summary:We present a Cauchy test for the almost derivability of additive functions of bounded BV sets. The test yields a full descriptive definition of a coordinate free Riemann type integral

Bongiorno, B.

Di Piazza, L.

Pfeffer, W. F.

English

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeB. Bongiorno; Luisa Di Piazza; Washek Frank PfefferA full descriptive definition of the BV-integralCommentationes Mathematicae Universitatis Carolinae, Vol. 36 (1995), No. 3, 461--469Persistent URL: http://dml.cz/dmlcz/118774Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1995Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 36,3 (1995)461–469 461A full descriptive definition of the BV-integralB. Bongiorno∗, L. Di Piazza†, W.F. Pfeffer‡Abstract. We present a Cauchy test for the almost derivability of additive functionsof bounded BV sets. The test yields a full descriptive definition of a coordinate freeRiemann type integral.Keywords: Perimeter, partition, gage, absolute continuityClassification: Primary 26B30, 26A39; Secondary 28A75The BV-integral, introduced in [5, Definition 5.1] under the name “variationalintegral”, is a coordinate free generalization of the Lebesgue integral defined onall bounded Caccioppoli sets. Unlike the Lebesgue integral, it integrates partialderivatives of differentiable functions and provides the unrestricted Gauss-Greentheorem. The purpose of this note is to present a complete characterizationof those additive functions of bounded Caccioppoli sets that are indefinite BV-integrals (Theorem 3.9).1. PreliminariesThe ambient space of this paper is Rm, where R is the set of all real numbersand m is a fixed positive integer. The metric in Rm is induced by the maximumnorm, and U(x, ε) denotes the open ball about x ∈ Rm of radius ε > 0. For a setE ⊂ Rm, we denote by clE, intE, ∂E, d(E), and |E| the closure, interior, bound-ary, diameter, and Lebesgue measure of E, respectively. The words “measure”,“measurable”, and “negligible” as well as the expressions “almost all” and “al-most everywhere” always refer to the Lebesgue measure in Rm. The symmetricdifference of sets A and B is the set A △ B = (A − B) ∪ (B − A).Let E ⊂ Rm. We say an x ∈ E is, respectively, a density or dispersion pointof E according to whetherlim infε→0+|U(x, ε) ∩ E|(2ε)m= 1 or lim supε→0+|U(x, ε) ∩ E|(2ε)m= 0 .The set of all nondispersion points of E is called the essential closure of E, denotedby cl∗E, and the set of all density points∗Supported in part by the Italian Ministry of Education (M.U.R.S.T.).†Supported in part by the Italian Ministry of Education (M.U.R.S.T.).‡Supported in part by the National Research Council of Italy (C.N.R.).462 B.Bongiorno, L.Di Piazza, W.F. Pfeffer- of E is called the essential interior of E, denoted by int∗E. The essentialboundary of E is the set ∂∗E = cl∗E − int∗E. Clearly, intE ⊂ int∗E ⊂ cl∗E ⊂clE, and so ∂∗E ⊂ ∂E. If E is measurable, the sets E △ cl∗E, E △ int∗E, and∂∗E are negligible [7, Chapter IV, Theorem 6.1]. The set E is called essentiallyor doubly closed whenever E = cl∗E or E = cl∗E = clE, respectively.The (m− 1)-dimensional Hausdorff measure in Rm is denoted by H, and a setT ⊂ Rm of σ-finite measure H is called thin. Each thin set is negligible but notvice versa. The perimeter (in De Giorgi’s sense) of a set A ⊂ Rm is the number‖A‖ = H(∂∗A). A bounded set A ⊂ Rm with ‖A‖ < +∞ is called a Caccioppolior BV set (BV for bounded variation — cf. [2, Section 5.11, Theorem 1]). EachBV set is measurable [6, Corollary 13.2.5], and the family BV of all BV sets