment of Refs. 6 and 7, together with a number of other familiar concepts such as total variation of a function of one real variable, Jordan length of a curve, Lebesgue area of a surface, and Lebesgue-Stieltjes integral of a [1-measurable function f: A - E1 in a measure space (A,3,,i.). Nishiura has shown in his thesis19 that this process also covers integrals over k-varieties in Em. Cesari"'s results in Refs. 6 and 7 are partially surveyed in Chapter 2 of this thesis. Chapter 3 complements the remark made by Cesari that the LebesgueStieltjes integral of a [>-integrable function f: A + E1 in any measure space (A,(,J1) can be interpreted as a BC-integral, B = f 0 of a convenient interval function - which is quasi additive with respect to a conveniently chosen mesh function 6. Here we show that the same result can be accomplished by means of an interval function which is quasi additive with respect to the mesh function of the usual theory of Lebesgue

5 Stieltjes integrals. In Chapter 4 we consider the bend 1, or total curvature, of a curve X in En, as defined by Iseki.l3 Under various sets of general assumptions on the curve, we show that Iseki's bend can be expressed as the BCintegral of an appropriate quasi additive function i. By this approach, it is shown that the bend RQ of the curve X is completely similar to the Jordan length L, and partakes with L the same formal properties and axiomatic treatment. An analogous result can be expected for possible extensions to bends Qk of any order k, 1 < k < n (k = 1 n, length; k = 2 < n, bend or total curvature; k = 3 < n, total torsion; etc.). In Chapter 5 we take into consideration a modified form of Weierstrasstype integrals I f(,4), where now both 5 and i are set functions and i is quasi additive and of bounded variation with respect to a given mesh function 6. Under general hypotheses on f,, and i, we show that the set function 0 = f( r,~) is again quasi additive and of bounded variation with respect to the same mesh function 5, and hence ff( f,~) = D is again a BC-integral. By this process, the line and surface integrals depending on differential elements of first and second, or higher orders, of the calculus of variations can be included in the same axiomatic treatment mentioned above. In Chapter 6 we discuss, in the frame of the same axiomatic treatment, invariant properties of the present integrals with respect to change of the generating set functions. In Chapter 7 we discuss the invariant

4 character of the same integrals with respect to linear orthogonal transformations in Em. Both the results of Chapters 6 and 7 extend results proved by Cesari and Turner in surface area theory, and show, therefore, that also these results hold in the present general axiomatic treatment. In Chapter 8 we discuss the difficult question of the semicontinuity of "regular" integrals. In the same present axiomatic treatment we prove that convenient properties of convexity on f (regularity) assure properties of semicontinuity of jf( 5,$) with respect to appropriate topologies. These general results do not require differentiability conditions on f, in harmony with recent work on line and surface integrals of the calculus of variations (L. Tonelli,25 and L. Turner29). The complexity of the present axiomatic treatment is to be expected in view of the generality of the results. In Chapter 9, we apply the results of Chapter 8 to the particular line and surface integrals of the calculus of variations. We deduce in each case corresponding sufficient conditions for lower semicontinuity proved by Tonelli, Cinquini, Turner, and others by a number of separate arguments. Chapter 10 (Part II of the thesis) deals with properties of harmonic functions in E3. It concerns the situation of a function i(P), PcE3, continuous in E3 and harmonic in an open connected set D C E3, such that the complement D' = E3-D is the union of two closed disjoint sets Co and C1, C1 compact, and = n, = 1 on C, b n C = on 3-Co are star-shaped with respect to the origin, that is, the intersections of every half-line from the origin with C1 and Co are segments, then the

5 regions (P:'(P) > k), 0 < k < 1, are also star-shaped with respect to the origin. In addition, if C1 and Co are convex, then each region (p:'(P) > k), 0 < k < 1, is also convex. This work was initiated and practically completed at the University of Otago. The research of this Chapter 10 was suggested to the writer by Professor D. B. Sawyer of the University of Otago, and continues previous work of R. M. Gabriel.31 The work on this chapter was completed-particularly the rigorous treatment of the counter example-at The University of Michigan.

PART I 2. THE BC-INTEGRAL 2.1 INTRODUCTION In this chapter, we partially review results of Cesari.6'7 The proofs are not given. Some of the results in Sections 2.3 and 2.5 will be extended in Chapter 5 and there proved in a modified setting. Consider a set A, a collection (I) of subsets I ("intervals") of A, and a non-empty family 09 of finite systems D = [I] of sets IE(I). We shall make the following general assumptions: (bl) either (bl) A is any set, and the sets I of each D c/ D = [I], are disjoint, that is, IcD, JeD implies I nJ = 0; or (bi) A is a topological space, "L is the collection of its open sets, the sets I of [I) possess interior points, and the sets of each D are non-overlapping, that is, IcD, JeD implies I nJO = IO~ J = 0 where - and ~ denote closure and interior respectively in the topology (A,7J). Let 5 be a real function ("mesh") on (that is, defined for every system De j)), such that (dl) 0 < (D) < oo; (d2) for each ~ > 0, there is a system De ) with 6(D) <. Let i be a vector function on (I) with values i(I) = [r(I)] = [1i(I),...,,k(I)] in Ek. We call such i an "interval" function, and denote its Euclidean norm function by ||i||1 6

7 Define B(,A,5) = lim inf E 4r(I) 5(D) ID IeD Br,iA,5) = uim sup E I) 6(D)0 IED B(~,A,8) = [B_ (,A,6)], B(i,A,5) = [Br(,A,)] If B(O,A,5) = B(I,A,5), we call their common value the Burkill-Cesari or BC-integral B(i,A,6) = fJ. We shall include i, A, and 6 in the notation for B only where necessary. We denote B(||I1$|) by V, and B( Ir|) by Vr. The number V is called the total variation of i. If V < oo, we say that d is of bounded variation. These integrals have the following obvious properties relative to a given system A, 2, (I), 7 5,6: (i) If i and i' have finite BC-integrals, and a,a' are constants, then o+a'o' has finite BC-integral aB(O)+c'B(' ). (ii) If 4 and V' have BC-integrals and are positive, and 3,' are positive constants, then BP+P'T' has BC-integral PB(r) +P'B(*' ). Hence, for r real, and defining r+ = ( I11+)/2, 1- = ( |1|-1)/2, B(I L) = B(4+) + B(4-), B(4+) = B(4)+B()-), where, in each case, the integrals on the right hand side are assumed to exist.

8 If 4,4' are real with *(I) < *'(I) for every I, then B(4) < B(4') when these integrals exist. Also Vr < V < 7Vr 1|B|| (V)1/2 < V. 2.2 QUASI ADDITIVE FUNCTIONS In the setting of Section 2.1, an interval function i on (I) is called "quasi additive" with respect to a mesh function 6 on cGif, (i) for every 8 > 0, there exists Tr() > 0 such that, for every Doe co with o(D0) < n( ), there exists (E.,Do) > 0 such that, for every DE c with b(D) < x(,Do), 1||( I) - (J)( < ~ IeDo and Z, 11l(J)11 < ~ where Z(I) is the sum over all JeD with J CI, and Z' is the sum over all JeD contained in no I. A real interval function 4 on (I) is called "quasi subadditive" with respect to a mesh function 5 on v'if, under similar conditions, (X) [lf(I) - (I)(J)]+ < ~. IeDo

9 For these properties relative to a given system A, (I, ([I,', 6, the following results have been proved:6 (iii) If 2,0' are quasi additive and a,a' are constants, then +a' 1T' is quasi additive. (iv) If V,4' are quasi subadditive and P,P' are positive constants, then 4r+p'r' is quasi subadditive. (v) If Z is quasi additive, then each Or is quasi additive, and conversely. (vi) If *+ and ~- are quasi additive, then 4 and I1| are quasi additive, and conversely. (vii) If / is quasi additive, then ||/II, 1rl, ~ r r are quasi subadditive. (viii) If each jr is positive and quasi subadditive, then 1||11 is quasi subadditive. (ix) If z is quasi additive, then 0 has a finite BC-integral. If 4 is positive and quasi subadditive, then 4 has a BC-integral. Hence if i is quasi additive, then ||/I|, ||rl, +, and gr have BC-integrals. Note that if r is positive, then the following strengthening of the quasi subadditive condition (r*) for each ~ < 0 and each Doe rd, there exists X(~,Do) > 0 such that, for every Dc:o with 6(D) < k(<,Do), [t(I) -I )r(J)]+ < IeDo

10 gives B(*) = sup t(I) D~e IED Under condition (r) only, B(f) may not be the supremum of the corresponding sums, but only the limit as & + O, as was proved by examples in Ref. 6. (x) If r is positive and quasi subadditive, and B(V) < o, then 4 is quasi additive. As a consequence, we have the following results. (xi) If a vector interval function ~ is quasi additive and ( 11i11) < a, then Il, I, rli, and ~r are quasi additive. (xii) If each |[rl is quasi additive, then |1j11 is quasi additive. Hence, if each r+, 8r is quasi additive, then I||/| is quasi additive. In Ref. 6. Section 4, Cesari shows how the following functionals can be expressed as BC-integrals of quasi additive interval functions with respect to appropriate mesh functions: The Jordan length of continuous and discontinuous curves in Eno The Cauchy integral in an interval in Em. The Lebesgue-Stieltjes integral of a p-integrable function f(x): A - El, in a measure space (A, {,p). The parametric line integrals over curves C in En assumed to be only continuous and rectifiable (or Weierstrass integrals on C). These integrals can be thought of as depending on generalized differential elements of order one of C.

11 The parametric surface integrals over surfaces S in E3 assumed to be only continous and of finite Lebesgue area (or CesariWeierstrass integrals on S). These integrals can be thought of as depending on generalized differential elements of order one of S. 2.3 THE WEIERSTRASS-TYPE INTEGRAL Jf(T,~) AS A BC-INTEGRAL In the setting of Section 2.1, let < be a vector interval function from (I) to Ek. Let T = T(w), wEA, be a mapping from A to Em. For each I (I}, define c(I) = sup I|T(u)-T(v)l|, u, yE I and for each De~c, define W(D) = max o( I) IeD We shall assume that the following condition holds: (aw) iJ(D) < b(D) Let f(p,q) be a real function on T(A) x Ek. We shall denote by U the unit sphere [q: I||q| = 1) in Ek. Assume that (f) f(p,q) is positively homogeneous of degree one in q, that is, f(p,tq) = tf(p,q) for all t > O, peT(A), qeEk; and f(p,q) is bounded and uniformly continuous on T(A) x U.

12 Define @(I) = f(T(T),(I)), where T is any fixed point of I. Cesari6 has proved the following fundamental theorem: (xiii) If ~ is quasi additive and of bounded variation with respect to 6, and conditions (X), (f) hold, then (D(I) = f(T(T)',(I)), TeI is quasi additive and of bounded variation with respect to 8, and the elements X, n of the definition (~) can be defined independently of the choice of the points T in I. Thus the BC-integral of < exists and is finite, that is J = ff(T,~) = lim Zf(T(T),/(I)) 6 ( D) -0 Also, ff(T,z) is independent of the choice of T in I. By means of this theorem, the Weierstrass-type integrals ff(T,;) relative to a mapping T and a quasi additive set function / are reduced to the standard BC-integrals of Section 2.1. Let us mention here that line integrals for rectifiable continuous curves have been treated as Weierstrass integrals by Tonelli23 in view of applications to calculus of variations, and again more recently by N. Aronszajnl G. Bouligand,2 K. Menger,l7 C. Pauc.21 Surface integrals

13 for continuous surfaces of finite Lebesgue area have been treated as Weierstrass-type integrals by Cesari.,56 2.4 INDUCED MEASURES We shall assume here that A is a topological space. As in Ref. 7, we localize the properties of the system A, 2, [I),}, 6 in Section 2.1 to a class of subsets of A, including A, as follows. For each G in, let DG = (I: IeD,ICG),d G = DG: Dc 6o. We require that (b2) OG is non-empty for each non-empty GE A. For G non-empty, let OG be a mesh function on QCG, such that (d3) for each T > 0, there exists v(T,G) > 0 such that, for every De~O with b(D) < v(T,G), 6G(DG) < T and DG is non-empty. For a vector function ~ on (I), we can consider, as in Section 2.1, the existence of the following limits: Br(G) lim r( 6G(DG)-O IEDG V(G) = lim I) G(DG) O IGDG Vr(G) = lim j Ir(I) 5G(DG)-0O IeDG V+(G) = lim X (I) 5G(DG) 0 IeDG We shall call V(G) the total variation of, in G. If these limits exist, then, by the relation (d3) between 6 and 6G, they also exist for 6(D)-O0.

14 The properties of BC-integrals given in Section 2.1 obviously apply also for each Ge ~. Also, if G1,G2e: with G1 C G2, then V(G1) < V(G2) whenever these exist, and similarly for Vr, Vr, Vr. From now on, we shall assume also that (, )O(I) is quasi additive with respect to each 6G. In other words, we assume that, given 5 > 0 and Ge ~, there is a number ((&,G) > 0 such that, if DOG = [I] is any system in (0G with SG(DOG) < r1(~,G), then there is also a number k( 6,DOG,G) > 0 such that, for every system DG = [J] in with SG(DG) < X(j,DoG,G), we have T 11^(1) - X 0(J)Il < ~E IEDOG JC I and, II;(J)l| < g. where Z ranges over all JEDG not completely contained in any IEDOG. If ~ satisfies condition (p8), then B(G), V(G), Vr(G), V+(G), and Vr(G) exist for each Ge A. If also V(A) < o, then ||Ij|, l, r+, - satisfy condition (b~), and all the integrals above are finite. From now on, we shall require (a) C 26; (c) each Ie(I) is -U -connected Under hypotheses (a), (b), (c), (d), (< ), and V(A) < o, every disjoint sequence (Gi) with Gi and UGie..has

15 V(UGi) = Zv(Gi) and similarly for B, Vr, Vr, Vr. Consider the following conditions for sequences Gi in. (H1) If Gi - 0, then V(Gi) -+ 0 and similarly for B, Vr, V+, V. (H2) If Gi C Gi+l and Gi - Ge ~ then V(Gi) + V(G) and similarly for B, Vr, Vr, V.. (H3) If (,Gie ~ for each n and UGiEt then V(UGi) < ZV(Gi) and similarly for Vr, V, Vr The condition (e) "For every pair of distinct sets G1,G2cE. with G = G1 U G2c, G1 fl G2 /, and any Ie(I) with I C G, I fG G1 #, I f G2 # ~, there exists X(I,G1,G2) > 0 such that any DGe G with 6G(DG) < X(I,G1,G2) and JEDG with J C I have J C G1 or J C G2 or both,' with (H2) and V(A) < o implies (H3) Also, the condition (g) "The sets Ie(I} are fL-compact with (e) and V(A) < co implies (H2). From now on, we require that (a'), is a subtopology of 7/. For each XC A, define t(X) = inf V(G), G X and similarly for pr, + fr, Vlr from V(A) <, Vall these are finite, and we can define

16 vr(X) = +4(x) - Ir(X), v(X) = [vr(X) It can be shown that, for every set X C A, there is a sequence (Gi], Gi E, XCGi, with V(Gi) -+ (X), Vr(Gi) + r(X), Vr Gi) + (X) Vr(Gi) + r-(X), and, if V(A) < co, B(Gi) + v(X). Also 1r(X) = A+(X) + I(X) and k Pr()< (X) < (X) (X) r=l ||v(X)l| < [Z2,r(X)]1/2 _< (X) In addition, for every G~ c!, P(G) = V(G), and similarly for r., pr, Ir-, and, if V(A) < oo,. (xiv) If condition (Hi) holds, then p(0) = 0, and similarly for (xv) If conditions (H1), (H2), (H3) hold, and V(A) < oo, then Ap, Pr, PIr and Pr are other measures. We shall then define measurable sets in the standard way. Consider the condition (H4) For every GE, there exists a sequence (Gi}, GiE-, such that GiC G, GiC Gi+l (where Gi is the v-closure of Gi), and V(Gi) + V(G), and similarly for B, Vr, Vr, Vr. The condition (P)'For every Ge l, there exists a sequence (Gi), Gi~,, such that

17 GiCG, GGi G i+l, and Gi C+ Gwith (H2) implies (H4). (xvi) If conditions (H1), (H2), (H3), (H4) hold and V(A) < o, then all Ge (and so all sets of the minimal a-algebra.0 containing;) are d, Mrc, I+r, and p.r measurable, so that the restrictions to _4 are measures; these measures on ) are -regular; and the vr on are signed measureso Also, for each r, there is a Hahn decomposition of A into two disjoint measurable sets A, Ar, such that, for every HE, vr(Ar H) > O, vr(Ar/)H) 0. Writing vr(H) = vr(Ar4 H), v(H) = -vr(Ar H), vr = Vr + V, we have 0 + + 0 < v _ r, 0 < v- < Mr,'- < Mr r 0 < vl < * ar From now on, we suppose that ~ satisfies a stronger quasi additivity condition (i') in which sums are taken over J C (oir/C) I1, the interior of I, rather than just Io (xvii) If (Hi), (H2), (H3), (H4) hold and V(A) < oo, or = Vr, i-r = Vr ir = v4 on p; and for any HE r (H) X sup (r(Hi))1/ = sup |v(H) [H] HiE[H] ]Hi[H] where [H] is any finite decomposition of H into disjoint sets Hi~ ~.