is analgebra in Rm. For E ⊂ Rm, we let BVE = {A ∈ BV : A ⊂ E}.The regularity of a BV set A is the numberr(A) ={|A|d(A)‖A‖if d(A)‖A‖ > 0,0 otherwise.The isoperimetric inequality ([2, Section 5.6, Theorem 2,(i)]) shows that a se-quence {An} of BV sets is regular in the sense of [7, Chapter IV, Section 2]whenever infn r(An) > 0.Let A be a BV set. The set of all x ∈ int∗A such thatlimε→0+H[U(x, ε) ∩ ∂∗A](2ε)m−1= 0is called the critical interior of A, denoted by intcA. According to [8, Section 4],we have H(int∗A − intcA) = 0; in particular, the set cl∗A − intcA is negligible.The next lemma, proved in [4, Lemma 1.2], gives an important property of thecritical interior.Lemma 1.1. Let A ∈ BV and x ∈ intcA. Suppose {Bn} is a sequence of BV setssuch that x ∈ cl∗Bn and r(Bn) > ε > 0 for n = 1, 2, . . . . Then x ∈ cl∗(A ∩ Bn)for n = 1, 2, . . . , and if lim d(Bn) = 0, then r(A ∩ Bn) > εm+1 for all sufficientlylarge integers n ≥ 1.2. The integralUnless specified otherwise, by a function we always mean a real-valued function.An additive function in a BV set A is a function F defined on BVA such thatF (B ∪ C) = F (B) + F (C)for each pair of disjoint sets B, C ∈ BVA. Such an F is called continuous if givenε > 0, we can find an η > 0 so that |F (B)| < ε for each B ∈ BVA with ‖B‖ < 1/εand |B| < η.A full descriptive definition of the BV-integral 463A partition is a collection P = {(A1, x1), . . . , (Ap, xp)} where A1, . . . , Ap aredisjoint BV sets and xi ∈ cl∗Ai for i = 1, . . . , p. When E ⊂ Rm and⋃pi=1Ai ⊂ Eor {x1, . . . , xp} ⊂ E, we say P is a partition in E or a partition anchored in E,respectively. Clearly, each partition in E is anchored in cl∗E. Given an ε > 0 anda nonnegative function δ on E, the partition P is called(i) ε-regular if r(Ai) > ε for i = 1, . . . , p;(ii) δ-fine if P is anchored in E and d(Ai) < δ(xi) for i = 1, . . . , p.It is convenient to denote⋃pi=1Ai by⋃P .A gage in a set E ⊂ Rm is a nonnegative function δ defined on cl∗E whosenull set Nδ = {x ∈ cl∗E : δ(x) = 0} is thin.Definition 2.1. Let A be a BV set and let f be a function defined on cl∗A.We say f is BV-integrable in A if there is a continuous additive function F in Asatisfying the following condition: given ε > 0, we can find a gage δ in A so thatp∑i=1∣∣∣∣f(xi)|Ai| − F (Ai)∣∣∣∣< εfor each ε-regular δ-fine partition {(A1, x1), . . . , (Ap, xp)} in A.In view of [5, Propositions 7.7 and 7.8], BV-integrability coincides with vari-ational integrability introduced in [5, Definition 5.1]. The function F , uniquelydetermined by f , is called the indefinite BV-integral of f in A. If f is Lebesgueintegrable in A, it is BV-integrable in A, and the two indefinite integrals coincide[5, Proposition 5.8].Let A be a BV set and let f be a function defined on A ∪ cl∗A. Since theBV-integral extends the Lebesgue integral, neither the BV-integrability nor theBV-integral of f is affected by the values f takes on negligible subsets of cl∗A;in particular, they are not affected by the values of f on A △ cl∗A. Thus, in theobvious way, we can and will define the BV-integrability and BV-integral for theextended real-valued functions defined almost everywhere in A.For additional properties of the BV-integral, including the Gauss-Green theo-rem, we refer to [5].3. BV-ACG∗ functionsLet E ⊂ Rm, let F be a function defined on BVE , and let x ∈ cl∗E. SetF (x) = infα>0supδ>0[infF (B)|B|]where the infimum in the brackets is taken over all sets B ∈ BVE with x ∈ cl∗B,d(B) < δ, and r(B) > α; furthermore, letˇF (x) = −(−F )(x). The extendedreal-valued functions