18 Earlier inequalities imply absolute continuity appropriate to the existence of the Radon-Nikodym derivatives Gr = dvr/dpt, Pr = dir/d), r = d-+r/d[, r = di/d pL-almost everywhere in A, and Y7 = dvr/ddr, r 4+ = dr/dr Yr = r/ Ir-almost everywhere, together with their respective measurability. We have'-l ~ ~Gr r < 1 ~ < r 7r, 7r r < 1. Let = [9r]. (xviii) If (H1), (H2), (H3), (H4) hold and V(A) < 0, then + + Pr = Pr + Pr' 3r P r P 3r 9 rr = 7rr Br = rr r = 7~, 13rl = r, ||J|| = = 1 [i-almost everywhere, and Yr + r = 1 r = r - 7r Y+Y = o, yr 1 +17r = 0 or 7r r = and either 7 = 1, Yr = 0 r Y+ = 1 r- almost everywhere

19 2.5 REPRESENTATION OF BC-INTEGRALS We mention here the following main theorems proved in Ref. 7. By T = T(w), weA, is meant a mapping from A into Em. (xix) Under hypotheses (a'), (b), (c), (d), (p'), (H1), (H2), (H3), (H4), V(A) < co, (c), (f), the integral ff(T,O) can be expressed as lim f(T(T),v(IO)) 8(D)O ID IeD (xx) Under the same hypotheses as in (xix), the function f(T(w),O(w)), weA, is defined k-almost everywhere in A, is k-measurable and k-integrable in A, and the integral Jf(T,O) has the following representation as a Lebesgue-Stieltjes integral in the measure space (A, 3, ff(T,) () f[T(w),9(w)]dk In particular, if we take T constant and f(T,) = $r, then u(D) = 0 for every D, so the relation (c) is certainly satisfied, and we have fSr = (A) f dr, r =,2,.,k, or, in vector form;f = (A) f dv

3. THE LEBESGUE-STIELTJES INTEGRAL AS A BC-INTEGRAL In Ref. 6, Section 4, Cesari has shown how the finite LebesgueStieltjes integral on any measure space can be expressed as the BC-integral of a certain quasi additive interval function r with respect to a certain mesh function 5. The objective of this chapter is to show that this can be accomplished also by another interval function r with the usual mesh function 8 of Lebesgue-Stieltjes theory 3.1 THE INTERVAL FUNCTION f Let (X, 2,p) be a measure space, and f a k-measurable real function on X, and let (X)ff di be the corresponding Lebesgue-Stieltjes integral. Since f can be decomposed into its positive and negative parts, we shall assume f > O. Let ip denote the k-measure of the set (x: f(x) > p)] In the setting of Section 2.1, let us take for the set A the extended set of non-negative real numbers, A = [0 < y < oo]. Let us take for [I) the collection of all half-open half-closed intervals I(p,q) = (pq] Let y be the collection of all finite systems D = ((pi-pi]: i = 12,.n-l}, 0 = Po < P1 < o.. < Pn-1 < C' n > 2, of non-overlapping intervals I covering some finite interval (O,Pn-l] Let (I) = V(p,q] be the interval function 20

21 (p,] = (q-p)-(x: f(x) > vpq, where vpq denotes any number with p < vpq < q. Let 5(D) be the usual mesh function of Lebesque-Stieltjes theory, (hD) = max (Pi-Pi-) + l/Pn-l i=l,.n-1 This function is obviously a mesh function. Theorem 3.1 If to < ~o, then the function r is quasi subadditive with respect to the mesh function 6, and the corresponding BC-integral coincides with the Lebesque-Stieltjes integral of f: Jf = (X) ff(x) di. If po = co, then the same is true with the particular choice vpq = q. Proof: The proof is divided into parts (a), (b), (c), and (d). (a) For po < co, r is quasi subadditive with respect to 6. In fact, we shall prove a more general result which will be used in part (b). For any t > O, take any Do = [(Pi-,Pi]: i = 1,2,...,n-1} with E(Do) < r( ~) = ~/lo; and any D = ((qj-lqj ] j = 1,2,..o,m-l1 with 6(D) < x(,Do), where

22 n-l %(, Do) = min E / [ Pi' 1 /Pn-l' Pi-Pi- for i=12,..n-l i=l Let qJ(i) = max (qj: qj < pPi qj(i) = min (qj: Pi1_ < qj these exist in [Pi_lPi] since qm-l > Pn-l and qj-qj-l < Pi-Pi-i Let rl,'2 denote the interval functions corresponding to two choices of vo Denote a sum over (qj l, q ] C (pi-li] by (i) Then (iPi-i-l)~pl - (qJ(i)-qj(i))IPi - (Pi-Pi-1)(l - ) + (Pi-(J(i)+ )j(i)-Pi-1) (P Hence (i) + E[8( pi_lpji ] - E *2(qj-_l qj] n-l < max(pi-Pi-l)(o- _Pn1) + 2 max(cj-qj_) k LPi 1 < 53~. The particular case'1 = *2 gives the required quasi subadditivityo (b) f1 is independent of the choice of vp in [p,q]o Let the notation be as in part (a), but denote (Pi_l,Pi] by Ii, and (qj_l,qj] by Jj.

25 Since W2 is non-negative, i(Ii) < C [F[('i) - (i)2(j)]J + Zr2(Jj) i Hence, for any ~ > 0, if 6(Do) < (E) and 5(D) < %( ~,Do), then 4Z(Ii) < 6+ Z2(Jj). Hence, if b(Do) < (E6), then Z(Ii)' + f2, so fr1 < ~ + f. Thus f1 < < S, so, by symmetry, fJ1 = fS2 (c) For o0 < 0o, f = (X)ff(x)dk. By part (b), we need prove this only for the choice vpq = q. For this choice, we shall not assume Uo < o. Consider the set n-1 S(D) = ) ([(x,y): Pi < < Pi, f(x) > pi i=l iLet m be the product measure Q x kt, where Q is real Lebesgue measureo Then m(S(D)} = r(I), IcD

24 where vpq = q, so lim m[S(D) = fJ 6(D)-O Now ((x,y): 0 < y < f(x)) C lim inf S(D). (D) -0 Hence (X) ff(x)dL < m[lim inf S(D)} ( D) -0 < lim m(S(D)). b(D)-+O Also S(D)C[(x,y): 0 < y _ f(x)). Hence lim m[S(D)) < (X) Jf d. 6(D) -K Thus fN = lim m(S(D)) = (X) ff d. F(D)0O Remark: The condition po < oo cannot be omitted here when Vpq is chosen arbitrarily in [p,q]. For example, let f(x) = x2 on (x > 1) in E1. Then If dpl = 1. However, if pi < 1, P2 = p-/4, and vp p = P1, then

25 (p2-pl)it(x: f(x) > vplp = (pV/p1/P)( p1/2l 1) + 00 as p1 +0. (d) For to possibly infinite, if vpq = q, then 4(p,ql = (q-p)4t(x: f(x) > q) is quasi subadditive with respect to 5. Let the notation be as in part (a). For any > O, take any Do; and take any D with b(D) < k(~,Do). Then (Pi-1, Pi] - (qjl qj] (Pi-j( i)+qj(i)-Pi-_l) i ~ Hence (Z (PilPi3] - Z j-l( )+ E17Pi-1-1Pi, - E q*j_,.qj ]] < 2 max (qj-qj_l) Z Pi < 2. 3o2 COMPARISON WITH PREVIOUS RESULTS For (X?,7,) as in Section 3.1, and f a non-negative t-integrable function on X, Cesari considers the following situation (Ref. 6, p. 101105)o The set A = X.

26 Intervals I(p,q) = Ix: xeX, p < f(x) < q}, where 0 < p < q < co Systems D = (I(Pi_l,pi): i = 1,2,.oo,n} with 0 = p < p1 < < Pn — The interval function K(p,q) = pP(xo p < f(x) < q) The mesh function n-l S) =l mn (Pi-Pi-1) + 1/Pn-1 + m Pilx f(x)= P i 1-() = o on-1 i=l Then is quasi additive with respect to 5 (Refo 6)0 Classical theory, for example (Ref. 22, p 117-121), (Refo 11, p 179-183), showsg though in a somewhat different and restricted form, that (X) ff(x) dp is the BC-integral of * with respect to bo Since (D) < 5S(D), the BC-integral of V with respect to 6 is also (X)ff(x) dp., In fact, provided. Lo < oo, (X) ff du (even possibly infinite) is also the BC-integral of the more general interval function h'(p,ql] = vp (xo p < f(x) < q3 with respect to 5(D), for any Vpq satisfying p < Vpq q,C poo O P P The following example shows that 4, and more generally 4t, need not be even quasi subadditive with respect to 6o Let (X,Tj1 ) be a measure space with i(X) = 1, and consider

27 f(x) = 1 for every x in X. Then r'(p,q) = vpq for p < 1 < q, 0 otherwise. If't were quasi subadditive with respect to 6, then y [rd~(]) -z(I)c4(J)]+ < ~ IeDC for ('Dc) less than some r( ) and 5(D) less than some X( E,Do) Take = 1/2; Do with 6(Do) <'q(1/2) and some member I(a,l) with a > 1/2 (so'(Io) = vl > 1/2); and D with 6(D) < x(1/2,Do) and some member J(p,y) with < < 1 < y (so r'(J) = 0 for JC IEDo)o Then ['(I) - ZI (J)] = v > 1/2 IeDo

4. THE BEND OF A CURVE AS A BC-INTEGRAL Isekil3 has introduced the concept of "bend" of a curve in En, as a generalization of total curvatureo He develops the theory of this bend in Refo 13-15, and other paperso Results relevant to this chapter are given in Section 4.1 belowo The objective of this chapter is to obtain the bend of a curve as a BC-integral from a generating interval function as simple as possibleo This leads to a systematic treatment of the bend. As is to be expected, the simplicity of the generating function that can be achieved depends on the strength of conditions imposed on the curve. Sections 4.2, 4.5, and 404 below deal with the problem under different sets of conditions. In the present chapter, the set A of Section 201 will be a fixed interval [a,b] with the Euclidean topology, and the systems D will be finite subdivisions of [a,b], so that the subsets I are closed subintervals of [a,b]o 4l1 THE BEND OF A CURVE Consider a curve X(t), a < t < b, in Eno Its bend 2 = Q(a,b) can be taken as the supremum of angle sums N-1 <X( ai)-X( ai-) X(ai+l)-X( ai) > i=l over all. subdivisions ao,al,.oo,aN of [a,b] with a = a0 < al <.. < aN=b, for which X(ail) # X(ai), i=l, oo No Here < A,B > denotes the geometric 28

29 angle between the non-zero vectors A, B, 0 < < A,B > < i; angles involving zero vectors are not defined. This is not quite the same as Iseki's definitions in Refo 13, p. 141 and Refo 14, po 115, but is easily seen to be equivalento We shall need a continuity property of the bend for continuous curves with finite bendo This is given essentially in Refo 13, Section 32. We shall cast it as follows- For every positive ~ and t in [a,b], there is a positive A(,t) such that 2(t-6,t) < e and Q(t,t+5) < & for 0 < 6 < A( Et) (When t = a or b, one of these must be omittedo) 4o2 CONTINUOUS LIGHT CURVES WITH FINITE BEND Let X(t), a < t < b, be a light curve, that is, X(t) is constant on no subinterval of [a,b] The simple interval function sup (X(t)-X(u), X(v)-X(t)> where the supremum is taken over all t in (u,v) with X(t) # X(u) or X(v), cannot in general generate the bend. This can be seen by considering a circle, where Q = 2it, while every D sum of the interval function has value Tr. Consider the interval function 4r(u,v) = lim sup {<X(u+6)-X(u), X(v-Sb' )-X(u+b)> ( -,A+)) 0+,0+) + <X( v —' )-X( + ), X( V) —X( v-6-' )>~ ~<~I~- ~ 1-~~~~~~i~ ~(~,-~ ~~~ ^

30 Here lim sup f(5,5') means inf sup f(5,5'). ( 5,' )+(O+,O+) r>O 11(,)11 r 6>0, 5 >0' is defined, because, if not, (i) X(u+5) = X(u) for all 6 in some positive neighborhood of 0, or (ii) X(v-3') = X(v) for all 5' in some positive neighborhood of 0, or (iii) X(v-36) = X(u+5) for all 5,6' in some positive neighborhood 11(56,6 )| < r, 6 > 0,' > 0, of (0,0). In all three cases, X could not be light. Note that the simpler symmetric lim sup, that is, with 5' = 6, need not be defined; for example, for u = 0, v = 1 with X(t) = (t-t,0) on 0 < t < 1. Using subadditivity of angle and lim sup, one can easily prove that V(u,-w) < 4(u,v) + 4r(v,w) + Y(v) where u < v < w, and Y(v) = lim sup <(v)X(v)-X( ), X(v+6)-X(v)> ( 5,5 )+(0+,o+) Now Z'(v) < Q for any slm over a finite number of v. Hence, if P < o0, y(v) = 0 except at a countable number of points Thus N-1 5(D) = max(ail-ai) + ) 7(ai) i=l is a mesh function on the family o) of finite subdivisions D = [Ii],

31 i = [ai,ail], a = ao < al <. < aN = b, of [a,b]. Theorem 4.1o If X(t), a < t < b, is a light continuous curve with finite bend,Q then * is quasi additive with respect to 6, and f= =. Proof: Consider any positive -. Take any Do = [Ii] = [aiai+l, a = al... < a = b, and any D [Jj], Jj = [bj,bj+1], a b = b < b b b, with 6(D) < min[ai+l-ai for 0 < i < N; A( ~ /N,ai) for 0 < i < N; ]. Here A is the function involved in the continuity of bend in Section 4.1. Let bj(i) = min(bj bj ai) b (i) max(bj: bj < ai+) These exist in [ai,ai+l] since bj+l-bj < ai+1-aio Then (Ii) - z ( (jj) J( i) < r(ai,bj(i)) + (bJ(i),ai+l) + y(bj) j(i) with special simplification at a for i = O and at b for i = N-l. Now

32 (u, v) < n(u,v), so Z[(Ii) -Z (i)(Jj)]4 < 35.~ Thus r is quasi subadditive with respect to 5, and so has a BCintegral fJ. Now fJ < immediately, so fr is finite. Hence, since V is non-negative, j is quasi additive with respect to 5. To prove that fV = Q, consider any angle sum lZ appearing in the definition of 2. For any positive, using continuity of angle and of X, shift the points of subdivision to where y = 0 (such points are dense), while keeping Z1 < L2+~, where Z2 is the angle sum for the adjusted subdivisiono Since fI < o, IZ((I)-fJ[ < ~ for 6(D) less than some ~(E). Subdivide the second subdivision further at points where 7 = 0 to obtain a subdivision D3 with 5(Ds) < 5(6). Now put in pairs of points about the points of subdivision of D3, sufficiently close to make the part of the new angle sum Z4 corresponding to the points of D3 less than E (this is possible since y = 0 at these points) and the rest of Z4 less than X t(I)+ ~ D3 By subadditivity of angle, a2 < Z3 < I4. Hence i < E4 + < ) I) + 3~ < S + 4 D3

33 Thus Z1 < Jf, so Q < fJ. This completes the proof of Theorem 4.1. Continuous light curves with finite bend have right and left tangents R, L at every point (with obvious restriction at a, b) (Refo 13, p. 162). Hence, by continuity of angle, l(u,v) = R(u), X(v)-X(u)> +<X(v)-X(u),L(v) and y(v) = KL(v),R(v)> 41.3 GENERAL LIGHT CURVES Let us consider the discontinuous plane curve X(t), 0 < t < 2, defined by X(t) = (t,O) on 0 < t < 1 and 1 < t < 2, (1,1) at t = 1. We have here sup Zt, = t. If we consider the modified interval function a' defined by t'(u,v) = sup iX(s)-X(u), X(t)-X(s)> + <X(t)-X(s), X(v)-X(t)>j where the supremum is taken over all s,t for which u < s < t < v, X(u) # X(s), X(s) # X(t), X(t) # X(v), we have sup' = 53J/2o Finally = 2 o The discrepancy between these values shows not only that we

34 cannot generate Q from r by adjusting the mesh function, but that any adjustment of 4 keeping to "two angles" will fail to generate R. Thus, in order to deal with discontinuous light curves X(t), a t < b, we shall consider the "three angle" generator k(u,v) = sup ~(u,t,v) u<t<v whe re ^(ut,v) = lim sup <X(u+5)-X(u), X(t)-X(u+b)> +0+ + <X(t)-X(u.+), X(v-6)-X(t)> + <X(v-5)-X(t), X(v)-X(v-5)>1 r is defined, because, if not, 0 would not be defined for any t. Then, for all 5 in some positive neighborhood of 0, (i) X(u+6) = X(u), or (ii) X(t) = X(u+6) or (iii) X(v-5) = X(t) or (iv) X(v) = X(v-5). In all four cases, X would not be light. Some manipulation shows that V has the same property

35 T(u,w) < k(u,v) + T(v,w) + 7(v) (u < v < w) as, but with 7(v) = lim sup <X(v)-X(v-6), X(v+6)-X(v) For a < u < b, define \ (u) = lim sup T(u,u+5), 6+0+ X (u) = lim sup i(u-6,u), 60+ X(u) = x+(u) + x-(u) Then, for any positive c,,(u,u+5) < X+(u) + and 4(u-6,u) < %X(u) + & for all positive 5 less than some A(,u). Define r = sup Z7(t) A = sup Z\(t) where the sums are over a finite number of distinct t. In order that N-l N-l 6(D) = max(ai+l-ai) )+ Y (ai) + (ai) 1 a be a mesh function on the finite subdivisions

56 D = [Ii], Ii = [aiai+], a = ao < al <... < aN = b, it is sufficient that (t: 7(t) = x(t) = O0 be dense in [a,b]o This will be true if r and A are finite. Since F and A < Q, 6 is certainly a mesh function for a curve with finite bend, Theorem 402~ If X(t), a < t < b, is any light curve (not necessarily continuous) for which 6 is a mesh function (in particular, for r < c, A < oo), then ~ is quasi subadditive with respect to 6, and fT = Q. Proof: For any positive t, take any Do = [Ii], I = [i,i+ a = a < a <... < aN = b with E(Do) < ~, and any D = [Jj], Jj = [bj,bj+l], a = bo < b <... < bM = b with 6(D) < min(ai+l-ai for 0 < i < N; A(5/N,ai) for O < i < N; L]. Then I*(I) -Z( J j) J(i) i(ai,b(ib)) + 7(b(i)ai+l) + (b) j(i) J( ) < \+(ai)+5/N + \-(ai+l)+E/N +! 7(bj) j(i) with special simplification at a for i = O and at b for i = N-l. Hence

37 [j(Ii1) - (i) (j)] N-1 N-1 < ( (ai) + E 7(bj) + 2~ 1 1 < 4 Thus V is quasi subadditive with respect to 5, and so has a BCintegral fWo Now 1f < S2 immediately, so fT = oo gives fJ = a. For fJ < oo, we shall prove that n < Kf. Consider any subdivision D1 of [a,b] with angle sum Z1. For any positive, IZT(I) - flI < ~ for b(D) less than some i( ~)o Subdivide D1 further to get D2 with max (ai+l-ai) < In(l()o Form a subdivision D_ with b(D]) < rn( ) by taking a point with y = = 0 in each subinterval of D2o From D2 and D2 combined, form D3 by adding pairs of points symmetrically about the points of D', close enough to make that part of the angle sum 3 corresponding to the points of D' less than ~, and the rest less than I (I)+ ~~ D2 Then Z1 <Z <Z< if ZC (I) + -+ < ST +- 3C De

38 Hence Z1 < t, so Q < fJ. 4.4 ANGLE SWEPT OUT BY DIRECTION Consider a field of directions on [a,b], that is, a function T(u) defined almost everywhere on [a,b], with values unit vectors in En. We wish to express the "angle swept out by T" as a BC-integral. The present discussion of the problem is similar to Cesari's treatment of the length of a discontinuous curve in Ref. 6, Section 4. Let U be the set in [a,b] on which T is defined. Define on U X (u) = lim sup T(u),T(u' as u'+- u+ on U, K-(u) = lim sup T(u'),T(u)> as u' + u7 on U, X+(b) = 0, X-(a) = 0 if these are relevant, \(u) = X(u) + X"(u) Then, for any ~ > 0 and any u in U, there exists A(~,u) > 0 such that KT(u),T(u)> <X (u) + ~- for 0 < u'-u < A(,u), u'eU and <T(u),T(u < (u) + - for 0 < u-u' < A( Eu), u'U. Define A = sup Z\(u), where Z is taken over a finite number of u. If A < oo then X(u) = 0 except at a countable number of pointso Hence, if A < ooa 5(D) = max( ai+l-ai)+al-a+b-aN++A-\( ai)

59 is a mesh function on the partial subdivisions D = [Ii], Ii = [aiai+l], a < al < a2 <... < aN _ b, aicU. Theorem 4_. The interval function 9(u,v) = (T(u),T(v)) is quasi subadditive with respect to 56 Proof: For any > 0, take any Do= [Ii], Ii = [ai,ai+l], a < al < a2 <... < aN K b, and any D = [Jj], Jj = [bj,bj+l], a < bl < b2 <.0. < bM + T(bJ(i)) T(ai+l)> < +(ai) + /N + - (ai+) + + /N with special simplification when b (i) = a. or bj(i) a.. ~(i) = bJ(i) =~

40 Thus X [G(ii) -l(i)(Jj)]+ i < 2E + Z(ai), where Z is taken over ai f any bj, so that ZX(ai) < - ZX(bj) < Hence the BC-integral of G with respect to 5 exists, and equals C(T) sup ZO(Ii). Note that A < C, so A < co if C < oo. Now consider the variation V(T) = sup Z IT(ai)-T(ai+1)|l of T, that is, the length of the curve traced out by T on the unit sphereo Since 20/T < A - ||T(u)-T(v)l| <, 2C/t < V < C. Simple examples in which T is discontinuous show that V can be less than C. However, for T defined everywhere and continuous on [a,b], we shall identify C with V. Obviously V = co if C = oo. Consider C < oo. For any

41 > o, 9 < JA/2 < % for |u-vl less than some( ). Since ZG - C as b(D) 0+, |Z9-C| < for 6(D) less than some n({ ) Take D with b(D) < min[S( ),q( )]o We have 9-A = 9-2 sin (9/2) < 93/24, so Zo < ZA + Zo3/24 Hence (- E < V + EC/24, which gives C < V. We now apply these results to any continuous curve X on [a,b] that has a right derived direction T (Ref. 13, Section 73) at all but a countable number of points of [a,b]. Then C(T) = ~(X) (Ref. 13, Section 95), so that we have another formulation for Q as a BC-integral. Also, if X is continuous with tangent directions T (Ref. 13, Section 42) everywhere in [a,b], and V(T) < o, then T is continuous (Refo 13, Section 67), so that we can identify C(T) with V(T).

5o GENERALIZED WEIERSTRASS-TYPE INTEGRALS Jf( S ) AS:BC-INTEGRALS In the definition of the Weierstrass-type integral Jf(T,$) in Section 2o3, the expression T(T) is essentially an interval functiono Our objective in this chapter is to replace T(T) by a new interval function ~(I), and so consider the BC-integral of the interval function ((I) - f( (I),( )) o The original reason for doing this was in order to express more conveniently integrals involving higher derivatives in the Weierstrass form, and thus as BC-integrals. Let f(pq,r) be homogeneous in q, and let D(w) be a point function which is a derivative. In the previous formulation, the integral (A) jf(T(w),G(w),G(w))d)L would be expressed in the Weierstrass form as lim f f(T(T), (l):D(T;)), TE~, (D)0O IeD with explicit use of the derivative D(w), In the new formulation, we can consider limits of the form lim f(T(T'),(I),A(I)) 5(D)-0 D 42

43 where A(I) is a quotient of interval functions giving the derivative D(w) in the limit Here T(T) and A(I) can be thought of as a single (vector valued) interval function ~(I) Thus we shall consider interval functions of the form @(I) = f( (I),(I)) where f(uv) is a real function of two vectors u,v, positively homogeneous of degree one in v, where ~(I), /(I) are vector-valued interval functions, and O(I) is quasi additive with respect to some mesh function 5o 5ol A LEMMA We shall need here and in later chapters a lemma that appears in a concrete form in Refo 6, p 109 (the first member of relation (5.1)). We shall state it here in an abstract form, and prove it directlyo Lemma 5 o Let ([iz i = 1,2,.o,n) and (1: j = 1,2,..,m be two sets of vectors in E. Define i/"lsi"ll for ~i' O, 1. any unit vector otherwise, and similarly a'jo Let J be a mapping from [1,2,..o,n) into the subsets of (l,2, oo,m). Denote by E(i a sum of terms over j for which jEJ(i), and by EZK a sum of terms over j for which jEJ(i) and IIai-j I > 7y Then

44 2 2. 11 %< ~ ~ 7+ I roof -We sha denote by a - theI inne product of t- kvectors ab Proofor We shall denote by ab the inner product of two k-vectors a For 7 < llaz-c3'll, we have 7Y < lai2 l - 2al. -~a + laj2 2 - aij, and hence 2 1ll ll < II1 -I al.-'j Qui.te generally, 0 < II jll ai0' v Hence 1: (.).~ h 1) e ail, + j 72 ) 11 11 JI l ( -)11 jll -11^ 1 I + 11+ ~l The required result follows with summation over i 5 2 EXISTENCE OF THE INTEGRAL ff(, ) In the setting of Section 2.1, let,6 be vector functions on ( I, 6 into Em, ~ into Eko Consider a real function f on K x Ek, [(I}CKCEd,E satisfying condition (f) of Section 2o3, namely f is positively homo

45 geneous of degree one on Ek, and bounded and uniformly continuous on K x U where U = [(q l jjq = 1, qeEk]J is the unit sphere of Eko We shall now formulate for the set function Y(I) a condition which extends condition (c) of Section 2o. For every Dc o, let i(Do) be the number )(Do)= max lim sup max ||11()-(J)|| o IEDQ b(D)-0 JCI JeD We shall assume that (~) 6(Do)-o0 We consider now the set function 0(I) = f((I),C(I)), Ie(I o If the limit lim (I) = i (D)-0 I SD) T eD IED IEd exists, we denote this limit by Jf(_)oG We extend now the result (xiii) of Section 2.3 Theorem. l If 5 is quasi additive and of bounded variation with respect to a mesh function 5 (Section 2~2), and conditions (6) and (f) hold, then (is) ve and of bf((ation with respect to ))s the is quasi additive and of bounded variation with respect to &, Thus the

46 BC-integral of 0 exists and is finite, that is, JO = ff(,) = lim f(( )) 6(D)-0 TiD Proof: The conditions on f give ||Ij| quasi additive also, so, for any > 0, X 11(1-) -EI) J)il < ~ IeDo I IA()DII - Z(I) I(J)I I <,!eDo IEDO and I' l/(J)II < for 6(Do) less than some X( ) and b(D) less than some rE(,Do); and ^ ~Il(J)II - v < E JeD for b(D) less than some o( ). The condition (~) gives, for any I > O0 that there exists A( ) > 0 such that, for every Do with b(Do) < A(~), there exists X( gDo) > 0 such that, if JCI, IeDo, JeD with 6(D) < X(,Do), then (T ) o- (J)co < The conditions on f give that there exist M and, for any ~ > O,

47 a(.) > 0 such that |f(p,q)l < M on K x U and lf(p,q)-f(p',q')l < - for llp-p'll and |lq-ql11 < S(E ), pp'EK, q,q'EU. Take any Do with 6(Do) < min[A(^(Q )),(I),x(& 2( ~))], and any D with 6(D) < min[o(S ),r (,Do),rI( I2(DE),Do),X( (~ ),Do)]. Denote by Z the sum over JCI for which |>(l)-a()|, and by Z_ the corresponding sum for which jIc(I)-c(J)I| < S. Then I |f( (i),)(I)) -Z)fyJ),q ))I IeDo r-i (I) K lf(S(I),lc(i))| I |I(i)II - J (J)f I IEDo + I (Zn- + z(^)+ c(I) I)f^J.(J))f IA'7)II leDo < M I II(I) - Z I(iJ)11 I +E I(J)1I leDo JeD *hM4-2(^ 1jI))-E 0(BJ) +i I(IA)II1 IEo IEDO (I) -^^ I|(J>)II < Mt + ((v+~.) + 8M = (9M+V+ ~lf(^(J),A(J))| = -1 |f(S(J),a(J)) | (J)|| ^ t1 11^(J)11 <M ~- *

48 Hence both sums are less than any positive ~ by taking = min(l, ET/(9M+V+l), /M). Since the conditions (f) on f carry over to If|, ||1 is also quasi additiveu Hence D is quasi additive and of bounded variation. Remark: By this theorem, the Weierstrass-type integral ff(5,~) depending on two set functions,/, of which ~ is quasi additive and of bounded variation with respect to a mesh function 5, is defined as a BC-integral. Only the axioms (v), (6), and (f) are used, as for the integral ff(T,~) only the axioms ((), (X), and (f) were used in Section 2.3. We shall show that Theorem 5.1 extends the result (xiii) of Section 2.3. There T(w), weA, is a map from A into Em. Take (I) = T(T) where T is some point of I. Then max |I(I)-(J)[| < sup I|T(u)-T(v)||, JCI u,v I JeD so that E(Do) < (i)(Do) where OD(Do) was defined in Section 2.3. Thus our condition (5) is satisfied if OU(Do) + 0 as 6(Do) - 0, which is certainly the case if (Do) < 5(Do).