x 7→ F (x) and x 7→ F (x) defined on cl∗E are denoted by Fand F , respectively. When F (x) = F (x) is a real number, we denote it by F ′(x)464 B.Bongiorno, L.Di Piazza, W.F. Pfefferand say F is BV-derivable at x. By F ′ we denote the function x 7→ F ′(x) definedon the set of all x ∈ cl∗E at which F is BV-derivable.The following lemma, proved in [4, Lemma 2,3], facilitates applications of Vi-tali’s covering theorem.Lemma 3.1. Let F be an additive continuous function in a BV set A. If x ∈intcA, thenF (x) = infα>0supδ>0[infF (A ∩ C)|A ∩ C|]where the infimum in the brackets is taken over all doubly closed BV sets C withx ∈ C, d(C) < δ, and r(C) > α. In particular, F (x) ≤ F (x) for each x ∈ intcA.Definition 3.2. Let F be an additive continuous function in a BV set A. Wesay F is BV-AC∗ on a set E ⊂ cl∗A if given ε > 0, there is an η > 0 and a gageδ in A such that∣∣∣∣F(⋃P)− F(⋃Q)∣∣∣∣< εfor all ε-regular δ-fine partitions P and Q in A anchored in E for which∣∣∣∣(⋃P)△(⋃Q)∣∣∣∣< η .If cl∗A =⋃∞n=1En and F is BV-AC∗ on each En, we say that F is BV-ACG∗.In a more general setting BV-ACG∗ functions were introduced in [1]. Ourresults below parallel some of those obtained in [3], where a concept closely relatedto the BV-ACG∗ functions has been applied to Perron type integrals.Following [4, Definition 2.5], we say an additive continuous function F in a BVset A is BV-absolutely continuous if given a negligible set N ⊂ cl∗A and an ε > 0,there is a gage δ in A such that |F (⋃P )| < ε for each ε-regular δ-fine partitionP in A anchored in N .Proposition 3.3. Each BV-ACG∗ function in a BV set A is BV-absolutely con-tinuous.Proof: Let F be a BV-ACG∗ function in A. With no loss of generality, we mayassume that there are disjoint sets E1, E2, . . . such that cl∗A =⋃∞n=1En and Fis BV-AC∗ on each En. Choose a negligible set N ⊂ cl∗A and ε > 0, and fix aninteger n ≥ 1. Letting Q = ∅ in Definition 3.2, find a gage δn in A and ηn > 0so that |F (⋃P )| < ε2−n for each ε-regular δn-fine partition P in A anchored inEn with |⋃P | < ηn. There is an open set Un containing N ∩En with |Un| < ηn.Making δn smaller, we may assume that each δn-fine partition P anchored inN ∩ En is a partition in Un; in particular, |⋃P | < ηn. Define a gage δ in A byA full descriptive definition of the BV-integral 465setting δ(x) = δn(x) if x ∈ En for n = 1, 2, . . . . If P = {(A1, x1), . . . , (Ap, xp)} isan ε-regular δ-fine partition in E anchored in N , we obtain∣∣∣∣F(⋃P)∣∣∣∣≤∞∑n=1∣∣∣∣∣∣∑xi∈EnF (Ai)∣∣∣∣∣∣<∞∑n=1ε2−n = ε ,which establishes the BV-absolute continuity of F . The following characterizations are called, respectively, the partial and fulldescriptive definitions of the BV-integral:1. Among all continuous additive functions that are derivable almost every-where in a BV set A, characterize those which are indefinite BV-integralsin A.2. Among all continuous additive functions in a BV set A (derivable or not),characterize those which are indefinite BV-integrals in A.A partial descriptive definition was given in [4, Theorem 2.6] employing theconcept of BV-absolutely continuous functions. Using the stronger concept ofBV-ACG∗ functions we shall present a full descriptive definition in Theorem 3.7below.Lemma 3.4. Let F be a continuous additive function in a BV set A, and letE = {x ∈ cl∗A : F (x) < r < s < F (x)} .If F is BV-AC∗ on E, then E is negligible.Proof: With no loss of generality we may assume E ⊂ intcA. Choose positivenumbers ε and η, and let ε′ = εm+1. If F is BV-AC∗ on E, we can find a positivenumber η′ ≤ η and a gage δ in A so