49 5.3 TRANSFORMATION OF THE INTEGRAL ff(,Z) We now wish to transform ff(,j) to the form lim f(;(I), B(I~)) b(D)O ID as in Section 205. For this we need a lemma that is essentially relation (6~2) of Ref. 7, p. 141. Lemma.2. Under hypotheses (a'), (b), (d), and (A') of Section 2.4, we have j ||(I)-B(I~)|| + 0 as b(Do) + 0 ieDo Proof: Our conditions give, for any E > 0, y Ii(I)(10(IO) (J) < ~ IeDo for 6(Do) < X( ) and b(D) < ( ( Do); Il(I )(J) -B(I~o)1 < g for 5io(DIo) < A(C,I~), and 6IO(DIo) < - for 6(D) < V(~,I~)

50 Now take any Do with 6(Do) < ( E /2), and having n members, say; and take D with 5(D) < min[i( ~/2,Do),v(A( ~/2n,I~),I~) for IEDo]. Then 86I(DIo) < A( /2n,I~), so II( I~)(J)-B(I~)I < /2n Hence j 1(I) - B(I~)II I < I( I) -Z(I~) (J) [ + I I(I(IO) ( J)-B( Io) I I < E/2 + n /2n = Corollaries: (a), B(IO) + B(A) as 6(Do) + 0. IcDo (b) Since j| (I)l) - IIB(I~)| < I)-B(I~), Z II |(I)l| - IB(I~)|| I + 0 as (Do) - 0, IcDo so I |B(I~)|| I V(A) as 5(Do) + 0 IeDo (c) If ||ll, satisfies axiom (,') (which is so if < satisfies axiom (v') and V(A) < oo), then X | ||(I)|| - V(~)| + 0 as (Do) + 0, IeDo

51 so J V(I~) + V(A) as 6(Do) 0. IlDo We are now in a position to extend the result (xix) of Section 2.5. Theorem.2 Under hypotheses (a'), (b), (d), (T'), V(A) < oo (;), (f), we have I f(((I),B(IO)) + ff(5,~) as 6(Do) + 0. IeDo Proof: We have, from Lemma 5.2, y ||I(:I)-B(I~)|| < &' IeDo for 5(Do) less than some p('). Let P(I) = B(I~)/I|B(I~)I|. Otherwise we use the notation of Theorem 5.1. For any ~ > O, let g' = min( g, e2( )), and take any Do with 6(Do) < min(p(~'), (.)). Then C!f((I),l(I)) - f(-(I),B(I~)) < Z |f(Y(I),p(I))| I ||~(I)|1 - IIB(I~)|I + Z If( (I),a(I)) - f( (I),P(I))I 1II(I)11 < M | ||,(I)1i - I||B(IO)|| | + ~EZ|(I)|| + 4M -2( ~a)z( I(I)-B(I~)1 + | ||Il()|| - ||B(I~)|| |) < Ma + ~((v+~) + 8M~ = ~(V+9M+) Hence Zf(( I),B(I~)) converges to the same limit as.f( (I),( I))

52 5.4 THE INTEGRAL ff(t,~) AS A LEBESQUE-STIELTJES INTEGRAL We now express, as in Section 2.5, ff(Q,~) as (A)ff(Z(w),G(w))di, where Z is an appropriate limiting point function of. For this we need a lemma, which is the result (5.iii) of Ref. 7. For a given D, define rv(IO)/i(Io) for weI0, IeD, i(Io) # O, D(w) = 0 otherwise. Lemma 5.. Under hypotheses (a'), (b), (c), (d), (H1), (H2), (H3), (H4), (vA) and V(A) < oo, (A) f l|9(w)-D((w)|I2 di + 0 as 6(D) 0. Proof: (A) f ||I - TDI|| dp (A) f dli - 2(A) Jf 9-TD dkt + (A) f DI ID12 dit = (A) - ||IIV(I)II2/(I~) = (A) - l |v(I~I| + L |II (IO)Il( (IO)-IIv(IO)l)/4(IO) < 2[i(A) - |lv(I~)|] =2[V(A) - I|B(I~)||] + 0 as 6(D) + 0 by Corollary (b) of Lemma 5.2. We are now in a position to extend the main result (xx) of Section 2.5, that is, to prove that the integral Jf(;, )-which was defined in Section 502 as a BC-integral-admits a representation as a LebesgueStieltjes integral.

55 We suppose that the interval function ~ converges to a mapping Z from A to Em in the following sense (Z) For [p-almost every w in A and any E > 0, there exists y(,w) > 0 such that if wcI~, IED with b(D) < y(,w), then'(I) - Z(w)|| < E o We also assume that K is closed. Theorem 5. Under hypotheses (a'), (b), (c), (d), (H1), (H2), (H3), (H4), (Z'), V(A) < o, (%), (Z), (f), the function f(Z(w), G(w))is u-integrable on A, and Jf(fQ,) = (A) ff(Z(w),G(w))d4 Proof: We have (A)fIJ||G-D||2d + 0 as 6(D) -+ 0. Take a sequence (Dn} with o(Dn) < 1/n. Then there exists a subsequence (Dnml with corresponding -m -+ p t-almost everywhere in A (see, for example, Refo 18, ppo 226, 229). Thus there exists a set A-C A with [1(A-) = [1(A), ||G0| = 1, and +m + G on Ao For wcA-, 1/2 < JJ|im(w)J for m > some N(w), so wcI0 for some IeDnmo Define (I) for wcEI~ IeDnm Zm(w) = an arbitrary point in K otherwise Then, for m > N(w), ||Zm(w)-Z(w)|| = Jis(I)-Z(w)|J < ~ for 1/nm < Y(,w), so Zm(w) + Z(w). Since K is closed, Z(w)EK.

54 Although f is assumed uniformly continuous and bounded on K x U only, its positive homogeneity gives f uniformly continuous and bounded (M') on K x (q: a < ||q|I < b) for any a,b satisfying 0 < a < b < oo In particular, f is continuous at each point of K x U, so, on A-, f(Zm(w),(W)) + f(Z(w),( w)). Now Zm, lm are obviously [i-measurable (coordinate-wise) on A-, so f(Zm,im) is lt-measurable, so f(Z,G) is v-measurable. Indeed, f is bounded and pI(A) < oo, so f(Z,9) is p-integrable on Ao Also (A)ff(Zm(w), m(w))dal + (A)ff(Z(w),G(w))dLi. But (A)ff(Zm(w), rm(w) )dk = (A-UI~)ff(Zm(w),lm(w))dlL + Zf((I),v(I~)). Now I(A-U)I~)f(Zm(w),qm(w))dli i MT(i1(A) - 1,(I~)) M'(V(A) - V(I~)) + 0 as m - oo by Corollary (c) of Lemma 5 2. Also v(I~) = B(I~), and, by Theorem 5.2, Hf( (I), B(I~)) -f+ f((, ) Hence (A)/f(Z(w),9(w))dB = Jf(,~).

55 Remark: The relation ff(~, ) = (A)ff(Z(w),G(w))dp applies to "non-parametric" integrals of the form (A) fg(Z(w))di For this, we put f(p,q) = g(p)lj ll, so that f(Z(w),G(w)) = g(Z(w)). Then (A) fg(Z(w))dp = lim g(g(l))() g()|| b(D)-) 0 ID IcD provided g(p) is uniformly continuous and bounded on Ko Of course, the conditions on ( and ~ must still be satisfied. 5o5 THE CONDITIONS (t) AND (Z) We wish to examine the relation between the condition ( ) used in Theorem 51l and the condition (Z) used in Theorem 5~3 We shall show that a slight strengthening of each implies the other. In order to do this, we need the following lemmao Lemma 40 Let 1t be a measure induced as in Section 2.4 by an interval function ~ under hypotheses (a'), (b), (c), (d), (Hi), (H2), (H3), (H4), (' ) and V(A) < oo Then for 1-almost every w in A and any > O, there exists D with 6(D) < ~ and wI~ for some IeD,

56 Proof: By Corollary (c) of Lemma 5o2, IP(I~) -+ i(A) as 6(D) + 0, so IeD for any E > 0, there exists p( ) > 0 such that (A) - 1 (UJI~) < & for 6(D) < p(F ). Take p( ) < ~. For each positive integer n, take Dn with 6(D) < p(l/n)o Let An = UI~: IEDno Then ji(A)-pL(An) < 1/n. Take B = lim sup An = n nU Ano We have 1t(A) < go, so m n>m 1i(B) > lim sup pt(An) = 1(A), so 1t(B) = |J(A)o For weB and any 6 > 0, w ~ some An with n > 1/a, so weI~ for some IeD with 6(D) < 1/n <. Now consider the following strengthening of condition ( )~ (Q') For any ~ > 0, there exists A( ) > 0 such that, for every Do with 6(Do) < A(S), there exists X(,Do) > 0 such that, if J~o IO~ 0, IcDo, JeD with b(D) < X(E,Do), then I1 I) -:(J)11 0, there exists I, weIO, IeDo with 6(Do) < A( )o By condition ('t), if wcJ~, JleD1 with 6(D1) < X(~,Do), and wcJ~, J2ED2 with 6(D2) < X( g,Do), then jII(J1)-~(J2)11 < 2Eo Hence Z(w) lim [S(J): weJ~eD] 6(D)+O existso Again from condition (~'), if weI0, IeDo with 6(Do) < A(,),

57 then ||R(i) - Z(w)|l < ~<. Now consider the following strengthening of condition (Z): (Z') For any > 0, there exists ( E) > 0 such that if wcI~, IcD with b(D) < y(~), then II|(I) - Z(W)|l < & Theorem. If there exists Z(w), weA, satisfying condition (Z'), then Y(I) satisfies condition (Q'). Proof: Consider any Do and D with b(Do) and 5(D) < y( ~/2). For IcDo, JeD with J~o)o i I, take wcJo(I~o. Then ItI(I)-Y(J)ll < 1||(I)-Z(w)ll + Ilz(w)- (J) I < ~.

6. INVARIANCE PROPERTIES OF INTEGRALS ff(?,O) In this chapter, we study relations between interval functions ~, and Q',' which ensure that the corresponding integrals ff(5,[), ff(Q',[') have the same value. The present discussion was suggested by Cesari's treatment of the invariance of surface integrals under Frechet equivalence in Refo 53. Integrals over rectifiable curves present an analagous invariance under Fr(chet equivalence. We consider a system A,7L, ([I]3,j, 6 as in Section 2.1, with t a vector function from [I) to KO Em and ~ a vector function from [I] to Ek; and then a second system A',I L, (I',,y, 61, with f' from (I') to K and O' from (I'3 to Ek. Let f(p,q) be a real function on K x Ek, satisfying the conditions (f) of Section 2.3. Sufficient conditions on 5,o and',' for the existence of the integrals ff(, ), ff(I',') were given in Section 5o2. 6.1 RELATIONS R BETWEEN INTERVAL FUNCTIONS We consider three relations (R1), (R2), (R3) of ~',.' to.,o with increasing strength~ (R1) For any ] > 0, there exists a homeomorphism h from A to A' and systems DEcj D'EcJv with 6(D) < 5, 6'(D') < j, satisfying conditions (a) and (P) below. (R2) For any g > 0, there exists a homeomorphism h from A to A' and a number x( ) > 0 such that, for every D'c' with 6'(D') < x(&), 58

59 there exists DE with 6(D) < E, satisfying conditions (a) and (P) below. (R3) For any > 0, there exists a homeomorphism h from A to A1 and a number k( &) > 0 such that, for every D' C' with 6'(D') < \( ), there exists a number r)(,D' ) > 0 such that every De 0 with 6(D) < r(,D' ) satisfies conditions (a) and (P) below. Condition (a): ||''(I')-5 (I)| < ~ for hIC I', IED, I'ED' Condition (p): (i) X ll' (I) -,[I'](I)l <, I cD' (ii) I l (I ) -Z[ I( II I < I'eD' and (ill) Z 11(I)11 < ~, where Z denotes a sum over all IcD such that hi C I', and Z denotes a sum over all IcD such that hi is contained in no I'cED' We shall say that j,' are related to 5,~ in the sense (Ri), or 5'', are (Ri)-related to,~, or (', )Ri(,~), i = 1, or 2, or 35 For point functions T(w), T'(w') respectively from A,A' to K, inducing interval functions %(I) = T(T) for some TEI, ~'(I') = T'(T') for some T'I', condition (a) becomes

60 IT'(T )-T(T)II < for hICI', IeD, I'cD' This condition is closely related to the standard condition in Frechet equivalence: (a') sup IfT'(hw)-T(w)jj < E weA Lemma 6.1. Assume that c'(D') - 0 as 6'(D') - 0 (see Sections 2.3, 5.2); that is, for any' > O, U'(D') < _ for 6'(D') less than some p(E ). Then, for D' with b'(D') < p(./2), condition (ac') relative to ~/2 implies condition (a). Proof: For hI C I', IrC'(I' )-(I)- = IlT'(T')-T (T)1| < IIT'(T')-T'(hT)II + |ITl(hT)-T(T)11 < c'(I') + e/2 < 6. 6,2 INVARIANCE OF INTEGRALS ff(,y) UNDER RELATIONS R Theorem 6.1. Let f(p,q) be a real function on K x Ek satisfying condition (f). Let I, I',T' be vector interval functions such that the integrals ff(S,9), ff(f',~') exist, and V = lim sup < II(I)|l < - 5(D)-0 IED Then, if t',' are related to 5, in the sense (R1),

61 Jf(,,', ) = Sf(,/). Proof: The conditions on f give that there exist M and, for any 8 > 0, 5(') > 0 such that If(p,q)| < M on K x U and If(pq)-f(p',q')l < 6' for Ilp-p' lI and Iq-q' | < ( 6 ), p,p'EK, q,q' EU Further, |Tf((I),s( I)) - ff(,()l| < E for b(D) less than some p( ); |Zf('( I'),'(I' )) - Sf(',~')| < ~ for b'(D' ) less than some p'( ~); and Z il1(I)11 < V+ ~' for 6(D) less than some o( ). In the relation (R1) in Section 6.1, take' = min(c,p( )'( ), ),( ), 2( E )) to get h, D, D' with 6(D) < p(') and a(.'); 6'(D' ) < p'('); and

62 conditions (a), (f) satisfied; in particular, II|'(I')-5(I)lI < g<S (~') for hi CI'. Then | f('(I'),'(I')) - y f((I), (I)) TED I' eD' ITD < ~ Jf('(I')'(I' ))[I ]f((I),(I)) I' eD' + I |f(((I),(I ))l =,}| f(d'(I' ),C'(I')) m'(I') [II-Z[I l.(I)] I' eD' + F_[I'] rf('(I'd),C'(I'd))-f(( I),(dI)3 ||(dI)l[| -. |f( (I),c(I))l I11(I)11 < M | /' (I' )-Z[ I( I)| + Z|d(z)|| I' eD' + MH-2(') t ( *(~I'}-Z['](i)l I' eD' + I lGd(I! I' )- I' ]||(1' I)i't i) + ~ ll(I)ll < Me + <'(v-+~') + 4M-2( ~')'e2 + MF < (10M+V-+')'. Hence I f( f',')-If(,) L < (dOM+V-+2+.' )'

63 for any' > O, so Jf(f, =) = ff(,). Note that we have assumed that ff(O,j),and ff(i',') are finite. This is certainly so for the first, by the conditions of f and V. The possibilities ff(Q',') + co are easily eliminated by examination of our argument above. In Theorem 6.1, we have assumed that both integrals ff(S,) and ff(',)') exist. If we use the stronger relation (R2), then the existence of ff(1',1') follows from the existence of ff((,4)O Therem 6.2. If ff(i,~) exists, V < oo, and',' are related to f,/ in the sense (R2), then ff(Q',') exists and equals ff(5,4). Proof: For any ~' > 0, take E = min(E',p( (), ( ( t), 2(F)) in relation (R2) in Section 61oo This gives h and X(~), such that, for every D'cE with 6'(D') > X' (g ) - ( ~), there exists D with S(D) < ~ < p( ~' ),(), satisfying conditions (a) and (P); in particular |I((I') - (I)|l < < (~' ) for hIC I. Then, as before, |X f('(I'),'(I'))- X f((I),(I)) < (loM+V-+~')Et I'eD' IeD so y E f('(I'), /'(I)) - ff(~,~)1 < (10M+V+l+~t')' I'eD'

64 For any ~ > 0, take (T = min(l, E/(10M+V+2)). Then, for &'(D') < T(T-) I'( E'), f{'(I'), )'(I'))- lf(~,) < I'eD' Hence Z f('t(I'),1'(II)) -+ ff(~ ) as 6'(D') 0. I eDt As particular cases, we consider in turn f(p,q) = qr, Iqrl, I|qlj The conditions on f are satisfied. Then, under the other conditions, B, Vr, and V are invariant. However, some of the conditions assumed in the general theorems are superfluous here. We shall prove directly the invariance of B and V under these relaxed conditions. Theorem 6.3. If B,B' exist and (' is related to ~ in the sense (R1) restricted to conditions (P)(i) and (p)(iii) only, then B' = B. If V,V' exist and ~' is related to / in the sense (R1) restricted to conditions (P)(ii) and (iii) only, then V = V. If B exists and I is related to i in the sense (R2) restricted to conditions (p)(i) and (iii) only, then BT exists and equals B. If V exists and i' is related to i in the sense (R2) restricted to conditions (P)(ii) and (iii) only, then Vt exists and equals V.

65 Proof: We shall prove only the first resulto The method of proof for the three other results will then be fairly obviouso For any t > O0 ||Z(I) - BI| < ~' for b(D) less than some p('), and 1Z'T(I') - B'Il < ~7 for 6'(Di) less than some p'(')o Take ~ = min(,Epp(' ),p'( )) in the restricted relation (R1), to get h, D, D' with ((D) < p('), b'(DT) < p'(2T), and conditions (p)(i) and (iii) satisfiedo Then lIB' - B|l < lIB' - Z' (I')|I + ) Il (It)- Z 1 (I)i IteDt + Z 11(I)11 + 11Z(I) - B|| < E + ~ + + < 4< t. Hence Bt - B. 6.53 SUBSTITUTION OF THE INVARIANCE OF V IN THE RELATIONS The following result is of some importance, in that the invariance of V can often be proved independently, for example in Ref. 5, p. 457 by semicontinuity. Theorem 6.4. Relations (R1), (R2), (R3) are equivalent respectively to the same relations with (p)(ii) and (iii) replaced by V' = Vo Proof: The forward implication has already been considered.

66 For the reverse, we shall make use of the relations Iml = 2m -m, [IIAlI-ZllBl]+ < I|A-ZB|I. We have, for any' > 0, |1Zl)(I)||-v| <' for ((D) less than some ca('), and I|||Z'(I')l-V' I < ~' for 6'(D') less than some a'('). In the case of (R1), for any ~ e > 0, take = min( ~'/4,~('/4),c' (~'/4)) in the adjusted conditions to get h, D, D' with &(D) < K < E' and r(E /4); bt(D)) < ~ < F and a'('9/4); II'(IT) - St /(I)11 < < E1; I'eD' and II|'(I() - f(I)|| < E < ~7 for hIC I' Now L ||['(I')|l - 7 [l[(I)') + 1 1(1)11 X 1 ~[I ] + 2E[|'(I')|| _ 1]((I)|| ]+v - S( (Is')1 + |[-(I)' -) v + S||U(I)|| ~ 2ZII (I')-Z /(I)|| + VT - TY^ (,,)II + Z||(i)|| - V < 2'/2 + ~'/4 + e'/4 Hence (P)(ii) and (iii). The (R2) and (R3) cases can be treated by similar techniques.

67 604. PROPERTIES OF THE RELATIONS R We shall discuss in this section some properties of the relations R. First, we consider the reflexive property; that is, we consider whether the relations (Ri) have the property (~,~)Rij(f,) for i = 1, or 2, or 3. For (S',s) = (5,) and h the identity homeomorphism, the relation (R3) reduces to condition (c) for ~ and quasi additivity of a and I||I[ with respect to &o Hence, if and ||A|| are quasi additive with respect to b (or, equivalently, 0 is quasi additive and of bounded variation) and 5 satisfies condition (s), then (~,)Rs3(5,), and so also (:,/)R^(S, ) and (5,)Ri(S,)o The relations (Ri) could be made symmetrical by adding the respective transposed relations. The relations (Ri) themselves in general do not appear to be transitiveo However, in the case in which:(I) = T(?), Tel for a point function T in the form discussed in Section 6.1, we can show that (R2) and (R3) are transitivee Theorem 6.5.. Suppose that T',' are (R2)-related to T,, and that T"," are (R2)-related to T','. Then T","1 are (R2)-related to T,ro Proof: For any > O, we get h and x(~); and for any t' > 0, we get h' and x'(g')o Now for any E > O, take E = ~/5, E= min ( /52,((~))o Then

68 sup IIT"(h'hw) - T(w)|I < s + ~< I o weA For any D" with b"(D") < %(T) = \1'( ), there exists D' with 6'(D') < < \(8) and satisfying the condition (W) on'4,". Hence there exists D with 6(D) < ~ < T and satisfying the condition (W) on yn'. Then, denoting by,[1 "]] a sum over all I with h'hIC I", we have II 1(r ) r r lMII I" - I1g"(( ") z[' (II",) + [ (I) ITt -[I'"] [ "] T T[I"] I"t I' l"" - (I + i I" I i~, Also V = d, V" ) - V, so Vt = V. We can thus use Theorem 6.4 (although it can be done directly) to conclude that TTT,J6T are (R2)(although it can be done directly) to conclude that T",j" are (Re)

69 related to T,. This proof can easily be adapted to prove transitivity in the (R3) case. 6.5. PARAMETRIC CURVE INTEGRALS As an application of the theory in this chapter, we shall prove the invariance of integrals over Frechet equivalent parametric curves. Let T, T' be continuous mappings of bounded variation into Em from closed intervals A = [a,b], A' = [at,bt] respectively, such that (F) for any ~T > 0, there exists a sense-preserving homeomorphism h from A to A' such that sup HITt(hw) - T(w)|| < El. weA Hlere(Ref. 6, p. 105), [I) is the class of all closed subintervals of A, z ~is the class of all finite subdivisions D = I. ~ i = 1,ooN], Ii = [ailai] a = ao < ai <. <aN = b, and b(D) = max(ai-ail). We take /(I) = T(v) - T(u) where I = [u,v]. A similar system is taken for T' We shall show that T',~1 are related to T, in the sense (R3) restricted to conditions (o') and (P)(i). Then, if we assume the invariance of the curve length L = V(f), the invariance of the curve integral Jf(T, ) follows from Theorems 6.4 and 6olo

70 Theorem 6.6. If the mappings T, T' are related in the sense (F), then T,~' are related to T,. in the sense (R3) restricted to conditions (a') and (p)(i)o Proof: For any ~ > O, consider any subdivision D' = [I': j = 1,ooM] of A', I' = [a,a'], a' = al <o,,<a' = b j j-lj M The condition (F) gives a sense-preserving homeomorphism h such that sup IIT'(hw) - T(w)I < E/4M weA Condition (Rs3)(C) is obviously satisfiedo The mapping h: A + A7 is increasing with h(a) = a', h(b) = b: It is continuous and so uniformly continuous on A, that is, for any, > O, |IAh <' for jAwl less than some X(')o Uniform continuity of T on A also gives I|ATII <' for |Awl less than some 7(~?)o Take any subdivision D [Ii: i = 1, o o oN] f A, Ii = [ail,ai], a = ao < al <o oo<aN = b, with b(D) < min[y(%/4M), X(aj-a'. ) for j = l,..oM] J j-i Define a(j) = max a.:ha < a' I() 1 I - j' a,= min a: ha > a?; i(j) i i- j-1

71 these exist with a' < ha < ha < a' by the condition on b(P). j-1- i(j) I(j) - j Then ^*(Ij) ES' ]i ) - j () T'(al) - TI(a ) - T(a ) + T(ai) j j-l I(j) J(j) =T'(aj) - T(hla) - T'(a 1) + T(hlaj ) + T'(hla') - T(a )) + T(a (() - T(h'la' ) ~ I(j) i(j) j-1 Hence so z li(i: ) - zIj ](i)l < f. Thus condition (W)(i) of relation (R3) is satisfied. 6 6. PARAMETRIC SURFACE INTEGRA.LS As a second application of the theory of this chapter, we shall prove the invariance of integrals over Frechet equivalent parametric surfaces. Let T,T' be continuous mappings of bounded variation into E3 from respective admissible sets A,A. in E2(Refo 5, po 27), such that (F) for any ~' > O, there exists an orientation-preserving homeomorphism h from A to A.' such that

72 sup IIT(hw) - T(w)|| < E weA Here(Ref. 6, p. 106), (I} is the class of all closed simple polygonal regions ICA, and S is the class of all finite systems D = [I] of nonoverlapping regions I e (I). Let Tr, r = 1,2,3, be the projections from E3 onto the coordinate planes Fr of E3. Put Cr = Tr T I*, where I* is the oriented boundary curve of I. Let O(p,Cr) be the topological index of the point p e Fr with respect to Cr. Then O(p,Cr) is integrable on rr with respect to Lebesgue 2-measure m. Put jr(I) = (rr)O(pCr)dm, r = 1,2,3, Ur = sup 1r(I)1 U = sup 1/()I)||. Define q(D) = max sup IT(u) -T(v)||, IeD u,vel m(D) = max m( J[Cr]) r=l,2,5 IeD I rD u,v vI ~(D) = max f U - 9 ||(I)||,Ur - Ir(I)l for r = 1,2,3. L IED IeD J

73 Then(Ref. 5, p. 358), 6(D) = d(D)+m(D)+it(D) is a mesh function. We shall use the result(Ref. 5, pp. 186,296) that Nr(p) = sup IO(p,Cr)| is m-integrable on Fr. De Cl) IeD Similar definitions and remarks apply to T'. We shall translate the first part of Ref 53, Section 3, to show that T, are related to T',/' in the sense (R2). Then the invariance of the surface integral ff(T,}) follows by Theorem 6.2 (the second part of Refo 3, Section 3, is essentially the proof of Theorem 602). Theorem 6.7. If T,T' are related in the sense (F), then T, are related to Ts,' in the sense (R2). Proof: By absolute continuity of the integrals, for any E> 0 and r = 1,2,3, (E)S[Nr(p) + Nr(p)]dm < & for m(E) less than some T(). Take T(8) < 6. Consider any D = [I] e with b(D) < T(6)/2. Let X\() < be such that, for r = 1,2,3, the closed k( E)-neighbourhood A of U[Cr] has m(A-.) < m(.J)[Cr]) + T(~ )/2 < T(&) Take an orientation-preserving homeomorphism h A -+ A' such that sup||lr(hw) - T(w)|| < x(6)/2. weA

74 Put Cr = TrT'hI-*; then the Frechet distance I|Cr,Crjl < (~E )/2. The sets hI are compact; hence T' is uniformly continuous on UhI, that is, for any j' > 0, [IT?(v )-T'(u' )I < ~' for |Iv'-u'|1 less than some 1(1'), v,u' eU hIo For each I c D, we can take I'C hi, I' E (I'}, with II'*,hI*ll <'(%(6)/2)o Since the hI are non-overlapping, D' = [I'] E'. Put Cr = TrT'IT. Then |ICr,C || < \(~)/2. Hence |[CrC| < \ (FQ). Thus O(p,CC) = O(p,Cr) for p Frr-Ar(Refo 5, po 85)o Hence r(I) - r(I ) = (/-)[o(p,Cr) - (p,C) ]dm, so Z|lr(I) - <r(I')l' (Ar)f[Nr(p))+N(p)]dm < ~ Thus S|Ir,(I) < |Ir(I)) +I, so Ur-6/2 < Ur+<. Hence Ur < Ut, so U' = Ur by symmetry. Also E||Z(I)-' (I' )| < 3E, so ZS||(I)II < Z|L' (I')1|+35, so U- d/2 < U'+53. Hence U < U', so U' = U by symmetry. From these relations, we have Ur - Zr(I' )l Ur - Zlr(I) + Zlr(I) - )r( )[ < -(D) +E, U' - Z||'(I')|| < U - Z[ )(I)I| + ZI (I) - (I')| < la(D) + 3+ Also

75 d'(DY) < max sup IIT'(hu) - T'(hv)I| IeD u,vcI < d(D) + ~x(), m'(DI) < m(D) + T(~)/2. Thus b'(D') < b(D) + k() + T()/2 + 356 < 5~. Hence for any ~ > 0,' (D') < g' by taking ~ = ~'/5. [I] We have taken I' so that h I'C I, so a sum L over I' with h lIT' I is over only one member. Thus (p) (i) Z(I)- E[I],(I')I = li(I)-~'(I')I<I3X<', (ii) 7[(i>I|-E|S[I]I[,(I,)1 = 2|||,(I)||-||'(I')|||[||[(I)-'(I')|l<, (iii) Zi (I,1) I = 0o -11 In addition, if w C I, w' e 1', h I'C I, then w' = hwo for some wo e I Hence (a) I|T(w) - T' (w')l < |IT(w) - T(wo)l| + |IT(wo) - T'(hwo)ll < d(D) + ~(6)/2 < ~ < ~'. We have thus shown that (T,,)R2(Tt,' ).

7. ROTATIONAL PROPERTIES OF INTEGRALS ff(, ) We now apply the results of the last chapter to the problem of the behaviour of ff(5,Z) under rotations in Em and Ek. This problem arises classically when the interval functions 5,Z are generated from a variety in En, that is, a mapping T from A to En. Any orthogonal transformation R in En will give a second variety T' = RT in En, which will generate corresponding Q,1, and often'. One expects simple rotational relations between B' and B, V' and V, Sf(s,'') and Jf(SI), but not beween the interval functions t' and, Q' and f, because of their approximative nature (see, for example, the remark in Ref. 28, p. 923). However, one would expect approximative rotational relations between ~' and, Q' and 5; such approximations are just what were considered generally in the last chapter. From these approximative rotational relations, we deduce the rotational relations between the BC-integrals. Relative to a system A, f2, [ IJ, -, we shall consider interval functions, ~j from (I) to Em and, X' from [(I to Ek, and mesh functions 6, 6' on 7.1o APPROXIMATIVE ROTATIONAL RELATIONS For P, Q orthogonal transformations on Em and Ek respectively, we consider three relations (01), (02), (03) between ~j, I' and:, a with increasing strengtho (O1) For any ( > 0, there exist systems D, D' e<with 6(D) and 76

77 6'(D) < ~ and satisfying conditions (p), (q) belowo (02) For any ~ > 0, there exists k(() > 0 such that, for any D' with Z'(D') < \( ), there exists D with b(D) < e and satisfying conditions (p), (q) below. (03) For any > 0, there exists k( ) > 0 such that, for any D' with 3'(D') < Xk(), there exists r(,D')) > 0 such that, for any D with b(D) < TI( 6,D'), conditions (p), (q) below are satisfied. Condition (p): ||K(I') - P(I)ll < for ICI:, IeD, I'eD'. Condition (q): (i) () - Q' IcD (ii ) I|'(I' )| - Q (I)(| <, I'eD' and (iii) z'V(I)l < e 7.2. RELATION BETWEEN INTEGRALS Let KC Em be such that (I) and p-1'(I) eK for each I e (I). Suppose that f(p,q) is a real function on K x Ek, satisfying the conditions (f) of Section 2.5. Define g(p,q) = f(P'p,Q q) Theorem 7.1. If jf(5,~), jg(Q',~') exist, V < oo, and t',~' are related to 5,i in the sense (01), then fg(?',~s) = Jf(S,)o

78 Proof: In the setting of Chapter 6, relation (01) states that P-'~, Q-~1' are related to E, ~ in the sense (R1), with h the indentity homeomorphism. Hence, by Theorem 6.1, ff(~,$) = ff(p-1',Q -1' ) = g(',') Theorem 7.2. If ff(~,$) exists, V < oo, and Q',' are related to 5,5 in the sense (02), then fg(Q',t') exists and equals ff(S,~). Proof: In the setting of Chapter 6, relation (02) states that p -l, Q-lh' are related to,f in the sense (R2), with h the identity homeomorphism. Hence ff(P-1l',Q'-1') = fg(Q',') exists and equals ff(S,/). As particular cases, we consider in turn f(p,q) = (Qq)r, l(Qq)rl, lqlo. The conditions on f are satisfied, so that, under the other conditions, B' = QB, Vr = B(()) VI = V. For the second, g(',/') = f(P,YQ /-1) = vr; the rest are obvious. However, corresponding to Theorem 6.3, the relations B' = QB, V' = V can be proved from fewer assumptionso These we now set out. If B,B' exist and I,-' satisfy relation (O1) restricted to (q)(i) and (iii), then B' = QB. If V,V' exist and ~,~' satisfy relation (O1) restricted to (q)(ii) and (iii), then V' = V.