that∣∣∣∣F(⋃P)− F(⋃Q)∣∣∣∣< εfor all ε′-regular δ-fine partitions P and Q in A anchored in E for which∣∣∣∣(⋃P)△(⋃Q)∣∣∣∣< 4η′ .Select an open set U containing E with |U | < |E| + η′, and let R and S bethe families of all doubly closed sets C ⊂ U such that d(C) < δ(xC ) for anxC ∈ E ∩ C, r(C) > ε, and respectively,F (A ∩ C) < r|A ∩ C| and F (A ∩ C) > s|A ∩ C| .In view of Lemma 1.1, making δ smaller, we may assume r(A∩C) > ε′ for each C ∈R∪S. Clearly, R and S are Vitali covers of E−Nδ . Since Nδ is a negligible set (in466 B.Bongiorno, L.Di Piazza, W.F. Pfefferfact, a thin set), applying Vitali’s covering theorem [7, Chapter IV, Theorem 3.1],we obtain collections {A1, . . . , Ap} ⊂ R and {B1, . . . , Bq} ⊂ S, each consisting ofdisjoint sets and such thatmin∣∣∣∣p⋃i=1(E ∩ Ai)∣∣∣∣,∣∣∣∣q⋃j=1(E ∩ Bj)∣∣∣∣> |E| − η′ .From the definitions of R and S, we see thatP = {(A ∩ A1, xA1), . . . , (A ∩ Ap, xAp)} ,Q = {(A ∩ B1, xB1), . . . , (A ∩ Bq, xBq )}are ε′-regular δ-fine partitions in A anchored in E. Since⋃P ⊂ U and |E∩⋃P | >|E| − η′, we have |E △⋃P | < 2η′; by symmetry, also |E △⋃Q| < 2η′. Thus|(⋃P )△ (⋃Q)| < 4η′, and we obtainε >∣∣∣∣F(⋃P)− F(⋃Q)∣∣∣∣≥q∑j=1F (A ∩ Bj)−p∑i=1F (A ∩ Ai)> s∣∣∣∣⋃Q∣∣∣∣− r∣∣∣∣⋃P∣∣∣∣> s(|E| − η′)− r(|E| + η′)= (s − r)|E| − (s+ r)η′ ≥ (s − r)|E| − |s+ r|η .The negligibility of E follows from the arbitrariness of ε and η. Lemma 3.5. Let F be a continuous additive function in a BV set A, and letE = {x ∈ cl∗A : F (x) = +∞} .If F is BV-AC∗ on E, then E is negligible.Proof: We may assume E ⊂ intcA. Proceeding towards a contradiction, suppose|E| > 0. If 0 < ε < 1/(2m) and F is BV-AC∗ on E, we can find a gage δ in Aand a positive η < |E|/2 so that∣∣∣∣F(⋃P)− F(⋃Q)∣∣∣∣< 1for all ε-regular δ-fine partitions P and Q in A anchored in E for which∣∣∣∣(⋃P)△(⋃Q)∣∣∣∣< 4η .A full descriptive definition of the BV-integral 467Let U be an open set containing E with |U | < |E| + η. Using [5, Lemma 3.4]and Vitali’s covering theorem, find disjoint closed cubes C1, . . . , Cp contained inU such that |E −⋃pi=1Ci| < η while r(A ∩ Ci) > ε and d(Ci) < δ(xi) for anxi ∈ E ∩ Ci and i = 1, . . . , p. Thus P = {(A ∩ C1, x1), . . . , (A ∩ Cp, xp)} is anε-regular δ-fine partition in A anchored in E, and |E △ (⋃P )| < 2η.In view of [5, Lemma 3.4] and Lemma 3.1, the family B of all closed cubesB ⊂ U such that r(A ∩ B) > ε, d(B) < δ(yB) for a yB ∈ E ∩ B, andF (A ∩ B)|A ∩ B|>2|E|∣∣∣∣F(⋃P)+ 1∣∣∣∣,is a Vitali cover of E. Using Vitali’s covering theorem again, find a disjointcollection {B1, . . . , Bq} ⊂ B so that |E −⋃qj=1Bj | < η. Then Q = {(A ∩B1, yB1), . . . , (A ∩ Bq, yBq )} is an ε-regular δ-fine partition in A anchored in E,and |E △⋃Q| < 2η. Observing that∑qj=1 |A ∩ Bj | ≥ |E|/2, we obtainF(⋃Q)=q∑j=1F (A ∩ Bj)>2|E|∣∣∣∣F(⋃P)+ 1∣∣∣∣·q∑j=1|A ∩ Bj |≥ F(⋃P)+ 1 .This is a contradiction, since |(⋃P )△ (⋃Q)| < 4η. Proposition 3.6. A continuous additive function F in a BV set A that isBV-ACG∗ is derivable almost everywhere in cl∗A.Proof: Let cl∗A =⋃∞n=1En, and let F be BV-AC∗ on each En. The set of allx ∈ cl∗A at which F is not derivable is the union of the setsEn,+∞ = {x ∈ En : F (x) = +∞} , En,−∞ = {x ∈ En : F (x) = −∞} ,andEn,r,s = {x ∈ En : F (x) < r < s < F (x)}where r and s are rational numbers. Lemma 3.5 applied to F and −F shows thatthe sets En,±∞ are negligible, and the sets En,r,s are negligible by Lemma 3.4.Since we have only countably many of these sets, the proposition follows.Corollary 3.7. Let F be a continuous additive function in a BV set A. If F isBV-ACG∗, then F′ is BV-integrable in A and F is its indefinite BV-integral.Proof: According to Proposition 3.6, the derivate F ′ is defined almost every-where in cl∗A, and by Proposition 3.3, the function F is BV-absolutely continuous.An application of [4, Theorem 2.6] completes the proof. 