79 If B exists and if' satisfy relation (02) restricted to (q)(i) and (iii), then B' exists and equals QB. If V exists and ~,' satisfy relation (02) restricted to (q)(ii) and (iii), then V' exists and equals V. 7.3. SUBSTITUTION OF SPECIAL RELATIONS Corresponding to Theorem 6.4, conditions (q)(ii) and (iii) in relations (01), (02), (03) can be replaced by V' = V. This is important because invariance of V and often be proved independently, for example in Ref. 5, p. 355. Furthermore, in the case in which S(I) = T(T), TeIfor a point function T, with T' = PT, we can deduce the relation between the Weierstrass integrals from the B relation, as in Refo 28, p. 925 and Ref. 20. Theorem 7..3 Assume conditions on f, T, and > as in Section 2.5 in order that Zf(T(T),B(I~)) + f(T,,) as 6(D) + 0, Zg(T'(T),B'(I~)) f g(T',T') as b'(D) + 0O Also assume that for any E > 0, there exists D with b(D) and 6'(D) <. Then, if B' = QB, fg(Tl,'t) = f(T=,). Proof: Zg(TT(T),B'(IO)) = Zf(P-lpT(T),Q-lQB(IO)),

80 The first converges to fg(T',') as b'(D) - 0 The second converges to f (T -) as b(D) - O0. Hence the result.

8. SEMICONTINUITY OF INTEGRALS We now prove semicontinuity theorems for our integrals relative to suitable topologies. For these, the form (A) ff(Z(w), G(w))dpl is most convenient. In this form, we shall relax many of the conditions that we have imposed previously. The measure t will be arbitrary. The continuity and boundedness condition on f will be relaxed to a measurability condition. Homogeneity of f will not be assumed at first. To distinguish this situation from the previous situation, we shall take the function f of the form f(r,s), reEn, s~Es. New requirements of convexity relative to s will be imposed on f in order to obtain the semicontinuity theorems. We shall show in particular cases in Chapter 9 that our general semicontinuity theorems give the standard ones. 8.1. THE TOPOLOGY T Consider a set A and a class /of triplets T = (p,a,k), where, in each T, ft is a measure on A, c is a pt-integrable mapping from A to El, and p is a kt-measurable mapping from A to En. By "measurable" here, we mean that each of the component functions is measurable. Denote the class of t-measurable sets in A by 1o0 We shall denote the distance of a point r from a set R in En by d(r,R). Let U be the unit sphere (s:||s|| = 13 in ERo Define an ecart on c /by 81

82 t(T1,To) = sup ||pi(w)-po(w)|| + weA sup inf rI4(M1)-40(Mo)1 + MoG74 MleX4 UeU: MiCMo [ (Ml)facldl-(Mo)fdo o].u J ucU M1C MO This has the properties t(To,To) = 0, t(T2,To) < t(T2,Tl) + t(T1,To); but t(T1,To) = t(To,T1) T, = To whenever t(T1,To) = 0 need not be valid. However, the two valid properties ensure that the neighbourhood s B~ (To) = (T: t(T,To) < 3 form a basis for a topology T on c/(Ref. 16, p. 47). Remark: If gto is regular with respect to a topology OC /o on A, as will be the case in our major theorem, then %O can be replaced by o On, the class of O-closed sets, in the expression defining t(T1,To). To prove this, denote the corresponding second parts of t(T1,To) by rl,'. Obviously i' < n. For any C > O, we have 1I - ~ < lU1(M1) - o(MO) + I [(M1)fajd1 - (MO)SaOdpO].UI

85 for some u E U, some Mo, and all M1C M. Now |I(E)Jfrod4ol < for 1lo(E) less than some K(&) by absolute continuity of the component integrals. Take FoC Mo, Fo 0 ~o, with po(Mo-Fo) < min(E, K(Q)). Then - - < (M) - o(Fo)I + + l[(M1)faild1 - (FO).faoddio].U I + for all M1C Fo. But for all u e U, all Fo, and some M1C Fo, n1 + > Il1(M1)-to(Fo)I + I[(M)fold4i - (Fo)feSodio].UI iHence T - C < l' + 35 SO. 8.2. THE FIRST SEMICONTINUITY THEOREM Let f(rs) be a real function on En x El such that, for each T = (pat) e (f') f(p(w),a(w)) is I-measurable on A. It will be obvious that it is sufficient for f to be defined on J (pxo)(A) only. In particular, condition (f') is satisfied if f is continuous. Theorem 8.1. Consider a particular triplet To = (Po,cro,go) satisfying the following conditions. (1) There exists 6 > 0 such that f(r,s) > 0 for d(r,po(A)) < &, s E E,.

84 (2) For j-o-almost every wo c A and any, > 0, there exist 6( jwo) > 0, w(~,wo) e E1, b(Ftwo) e El, such that, for ||r-po(wo)|| < &b(,wo) (and (r,s) eU (pxc)(A.)-this will be implicit throughout), (a) f(r,s) >f ) (,wo) + b(,wo),s for all s, (b) f(r,s)< P(q,wo) + b(L,wo).s + C for IIs-ao(wo)lI < ((~,W o). (3) o(A) < co. (4) The measure plo is regular in some topology o on A, C/fo (We use "regular" in the sense that, for any > 0 and any M c 1, there exists a set Ffo-closed, FC M, with io(M-F) < )o (5) For any ~ > 0, there exists a set KCA, o-compact, K e /lo, with [lo(A-K) < Then I(T) = (A) ff(p(w),o(w))d. is lower semicontinuous at To in the topology To Proof: The case I(To) < oo Take any 8 > O. By absolute continuity of the finite integral, there exists K > 0 such that (E)ff(po(w), co(w))djLo < for EC A with. o(E) < Ko Since po and ro are pio-measurable, pto(A) < oo, and [o is 0-regular, we ean apply Lusin's theorem(Refso 1.2 and 1.8) to the set (wo: WOC A, condition (2) holds } to obtain aj /o-closed set K with |o(A-K) K<, po and

85 ao'o-continuousonK, and (2) holding for every woeK. By (5), we can take K o0-compact By p o-continuity of Po and ao on K, for each wo in K, there is a set H(wo) C7o, containing wo, such that, for w in H(wo)QK, jlpo(w) - po(wo)ll < s(,Wo)/2 and lcao(w) - ao(wo)ll < 6(~,wo) Consider any r such that d(r,po(H(wo)nK)) < b(~,wo)/2, Then there is a w' in H(wo)nK such that llr-po(w')Jl < (K',wo)/2, so that Ilr-po(wo)ll < (Iwo)o Hence (2t) (a) f(r,s) > P(E,wo) + b(,wo)oS for all s; (b) f(r,ao(w)) < (E,wo) + b( ~wo)oao(w) + E for weH(wo)QK, The collection (H(wo)~ wo in K] covers K, which is'-compact. Hence we can take a finite sub-cover (H(wi): i = 1,2,o..v}o We shall write H(wi) as Hi, 6(E,wi) as 6i, y(~wi) as Pi, and b(, wi) as bio Put Ei = Hi-H1-H2-oo-Hilo Then the Ei are disjoint, EiC Hi, and KC UEi = U1Hi. Next, put Bi = EiOK, so that UBi = K. Also Bi HiflK,

86 so that for weBi, d(po(w),po(HinK)) = 0 and weHirlKt Hence, from (2'b), f(po(w), ao(w) ) Pi +bi ao(w) + ~ Consequently I(T0) < (UBi)ff(Po(W)oo(w) )dio + ~ < Z (Bi)f[Pi + bi'oo(w) + F]dao + ~ < Z Pio(Bi) + Z bi.(Bi)fSo(w)dko + ~ +o(A) + ~. Consider any T in < such that t(T,To) < min(6; &i/2 for i = 1,2,.,.v.; 6( Il/il; C/Z!lbill)Then I|p(w)-po(w)II 0. Also I|p(w)-po(w)I| <'i/2 for all w in A, so that for w in Bi, d (p(w),po (HinK)) < 5i/2. Hence, by (2'a), f(p(w),s) > Pi + bias for w in Bi and all s. We can take Mi in'2, MiC Bi, with al(Mi) - o(Bi)l < il, and

87 [ (Mi)Jf(w)dn - (Bi)fao(w>)dio1]'bi/|bi||i < (/Zl|biI| Then I(T) > Z (Mfi)f(p(w),a(w))da > Z (Mi)J (pi + bi ~(-w))dt = Z Pi(Mi) + Z bi(4c(f(w)d i > Zipo (Bi) - IZ il/Zlpil +, bi (Bi);fo(w)d)o - ZI|bil|| /ZlbiI > I(To) - 6 o(A) - 53. The case I(To) - oo The essential difference from the treatment of the first case lies in getting a substitute for absolute continuity of I. Lemma 8.1. Suppose a non-negative function g(w), weA, is i-measurable on A, but (A)fg(w)dCt = oo. Then, for any ~, there exists K > 0 such that (E)fg(w)d[ > ( for all E such that pi(A-E) < Ko Proof: (A.)gm(w)dj. > 2~ for m greater than some m(~), where gm(w) min(m,g(w)). Then (E)jg(w)di > (E)fgm(w)dt =(A)fgm(w) - (A-E)fgm(w)dd > 2D - I (A-E)m for m > m().o Hence, for. (A-E) < ~/(m(0)+l) we have the required

88 inequality. Returning to the proof of Theorem 8.l in the second case, take any ~o By Lemma 8ol, there exists K > 0 such that, for [io(A-E) < K, (E)Sf(po(w),co(w))dio > C Now apply Lusin's theorem to get K as in the first case. Get H(w), Hi, bi, Pi, 6i as in the first case, but with ~ = (/2(2+p.o(A))o Copstruct Ei, B as in the first case. Then i < (UBi)ff(po(po), (w))djo < piio(Bi) + Zbi~(Bi)fco(w)dio + 1 o(A.) Consider T as in the first case, but with = //2(2+Po(A)), to get I(T) > ^ - o(A) - 2C = 5/2 Remark In Chapter 9, condition (1) will be described by saying that f is "non-negative near po" and condition (2) by saying that f is "Toconvex" o 803 THE SECOND SEMICONTINUITY THEOREM In applying our general results to particular systems, we shall wish to obtain the standard semicontinuity theorems of the calculus of variations. In its present form, our topology on is too fine for thiso With additional conditions, we can use a coarser topology on ~/ to obtain a theorem (802) which actually contains the standard semi

89 continuity theorems as we shall see in Chapter 9o We shall consider a neighbourhood system 2of To satisfying the condition ( Y?) sup |lp(w) Po(w)ll ~ 0 weA. and inf rI|(M) - o(Go) I+ 1 McT~M 0 MC Go [ (M)faSod - (Go)fcodo] uI for each Go e;o and u c U, as T - To in 2/ Remark: The "local" ecart t(TTo) = sup IIp(w) - po(w)ll + weA. sup inf r |(M) - o(Go) + ucUT MCGo I[(M)f odr - (Go)faodlo.u gives a neighbourhood system 2t coarser than that given by t(T-To), since t'(T,To) < t(T,To ) The neighbourhood system 2 satisfies condition ( ), but it is still too fine for our purpose in Chapter 9 because it involves uniform convergence with respect to Go and uo Theorem 8.20 Assume that the conditions of Theorem 81l are satisfied, together with the following conditions. (6) The mapping po is ~ o-continuous on Ao

90 (7) For any G E 0 ad a ny > 0, there exists B e with its'o-closure BC G and to(G-B) < o Then I(T) is lower semicontinuous at To in any neighbourhood system?fof T satisfying condition (?o Proof: We proceed at first as in Theorem 8.1, except that in the construction of K, we use K/2 instead of K. Also, since po is continuous on A, Ilpo(w)-Po(wo)jl < b(,wo)/2 for w in H(wo), instead of H(wo)qK. This adjustment is to be made throughout. Having reached the construction of Hij bi, Pi, bi, we put m = max[i ||Jlbijl: i = 1,2,..v]. For any > O, we have (E)fI||lo||d < T' for jo(E) < some x(l'). Take G eC o KCG, with?,o(G-K), < min[(E^'/m), /m,K]; this is possible by the regularity condition (4) in complementary form. Let Gi = GflHi. Use condition (7) to take B1 e -"" with BC G1 and po(Gi-Bi) < K/2v. Then Gi-Bi E c..oo Inductively, take Bi C 4o with i-1 i-1 BiC Gi- U B, o(Gi-U B- - Bi) < K/2v 1 1 for i = 2,3,.o.v. The sets Bi are disjoint and contained in G, while BiC Hi Note that they are not the same as in Theorem 8.1o

91 Then to (A-UBi) < 4o(K) + K/2 - Zo(Bi I< o(UGi)-,o (Bi) + K/2 i-1 = oG - U Gj) - G o(B)+ </2 1 i-1 < ~(Gi U Bj - Bi) +-/2 1 < K. Also IZ(Bi-K)j(pBibio o)dio I K< (Bi-K)fm(i+j||o1( )d 4 r(G-K)f (+jc1 o ll )djo < 2o Hence I(To) < (UBi)ff(po,po)d o + < (UBi))ff(po, o )do + 2 < Z(BinK)f[i+bio,+]d ]do +, 2 < Z(Bi)f [i+bi.Odlo - Z(Bi-K) 1J [i+bi6j * do (A.) + < CsiWo(Bi)+ bi Z (Bi)/odo f2 o(a) + 2~T Now consider a yneighbourhood of To such that, for any T = (p,a,,) in that neighbourhood,

92 sup l|p(w) - Po(w)|| < min(b; 6i/2 for i = 1,2,...v), weA and inf (|(M) - [o(Bi)l + My MC Bi |j[(M)Sadt - (Bi);obdji]-bi/lbij < min ( ~/Z 7 I, ~/ZII1bI) for i = 1,2,...v. We now proceed as in Theorem 8.1, noting that in using inequality (2'a), it does not matter that Bi is not contained in K, because po is continuous on A. Remark: Condition (7) is related to axiom (H4) on V. 8.4. CONVEXITY CONDITIONS The following condition is closely related to convexity of f in so (2) For any (roSo) in R x El.(RCEn) and any > 0, there exist 6 >0, O e E1, b E, such that, for I|r-rol| P + b.s for all s, (b) f(r,s) af(ro,sl) + (l-a) f(ro,S2). Put = (l/2)[f(ro,so) - af(ro,sl) - (l-a) f(ro,S2)] in condition

93 (2) to get P, b, b. Then f(r,si) > B + b.si, f(ro,s2) > P +b.S2. But then f(ro,so) = af(ro,sl) + (1-a) f(ro,s2) + 2 > + b.so +. However, the condition (2) is stronger than convexity, as the example f(r,s) = rs on E1 x E1 shows for ro = 0. If f(r,s) > 0 is required, take f(r,s) = [rs+l] For f continuous, condition (2) is weaker than the following strengthened convexity condition: f is convex in s; and for each ro, the graph of f(ro,s) contains no whole straight lines. This strengthened convexity condition can be put in the analytic form: For every r c R, sl, s2 c E~, 0 < a< 1, f(r,asl+(l-a)s2) < af(r,sl) + (l-a)f(r,s2); and for no r c R, s e EI, s' f 0 in E~, is f(r,s) = (l/2)f(r,s+\s') +(1/2)f(r,s-ks')

94 for all h. First, for f continuous, the strengthened convexity condition is equivalent to: For any (ro,so) e R x El and any C> 0, there exist 6 > O, v > 0, D e El, b C EQ, such that for I|r-rolI P + b.s + v Ils-soil for all s, (b) f(r,s) < P + b.s + E for I|s-soil < (see Ref. 27, p. 9). This condition obviously implies condition (2). Second, f(r,s) = P + b.s for 3,b constant satisfies condition (2), but its graph contains straight lines. Thus, when f is continuous, the strengthened convexity condition can replace condition (2) in Theorem 8.1. However, on certain subclasses of7, the semicontinuity theorem is valid when condition (2) is replaced by only convexity of f in s. Theorem 8.3. Suppose the hypotheses of Theorem 8.1 hold, except that condition (2) is replaced by f continuous and f convex in s. Then I(T) is lower semicontinuous at To on any subclass of / with (A)f lal|dl bounded. Proof: Take any e( > 0, and put F(r,s) = f(r,s) + ri||sl where r = ~/2m, m is the upper bound of (A)fjlalldp. Then F is continuous and satisfies the strengthened convexity condition. Hence (A.)fF(p(w),a(w))dII is lower semicontinuous at To on the class mentioned. Thus (A)fF(p(w),a(w))di > (A.)F(po(w),ao(w))dIo - /2

95 for t(T,To) less than some 5. That is, I(T) + Sfia||||d > I(To) + f|llaollddo - 6/2, which gives I(T) > I(To) - ~& A similar adjustment applies to Theorem 8.2. 8.5. THE HOMOGENEOUS CASE If f(r,s) is positively homogeneous of degree one in s, then condition (2) reduces to: for any (ro,so) e R x El and any c > 0, there exist b > 0, b C El such that, for I|r-roll b.s for all 6, (b) f(r,s) < b.s + < for Ils-soll < b. To prove this, condition (2a) with homogeneity gives cf(r,s) > P + a bs for all a > 0, so 0 > A. Hence f(r,s) < b.q + for ||s-soll < 6, llr-roll < b, r C R. Also, f(r,s) > P/0 + b.s for all a > O, so

96 f(r,s) > b.s for all s,1|r-roll < &, r c R. Now the term |1i(M1)-4o(Mo)l in t(T1,To) is brought into the proof of Theorem 8,1 by the P terms. Hence, if f(r,s) is positively homogeneous in s, Theorem 8.1 is valid with the improved ecart obtained by omitting the above term. A similar adjustment can be made to condition (V7) for Theorem 8.2.

9. SEMICONTINUITY IN PARTICULAR CASES In this chapter, we shall obtain known semicontinuity theorems for curve and surface integrals from our general theorems. in each case, we shall have the particular topology on J'which is used in the corresponding section of the calculus of variations. In each case, we shall verify the conditions (3), (4), (5), (6), (7), and (7/) of Theorem 8.2. Hence, if the conditions (f'), (1) and (2) on f are satisfied, we have semicontinuity theorems of the corresponding section of the calculus of variations. 9.1 PARAMETRIC CURVE INTEGRALS ff(XX')de Let the set A be a finite closed interval (w: a < w < b) in El, with Euclidean topology. Let the mappings p: A - En be continuous in 2C and of bounded variation. Take C = - for all T. The measures i. and corresponding signed measures v are constructed for the interval function $[uv] = p(v) - p(u) on the intervals I = [u,v] in A, by the process described in Chapter 2. The conditions for this are easily verified in this case. Let a = 0 = dv/dl; thus a is.-integrable. The triplet T = (p,a,p.) is now determined by p. 97

98 Since the mappings p are of bounded variation, pI(A) < oo The measures p are -regular by the general theory of Ref. 7. Condition (5) is trivial here. Condition (7) follows from (H4) of Ref. 7. The function f(r,s) is assumed to be positively homogeneous of degree one in s, so we shall consider the adjusted condition (2) mentioned in Section 8.5. We shall prove that the neighbourhood system induced by the uniform topology on p satisfies that condition. Note that fadp = v. Theorem 9.1. For each Ge2Z inf IIv(M) - Vo(G)(|| 0 ME: - MC.G as sup lIp(w) - po(w)) - 0 wcA Proof: We have X o(I) - vo(G) IEDG as bG(DG) o0 Hence, for any 6 > 0, |Ivo(G) - o(I)l11 < 5

99 for some finite number (m, say) of non-overlapping intervals ICG. Here, $[u,v] = p(v)-p(u) = v(u,v). Consider any p with sup |Ip(w)-po(w) || < (/m. Then weA |Iv(IO) - vo(IO) 1 < 2 6/m Hence I|v(UI~) - v (G) | < 3 e For the purposes of Theorem 8.3, (A)f lj|fd = (A) = L, the length of the corresponding curveo We can treat each mapping p as a continuous rectifiable curve C in En. The measure 4 corresponds to the arc length I on C, and.(A) is the Jordan length L of C. Thus C also has the representation X(1): 0 < I < L Since v[.,V'] = X(')-X(2), we can take a also as X' = dX/d~. Thus our integral has the form I(C) = (A) f(p(w),O(w))d = f(X(1),X'(1))d As proved in Ref. 7, if f is also bounded. and uniformly continuous on KxU, then our integral also has the form of a BC-integral I(C) = /f(p,p)

100 We can now deduce from Theorem 8.2 the Tonelli-Turner theorem: Theorem 9.2. Let f(r,s): KxEn + E1, KCEn, be positively homogeneous of degree one in s. Let C be the class of all continuous BV mappings p(w): A + K, weA = [a,b]CE1 (in other words, continuous rectifiable curves C in K) for which f(p(w),Q(w)) is measurable in the corresponding measure 1i on A. If CoE fis such that f is non-negative near Co and is Co-convex, then the integral I(C) is lower semicontinuous at Co in C, with the uniform topology. 9.2 PARAMETRIC SURFACE INTEGRALS Sf(X,J)dudv We shall show that Theorem 8.2 covers the semicontinuity results of Cesari in Ref. 4 and Turner in Ref. 29. In the latter paper, our system has the following form. The set A is any admissible set in E2 (see Ref. 5, p. 27). The dimensions n and ~ are both 35 The mappings p: A + E3 are continuous in the Euclidean topology 2 and of bounded variationo Each p determines a topology on A, namely the class of 2,( -open p-whole sets in A (see Ref. 5, Section 10.2). The measures t and signed measures v are constructed by the process described in Chapter 2, from an interval function ~ defined from p (see Ref. 6, p. 107). Let o = dv/d. = Q; thus a is.-integrableo The triplet T = (p,c,p.) is now determined by p.

101 Since the mappings p are of bounded variation, t(A) < o. The measures Bt are i-regular by the general theory of Ref. 7. Condition (5) is satisfied (see the remark at the bottom of p. 196 in Ref. 29). Each p is i-continuous. Condition (7) follows from (H4) of Ref. 7. The function f(r,s) is assumed to be positively homogeneous in s, so we shall consider the adjusted condition (CY) mentioned in Section 8.5. We shall prove that the neighbourhood system induced by the uniform topology on p satisfies that condition. Theorem 953. For each Goe o and each unit vector uEEA, inf |[v(M) - vo(Go)].ul + 0 Me YY> MCGo as sup 1 p (w) - p(w)iI 0 weA Proof: Let P be a rotation taking u to the z-axis. Then Vo(Go).u = Pvo(Go).Pu = v3(GoPpo) = v(Gopo) where we have expressed v explicitly as a function of p, v3 is the zcomponent, and po is the projection of Ppo on the (xby) plane. The second equality follows from a rotational property of v, Ref. 28, Theorem 3o

102 According to Ref. 29, Lemma 3, simplified for our purpose, for any > 0 and any plane mapping po of bounded variation, there exists 5 > 0 such that, for any other plane mapping pT with sup Ilp'(w) - po(w)II < 5, weGo there exists MCGo with Iv(M,p') - v(Go,p)| < Now, if we take any p with sup lip(w) - po(wv)l < weA and p' is projection of Pp on the (x,y) plane, then Ilp'(w) - po(w)ll < IPp(w) - Ppo(w)l =IIP(w) - po(w)II < 6 on A and so certainly on Go. Hence I[v(M)-Vo(Go) ].u = |v(M,p') - v(Go,po) < C Note that here, for the purposes of Theorem 8.5, (A)/||a||du = (A)

103 = V = L, the Lebesgue area of the corresponding surface. We can treat each mapping p as a continuous surface S with finite Lebesgue area (see.Ref. 5). The measure 4 is the same considered in Ref. 5, Section 25.5, and A(A) is the Lebesgue area of S. As proved in Ref. 7, if f is also bounded and uniformly continuous on KxU, then our integral I(S) = (A)f (p(w), (w))d4 also has the form of a BC-integral I(S) = ff(p,~) As proved in Ref. 5, Section 37 and Appendix B)5(ii), under the same conditions on f, S always admits a representation X(w): w = (u,v)eAcE2 such that I(S) = (A)/f(X(w),J(w))dudv, where J = (J1,J2,J3) are the usual Jacobians. We can now deduce from Theorem 8.2 the Turner theorem (see Ref. 29, Theorem 1): Theorem 9.4. Let f(r,s) KxE3 + E1, KCE3, be positively homogeneous of degree one in s. Let 9be the class of all continuous BV mappings

104 p(w): A -* K, wcA(in other words, continuous surfaces S in K with finite Lebesgue area) for which f(p(w),@(w)) is measurable in the corresponding measure kt on A. If So~ is such that f is non-negative near So and is So-convex, then the integral I(S) is lower semicontinuous at So in r- with the uniform topology. 953 NON-PARAMETRIC INTEGRALS ff(w,X,grad X)d4 Let A be any open set in Ek with finite Lebesgue k-measure pi. Let.. be the Euclidean topology on Ao Consider mappings X(w): A - Em, w = [wi]eA, absolutely continuous in the sense of Tonellio Thus 6X/wwi exists t-almost everywhere in A for each coordinate wi, is a-integrable, and wi dwi = X(wi=p) - X(wiU) for each segment (a < wi < P) in A on p*-almost every line parallel to the wi-axiso Here t* is Lebesgue (k-l)-measure; if k = 1, we take i* as enumerationo Let the mappings p be of the form (w,X(w))o Thus the dimension n = k+m. In the case k = 1, our mappings p are essentially non-parametric curves in Em+l on Ao In the case m = 1, our mappings p are essentially non-parametric hypersurfaces in Ek+l on A. Let a = grad X = [LX/wi ] This is a kxm matrix, but here we have to treat it as a km-vector; thus the dimension I = km. The vector-valued function a is >-integrable by the condition on Xo

105 The measure p is a -regular. In fact, the closed set F in the regularity condition can be taken compact, since its compact intersections Fn with the spheres (w: Ij||w < n) have 4(Fn) + p(F). Thus condition (5) of Section 8.2 is also satisfied. To show that condition (7) is satisfied, we use again the compact regularity of p. For any Ge 2' and any d > 0, we take FCG, F compact, with p(G-F) < C. G is the union of a countable number of closed intervals, and also the union of the corresponding open intervals. Since F is compact, we can take a finite number of the open intervals Ii covering F. Put B = UIi Then B =UIiCG. Of course, B is open, and i(G-B) < 4(G-F) < C. We shall now show that the neighbourhood system induced by the uniform topology on X satisfies the condition (2/). Theorem 9.5. For each Ge 2/ and each Xo, inf ( |(M)-p(G) |+||(M)J grad Xdt-(G)f grad XodJ||) + 0 ME: 1? McG as sup ||x(w) - Xo(w)ll + 0 weA Proof: By absolute continuity of the integral, for any (f > 0, I|(E)S grad XodJlI < ( for ((E) less than some A(6). Let F be a compact set in G with p(G-F) < min(,,A6(C)). G is the

106 union of a countable number of open intervals. Because F is compact, we can take a finite number of these intervals covering F. We can contract them to closed intervals still covering F, and decompose these into closed non-overlapping intervals Jio Then k(G-UJi) < C an (C) Hence II(G-UJi)f grad XodtI\ < C Consider any X with sup IIX(w) - Xo(w) l < /Z4(J) weA where Jt is the boundary of J.i Then |I(UJi)f(grad X-grad Xo)dI|| II(J*) J(x-o)d*l|| < 0 Hence I|(UJi) - p(G) | + II(UJi)J grad Xdp - (G)f grad Xodkll < 3(f The triplet T = (p,a,i) is determined by X. Thus we can describe conditions in terms of X. Specifically, "f is Xo-convex" will mean that f is To-convex in the sense of Section 8o2.

107 We can now deduce from Theorem 8.2 the following theorem. Theorem 9.6. Consider f(p,q,s): AxEmxEkm + El, A open in Ek. Let X be the class of all ACT mappings X(w): A + Em, weA, for which f(w,X(w), grad X) is measurable with respect to Lebesgue k-measure p. on A. If Xoex is such that f is non-negative near (A,Xo(A)) and is X-convex, then the integral I(X) = (A)Jf(w,X(w),grad X)dp is lower semicontinuous at Xo in X with the iniform topology. 9.4 CURVE INTEGRALS INVOLVING HIGHER DERIVATIVES Cinquini, in Refs. 8 and 9, deals with variational problems for curve integrals of functions involving derivatives up to the third order. We shall show how our theorems cover Cinquini's results for semicontinuity in these cases. Second Order Problems: Corresponding to the second order problems of Ref. 9, we have the following system. The set A is a closed interval in El, with Euclidean topology 2. Let X be any absolutely continuous mapping from A to E3 such that, when parametrized by its arc length -s, X' = dX/ds is also absolutely continuous. We put p = (X,X'), and a = X'^XT, where A denotes the vector product in E3. The measures p correspond to the arc lengths::s. -.(A) = L < c by absolute continuity of X.

108 Cinquini calls curves X satisfying the above conditions "ordinary," and uses a topology on them defined by the following neighbourhoods. If Lo > 0, a 6-neighbourhood of Xo is the class of ordinary curves X for which, considering s as a function of so, Is-1 < 6, IIX(s) - Xo(so)| < 6, IIX(s) - Xo(so)|| <6 To avoid confusion, we use the dot to denote differentiation with respect to so. If Lo = 0, so that Xo is constant, a b-neighbourhood of Xo is the class of ordinary curves X for which either L > 0, IIX(s) - XoII < 6, IX'(a) - X'()11 < 6 for all a,P in [O,L]; or L = 0, IIX - xoll < 8 If we restrict our considerations to a class of curves for which (A)SIIX"'I||d is bounded (by C, say), we can show that Cinquini's neighbourhoods satisfy condition (A), This is essentially a result of Refo 9,

109 po 33, but we prove it in our form. Theorem 9.7. For any ordinary curve Xo, any C > 0, and any Ge v, there exists 5 > 0 such that I -- inf ( [I(M)-ko(G) I|+1(M)JX'AX"d~-(G)fXo.AXodtoll) Mc 7 Me G < gf for all X in the 5-neighbourhood of Xo. Proof: First, consider the case in which Lo > 0. By absolute continuity of the integral, for any C' > 0, |I(E)fXoAXodIto.J < 6' for po(E) less than some 7(&'). As in Theorem 9ol1 we can construct a finite number (m, say) of closed non-overlapping intervals J in G with po(G-UJ) < (E /3) and (/53 For the moment, take any 6 < 1, and consider any X in the 5-neighbourhood of Xo. From the condition Is-l| < 6, we have iS-SoI < So <6o Now (J )fx'AX"d - (J)f JXoXod = (J)J(x'-io)A^X"d + (J)f(x -io)AXOdlo + (J)S(Xo"Xt s+Xo AX )dLo

110 In this, || (J) f (X' -o )^x"di|| < 5(A)f.||X"jl|d < 5C Similarly | (J) (xt -Xo)AxXodCol <aC C And I|(J)J(XAX"s+XoAX' )diL | = 1(:XoAX')(J) | = | (XoA[X'- 1o) (J) | < 2 Also t(J) - o(J) = (J)J'(s-l)do, so I|(J) - 0(J) I < 5Lo Then ||(UJ)fX'AX"dh - (G)fJXoXod || <_ Z||(J)x'AXt/X"d - (J)fXoXod|~ol| +C /3 < m(2C+2)6 + /35;