468 B.Bongiorno, L.Di Piazza, W.F. PfefferProposition 3.8. Let f be a BV-integrable function in a BV set A. If F is theindefinite BV-integral of f in A, then F is BV-ACG∗.Proof: We may assume that f is a real-valued function defined on the wholeof cl∗A, and let En = {x ∈ cl∗A : |f(x)| ≤ n} for n = 1, 2, . . . . Since cl∗A =⋃∞n=1En, it suffices to show that F is BV-AC∗ on each En. To this end, fixa positive integer n and let E = En. It follows from [5, Corollary 5.12] that Eis measurable and f is Lebesgue integrable in E. Hence, if χ is the characteristicfunction of E restricted to cl∗A, then fχ is Lebesgue integrable in A. By [5,Proposition 5.8], the function fχ is also BV-integrable in A, and we denote by Gits indefinite BV-integral in A.Choose an ε > 0. Using the absolute continuity of the indefinite Lebesgueintegral, find an η > 0 so that |G(Z)| < ε for each BV set Z ⊂ A with |Z| < η.There is a gage δ in A such thatr∑i=1∣∣∣∣f(zi)|Ci| − F (Ci)∣∣∣∣< ε andr∑i=1∣∣∣∣f(zi)χ(zi)|Ci| − G(Ci)∣∣∣∣< εfor each ε-regular δ-fine partition R = {(C1, z1), . . . , (Cr, zr)} in A. If such a par-tition R is anchored in E, then χ(zi) = 1 for i = 1, . . . , r, and we have∣∣∣∣F(⋃R)− G(⋃R)∣∣∣∣≤r∑i=1|F (Ci)− G(Ci)| < 2ε .Now choose ε-regular δ-fine partitions P = {(A1, x1), . . . , (Ap, xp)} and Q ={(B1, y1), . . . , (Bq , yq)} in A anchored in E for which |(⋃P )△(⋃Q)| < η. LettingX =⋃P and Y =⋃Q, observe that|G(X)− G(Y )| = |G(X − Y )− G(Y − X)|≤ |G(X − Y )|+ |G(Y − X)| < 2ε ;for max{|X − Y |, |Y − X |} ≤ |X △ Y | < η. Thus|F (X)− F (Y )| ≤ |F (X)− G(X)|+ |F (Y )− G(Y )|+ |G(X)− G(Y )| < 6ε ,which establishes F is BV-AC∗ on E. Combining Corollary 3.7 and Proposition 3.8, we obtain the following full de-scriptive definition of the BV-integral.Theorem 3.9. A continuous additive function F in a BV set A is BV-ACG∗ ifand only if F ′ exists almost everywhere in A and F is its indefinite BV-integral.A full descriptive definition of the BV-integral 469References[1] Bongiorno B., Differentiation of set functions, Rend. Circ. Mat. Palermo 26 (1977), 37–51.[2] Evans L.C., Gariepy R.F., Measure Theory and Fine Properties of Functions, CRC Press,Boca Raton, 1992.[3] Kurzweil J., Jarńık J., Equi-integrability and controlled convergence of Perron-type inte-grable functions, Real Anal. Ex. 17 (1991:92), 110–139.[4] Pfeffer W.F., A descriptive definition of a variational integral and applications, IndianaUniv. Math. J. 40 (1991), 259–270.[5] , The Gauss-Green theorem, Adv. Math. 87 (1991), 93–147.[6] , The Riemann Approach to Integration, Cambridge Univ. Press, Cambridge, 1993.[7] Saks S., Theory of the Integral, Dover, New York, 1964.[8] Volpert A.I., The spaces BV and quasilinear equations, Math. USSR-SB. 2 (1967), 255–267.(B. Bongiorno, L. Di Piazza)Department of Mathematics, University of Palermo, via Archirafi 34,90123 Palermo, Italy(W.F. Pfeffer)Department of Mathematics, University of California, Davis, CA 95616, USAandCenter for Theoretical Study, Charles University, Ovocný trh 3,116 36 Prague 1, Czech Republic(Received September 12, 1994)

A full descriptive definition of the BV-integral

http://dml.cz/bitstream/handle/10338.dmlcz/118774/CommentatMathUnivCarolRetro_36-1995-3_7.pdf

A full descriptive definition of the BV-integral

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.