111 and I (uJ) - o (o) I <_ ZI(J) - o(J) I + (5/3 < mLo + C/3 Hence r < 6 if 8 < /3m(Lo+2C+2) Next, consider the case in which Lo = 0. Then Cpo(G) = (G)fkXoAxodto = 0. For the moment, take any 6 < 1/2 NI, and consider any X in the 5neighbourhood of Xo. If L = 0, then the required result is trivial. If L > 0, then, as in Ref. 9, p. 32, L < 4 J 6. Take any JCGo Then kt(J) < 4 3 6, and II(J)fx'AX"dl1 < 4 < S6C Hence Tj < ( if 6 < 5/4 4 (C+1) We can now deduce from Theorem 8.2 the following theorem. Theorem 9.8. Consider f(p,q,t):E3xE3xE3 - El. Let A be a closed interval in Elo Let J2 be a class of AC mappings X(w): A - E3, weA, such that (i) when parametrized by the arc length s, X' = dX/ds is also AC;

112 (ii) (A)fIIX" lds is bounded; (iii) f(X,X,X'XX") is measurable with respect to s on A. If XoE E 2 is such that f is non-negative near (XoxXo)(A) and is Xoconvex with respect to t in the sense of Section 8.2, then the integral I(X) = (A)ff(X, Xs,X'A X")ds is lower semicontinuous at Xo in f2 with the Cinquini topology. Third Order Problems: In this case, let X by any absolutely continuous mapping from A to E3 such that, when parametrized by its arc length s, X' and X" are also absolutely continuous. We put p = (X,X' X'AX"I), and a = X'AX1. Cinquini defines a topology on these curves by the following neighbourhoods. If Lo > 0, a 5-neighbourhood of Xo is as in the second order case, but with the extra condition IIX (s) - o(sO) 11 < If Lo = 0, a b-neighbourhood of Xo is as in the second order case, but with the extra condition IIX"(a) - X"() 11 < for all a,5 in [O,L] when L > 0. We shall now show that Cinquini's neighbourhoods in the third order

113 case satisfy condition (2y). This is essentially a result in Ref. 9, p. 54. Theorem 9.90 For any Xo, any S > 0 and any Ge 2/, there exists 5 > 0 such that --' inf (!|(M)-ko(G) I+I|(M)fX'AX"'d -(G)jXoAo diolII) Me r MC G < C for all X in the third order 5-neighbourhood of Xo. Proof: Consider first the case Lo > 0. By absolute continuity, for any > 0, 1 (E) XA f Xod|o|d i <6? for po(E) less than some?\(' ) Construct a finite number (m, say) of closed non-overlapping intervals J in G with [io(G-6UJ) < X(C5/3) and ^/35o Denote by K the maximum of I]XOll at the end points of the Jso For any 6 < 1, consider any X in the 5-neighbourhood of Xoo We have II(J)fxVAxl"'t - (J)fJAXd^doLI = (X X", X-oAXYo) () II = I[X'A(Xl-XO )+(x?- o)Xo ](J)I| < 28 + 26K. Also |l(J)-yo(J) <_ 6Lo as in Theorem 9c7o

114 Then II(UJ)fx'Ax"'ad. - (G)fiXoAXodoll < Z|i(J)fxx11,^ x" d (J)JXoAXodtoll + C//3 < m(2+2K)b + 6S/3 and i1(LUJ) - o((G) I < mLo + 5/3 IHence r < ( if 6 < (/3m(LO+2K+2) Next, consider the case Lo = O0 For the moment, take any 5 < 1/2 " 5, and consider any X in the 5-neighbourhood of XO. If L = 0, the required result is trivialo If L > 0, take any closed interval J in Go Then p(J) < 4 53 5, and (J)fxAX"'d. = (X'AX")(J) = X' ()A x"(P) - X ()AX)" a) x' (P)A (X" ()-X"(a)) + (x (P)-x' (a) ),X" (a) where CaP are the s-coordinates of the ends of Jo Considering each component, we have Xi(L) - X(0O) = LXj(~)

115 for some 0 in (O,L), so I|X(s) | < b + 6/L and JtX"(s)|| < j3 3(1l+l/L) Also (X' ()-x'(a))AX"t(a) = (-a)[X(), ),(3)] - x"(a X (a) for some Q1, 02, 03 in (a,)), so Jl(x' (P)-x' (a))A)x" (a) < LJ36 Ni S(l+l/L) < 35 (4 J3 s+l) < 9 2 Thus Il(J)fkX"'1 d.1 < & + 962 < 4 Hence ri < 6 if 6 < < /4(1++ ) 0 We can now deduce from Theorem 8.2 the following theorem. Theorem 9.10. Consider f(p,q,r,t): E3xE3xE3xE3 + El. Let A be a closed

116 interval in El. Let g 3 be the class of all AC mappings X(w): A - E3, weA, such that (i) when parametrized by the arc length s, X' and X" are also AC; (ii) f(X,X',X'TX",X'AX"') is measurable with respect to s on Ao If XoE C3 is such that f is non-negative near (XoXxXXAXX ) (A) and is Xo-convex with respect to t in the sense of Section 8.2, then the integral I(X) = (A)ff(XX,X'X",X'AX"ds is lower semicontinuous at Xo in w3 with the Cinquini topology.

PART II 10. THE SHAPE OF LEVEL SURFACES OF HARMONIC FUNCTIONS IN THREE DIMENSIONS 10 o1 INTRODUCTION Let $ be a harmonic function in E3. We shall describe the shape of the level surfaces (P: /(P) = k) of ~ in terms of the corresponding regions (P: O(P) > k} of higher potential, or "regions of potential." The results of this chapter can be summarized as follows. If two regions of potential are star-shaped relative to some point, then every intermediate region of potential is similarly star-shaped. If two regions of potential are convex, then every intermediate region of potential is convexo On the other hand, we prove by an example that if two regions of potential are merely simply connected, the intermediate regions of potential need not be simple connected These results extend previous work of Gergen33 and Gabriel3 for Green's functions in three dimensionso HYPOTHESIS H: Let C1 and Co be two closed subsets of E3 (C1 not empty), and let i(P) be a real-valued function of E3, subject to the following conditions. (i) /(P) is continuous on E3, (ii) /(P) = 1 on C1, (iii) /(P) = 0 on Co, 117

118 (iv) /(P) -+ O as P -+ 0o, (v) /(P) is harmonic on D = (CO LC1)' = E3-(CoUC1l) Since the set Co may be empty, the situation just described includes the case where /(P) = 1 on a closed non-empty set C1,,(P) -+ 0 as P -+ o, and $(P) is harmonic on C1 = E3-C1 (see Ref. 31, pp. 3979 401)o We assume the existence of a function satisfying the stated conditions; some conditions on C1 and Co sufficient for the existence are given, for example, in Refo 30, pp. 290-312)o Note that C1 and Co are disjoint because of conditions (ii) and (iii), and that C1 is bounded because of conditions (ii) and (iv). In addition, by an application of the principle of the maximum in the strong form, we can deduce from our conditions that 0 < /(P) < 1 on E3. We shall denote the Euclidean distance of a point P from the origin by IIP I, the Euclidean distance of a point P from a set C by d(P,C). 10.2 STAR-SHAPED REGIONS By definition, a set C is star-shaped relative to the origin 0 if \P is in C whenever P is in C and 0 < 7 < lo Theorem 101.olo Let C1, Co, and $ satisfy Hypothesis H, and let C1 and Co = E3-Co be star-shaped relative to Oo Then the regions Dk =P: /(P) > k) are star-shaped relative to 0o Lemma 101.o Under the hypotheses of Theorem 10.1, D is connected. Proof: Let & be the distance between Co and C1 (for Co empty, let 5 be any positive number). Since Co is closed and C1 is compact, 5 is

119 positive. Take a point R in C1 at maximum distance from 0, and any real number A greater than 1||Rl| On each plane through OR, start from OR to divide the disc (P:IIPII < A) into closed acute sectors Ai determined by circular arcs of length less than 5o Let Ri be a point on the compact set Ai/C1 at maximum distance from Oo Since no point of Co is at distance less than 5 from Ri, there exists an arc Li across Ai not meeting Co UClo Since C1 and CO are star-shaped, the arcs Li can be joined by radial segments to form a curve K not meeting CoUCo1 For the same reason, every point of D can be joined by a radial segment to some K, and the curves K can be joined by a segment on the extended segment ORo Hence D is arc-wise connected. Lemma 10o2. Under the hypotheses of Theorem 101o, 0 < i(P) < 1 on Do Proof: Since D is connected, the strong form of the principle of the maximum gives both inequalitieso Lemma 10 3o Under the hypotheses of Theorem 10.1, l is non-increasing on each radius. Proof: Suppose Lemma 10o3 is falseo Then there are points Po, \oPo in D with.(0oPo) < z(Po), < \o < 1. Hence the function *(P) = $(P) -(?\oP) has a positive least upper bound m on E3. By condition (iv) in Hypothesis H, g|(P) I < m/2 for IIPII greater than some positive 5o Hence *(P) < m/2 also for JIIPI > 5. Hence m is the least upper bound of 4 on the compact set (P: IIPII < 63, and so is attained there. But m is not attained when P is in Co, since <_ 0 in Co. Nor is m attained when P is

120 in C1, since C1 is star-shaped so that I = 0 in C1. Also, when \oP is in Co, 4(P) = 0 since CO is star-shaped; thus m is not attained in that case. Lastly, m cannot be attained at P when?oP is in C1, since r(P) < 0 then. Hence m is attained at some point P1, where P1 and \oP1 are in D. Let d be the lesser of d(P1,Co), d(\oP1,C1)/\o; the second is certainly finite. Then the set N = (P: I|P-P111 < d) is contained in C'. Also \oN = (koP: P in N) is contained in C~, since C' is star-shaped. But \oN = (Q: IIQ-XoPllI/O < d), and hence is contained in C1, Therefore N:iscontained in Ci, since C1 is star-shaped. Thus 4(P) is harmonic in N. By the principle of the maximum, *(P) = m on N. Now either (a) |IP1-RII = d for some R in Co, or (b) I|P1-RII = d for some \oR in C1. In case (a), *(R) = 0-j(7oR) < 0, while in case (b), *(R) = ~(R)-l < 0. However, *(P) = m for some points in any neighbourhood of R. This contradicts continuity. Theorem 10.1 follows immediately from Lemma 10.3. Corollary: Under the hypotheses of Theorem 10.1, the radial derivative 6/3r is strictly negative in D. Thus grad 0 / 0 throughout D. Proof: The function ra//ar is harmonic and non-positive in D. Thus if r3^/3r were zero at some point of D, rco/3r would be zero throughout D, so that A would be radially constant in D. Since each radius meets the set C1, it would then follow that $(P) = 1 throughout D, contrary to Lemme 10.2.

121 10.3 CONVEX REGIONS Theorem 10.2. Let C1, Co, and ( satisfy Hypothesis H, and let C1 and CO be convex. Then the sets Dk = (P: /(P) > k) are convex. Lemma 10.4. If the hypotheses of Theorem 10.2 are satisfied, and if P and Q are two points in D such that /(P) = /(Q), then /(R) > /(P) for every point R on the open segment PQ. Proof: For all point pairs P,Q with /(P) = /(Q) and for all points R on the corresponding closed segment PQ, define Q(P,Q,R) = /(P) + /(Q) - 2/(R) The function Q(P,Q,R) is continuous and bounded on its domain of defipition, and its least upper bound m is non-negative. If m = 0, then /(R) > /(P) = ((Q) for all P,Q,R in the domain of 0. If we assume that Lemma 10.4 is false, then there would be some Po, Qo in D, and Ro in the open segment PoQo, with (Ro) < (Po) = /(Qo). Thus if m = 0 and Lemma 10.4 is false, we have Z(Ro) = i(Po) = A(Qo). Hence Q(Po,QoRo) = 0, and m = 0 is attained. Since Po,Qo are in D, then 0 < j(Po) = (Qo) < 1 by Lemma 10.2, hence 0 < Z(Ro) < 1o Thus Ro is in Do If m > 0, condition (iv) in Hypothesis H implies the existence of 6 > 0 such that Q(P,Q,R) < m/2 whenever |lPII > 5, or IIQII > 5, and therefore m is the maximum value of 0 on the compact set ((P,Q,R): IIPl, < IQ < 5, (P) = ) (Q),REPQ}

122 Now @(P,Q,R) = 0 whenever two of the points P, Q, and R coincide. Also, G(P,Q,R) < 0 whenever P or Q lies in Co; and O(P,Q,R) = 0 when P or Q lies in C1, since C1 is convex. If R lies in Co, then (by convexity of Co) either P or Q lies in Co, hence /(P) = /(Q) = $(R) = 0, and again Q(P,Q,R) = 0. If R lies in C1, then G(P,Q,R) < 0 because g(R) = 1. Thus, for m > 0, ~ takes its maximum at some P, Q, R distinct and in D. It follows that in both cases, either m = 0 and Lemma 104 assumed false, or m > 0, we could conclude that Q takes its maximum at some P,Q,R with P,Q,R distinct and in D, R in PQ, and /(P) = O(Q). By the corollary in Section 10.2 we have, on the other hand, grad ~ # 0 everywhere in D. Then, by a theorem of R. M. Gabriel (see Ref. 31, po 389), 8 is radially constant in D with respect to some centre. For any point S in D, consider a half straight line J from S on which f is constant on each segment lying in D. If J is completely contained in D, then j is constant on J, and, by condition (iv), then 8 = 0 on J and 6(S) = 0 If J is not contained in D, then the minimum of I|S-PII for P in J ((CoLUC) is attained either in Co, in which case /(S) = 0, or in C1, in which case /(S) = 1o Hence, in all cases, the results contradict Lemma 10.2o This proves that m = 0 and Lemma 10.4 is true. Proof of Theorem 10 2. If $(P) > /(Q) > k and /(R) < k for some R in PQ, then there exists a point P' in PR with /(P') = 6(Q)o This situation is impossible by Lemma 10.4o

123 10o4 A COUNTER EXAMPLE In relation to the results in Section 10.2, it is appropriate to consider an example suggested by W. J. Wong, which shows that if C1 and C' are merely assumed to be simply connected, then the regions Dk need not be simply connected, and grad 0 can be zero in D. We shall require bounds for the change in 0 with change in C1. The technique used is an adaption of a method used by Gergen in Refo 32. Suppose Ci is C1 with a piece removed, with corresponding /-. Then 0(P) —"(P) is harmonic in D, continuous in E3, 0 on Co, and non-negative on C1. Hence O(P)-'-(P) is non-negative on Do Let A be the piece of the boundary D* of D removed in forming Ci, and g(Q;P,D) the Green's function of D with pole P. If D* is sufficiently smooth (see Refo 34, po 237), then, for P in D, (P) - -(P) = (D*)f[)(Q) — (Q)] g(Q;P, d < (A)f ag(Q;P D)' - -4tan Let K by any compact set in D. Again provided D* is sufficiently smooth (see Refo 55, p. 259), g(Q;PD) has finite upper bound MK for P in K -4jt6n and Q in C*. Hence 1 ( (P) -'(P) < MKa(A) where a(A) is the area of A. Now apply this result to the following system. Let the set Co be an

124 open sphere with centre X, and the set C1 a solid torus inside Co, with the same centre of symmetry X. We form C1 from C1 by removing a section bounded by two half-planes having the major axis of C1 as common edge. Then C_ is a simply connected continuum. It has only one axis of symmetry, which cuts the inner surface of C1 at Y and Z, say, the latter being removed in forming Co. Since /(X) < 1, we can take k between /(X) and 1. First, take K = (P,P'1, where P is in XY and P' is in XZ with k < j(P) = ((P') < 1. By forming C1 appropriately, make MKa(A) < ((P)-k. This gives ~-(P) > k and 0-(P') > k, while Z-(X) < k. Hence the component of grad -z along YZ is zero somewhere in PP'. With symmetry, this shows that grad 0- = 0 there. Second, take K = (P: /(P) = k). For suitably formed C1, MKa(A) < k- (X)o Hence, ~-(P) > /(X) on K. On the major axis of C1, p (P) <_ (P) < O(X). This shows that [P: i (P) > /(X)) is not simply connected.

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