CHARGES, BV FUNCTIONS, AND MULTIPLIERS

FOR GENERALIZED RIEMANN INTEGRALS

Zoltán Buczolich^{∗}

Department of Analysis, Eötvös Loránd University,

Múzeum krt. 6–8, H–1088 Budapest, Hungary buczo@ludens.elte.hu

Thierry De Pauw

Department of Mathematics, University College London,

Gower Street, London WC1E 6BT, UK thierry@math.ucl.ac.uk

Washek F. Pfeffer

Department of Mathematics, University of California, Davis, CA 95616, USA wfpfeffer@ucdavis.edu

Abstract

For an invariant generalized Riemann integral in ℝ^{m}, we obtain the following results. (1) A function is a multiplier for the space of locally integrable functions if and only if it is locally bounded and locally BV.

(2) The dual of the space of all functions integrable in a bounded BV set A is linearly isomorphic to the space of all bounded BV functions vanishing outside A, and each element of the dual has the usual integral representation. (3) On Lipschitz domains an integration by parts formula holds for any continuous function that is pointwise Lipschitz everywhere except on a set of σ-finite (m − 1)-dimensional Hausdorff measure.

∗This author was supported by the Hungarian National Foundation of Scientific Research, Grant No. T019476 and FKFP 0189/1997.

1 Introduction

In dimension one, the problem of recovering derivative by integration has been solved a long time ago by Denjoy and Perron [20, Chapters 6 and 8]. In higher dimensions, however, a substantial progress was made only in the eighties. Following the ideas of Henstock, Kurzweil, and McShane, several authors suggested various Riemann type extensions of the Lebesgue integral which provide the Gauss-Green theorem for any differentiable (not necessarily continuously) vector field. Among these generalized Riemann integrals, the most viable appear those introduced in [15, Chapters 12 and 13]: they are defined over bounded BV sets, are invariant with respect to lipeomorphic changes of coordinates, and integrate functions with large, topologically unrestricted, sets of singularities. They have a common extension, referred to as the R_{∗}-integral in the present paper, and as the continuous integral in [13, Section 9].

Our main result is a complete solution of the multiplier problem for the R_{∗}-integral. Specifically, we show the family of all functions, called multipliers, whose product with every locally R_{∗}-integrable function is again locally R_{∗}- integrable is the algebra of all locally bounded locally BV functions. Using this, we obtain an integral representation of continuous linear functionals on the space of all functions that are R_{∗}-integrable in a bounded BV set. In the remainder of this section, we outline briefly the main points of our work for a fixed bounded BV set A ⊂ ℝ^{m}.

In the linear space BV^{∞}(A) of all bounded BV functions vanishing outside A, we consider a sequentially complete locally convex topology τ induced by the following convergence: f_{i} → 0 if and only if {f_{i}} converges to zero in L^{1}(A), and the numerical sequences formed by the L^{∞} norms and the total variations of the functions f_{i} are bounded. The family BV(A) of all BV subsets of A is topologized as a subspace of (BV^{∞}(A), τ).

A charge is an additive τ-continuous function defined on BV(A). The linear space of all charges, topologized by the uniform convergence on τ-compact subfamilies of BV(A), is a Banach space denoted by CH(A) (Proposition 3.4). Following [12] and [16], with each charge F, we associate a function F × L^{1} defined by the formula

(F × L^{1})(B) = ∫_{ℝ} F (B^{t}) dt ,

where B is a bounded BV subset of A × ℝ and B^{t} = {x ∈ A : (x, t) ∈ B}. Observing the subgraph Σ_{g} = {(x, t) ∈ A × ℝ : 0 < t < g(x)} of a nonnegative function g ∈ BV^{∞}(A) is a bounded BV subset of A × ℝ, we obtain a nondegenerate, bilinear, and separately continuous pairing

⟨F, g⟩ = (F × L^{1})(Σ_{g^{+}}) − (F × L^{1})(Σ_{g^{−}}) ,

which puts the Banach space CH(A) and the space (BV^{∞}(A), τ) in duality. The map Θ that sends a charge F to the functional

Θ(F) : BV^{∞}(A) → ℝ : g ↦ ⟨F, g⟩

is a linear homeomorphism from CH(A) onto the dual of (BV^{∞}(A), τ) equipped with the topology of uniform convergence on τ-compact subfamilies of BV^{∞}(A) (Theorem 4.3 and Remark 4.4).

It is well known each absolutely continuous (abbreviated as AC) charge F is derivable almost everywhere, and F is the indefinite Lebesgue integral of the derivate F′. Given a charge F, there is a Borel regular measure V_{∗}F, which is absolutely continuous and finite if and only if F is AC [2, Proposition 5.11]. If V_{∗}F is merely absolutely continuous (not necessarily finite), we say F is an AC_{∗} charge. It was proved in [3, Corollaries 4.3 and 4.7] each AC_{∗} charge F is derivable almost everywhere, and F is uniquely determined by the derivate F′. Thus, in analogy with the Lebesgue integral, we say a function f defined on A is R-integrable whenever there is an AC_{∗} charge F such that F′ = f almost everywhere. If A is a compact interval, this definition produces the generalized Riemann integral of [15, Definition 12.2.1]. We denote by R(A) the linear space of all R-integrable functions. Since the R-integral extends the Lebesgue integral, writing ∫_{A} f dL^{m} for f ∈ R(A) leads to no confusion.

If F is an AC_{∗} charge, g ∈ BV^{∞}(A), and χ_{B} is the indicator function of B ∈ BV(A), we let

G(B) = ⟨F, gχ_{B}⟩ ,

and show G is an AC_{∗} charge with G′ = F′g almost everywhere. Rewriting this in terms of the R-integral solves the difficult part of the multiplier problem (Theorem 6.4): if f ∈ R(A) and g ∈ BV^{∞}(A), then fg ∈ R(A) and

∫_{A} fg dL^{m} = ⟨F, g⟩ .

Denote by CH_{AC}(A) and CH_{AC_{∗}}(A) the linear spaces of all AC and AC_{∗} charges, respectively, and topologize R(A) so that the linear bijection

F ↦ F′ : CH_{AC_{∗}}(A) → R(A)

becomes a homeomorphism. We show CH_{AC}(A) is dense in CH(A) (Corollary 4.9 — a different proof presented in [4, Corollary 4.1] is incomplete) and infer (BV^{∞}(A), τ) is dual to each linear space X ⊂ CH(A) containing CH_{AC}(A). In particular, every g ∈ BV^{∞}(A) defines a continuous linear functional

T_{g} : R(A) → ℝ : f ↦ ∫_{A} fg dL^{m} ,

and the map g ↦ T_{g} is a continuous linear bijection from (BV^{∞}(A), τ) onto the dual of R(A) equipped with the w^{∗}-topology (Proposition 7.6).

Since the R_{∗}-integral is obtained by a transfinite iteration of improper Rintegrals (Definition 8.1), the extension of the previous results to the R_{∗}-integral is easy. We prove a very general Gauss-Green theorem (Theorem 8.6) and, in case A is a Lipschitz domain, we establish the integration by parts formula (Theorem 7.17) for continuous vector fields that are pointwise Lipschitz except on a set of σ-finite (m − 1)-dimensional Hausdorff measure.

2 Preliminaries

The set of all real numbers is denoted by ℝ, and the ambient space of this paper is ℝ^{m} where m ≥ 1 is a fixed integer. In ℝ^{m} we use exclusively the metric induced by the maximum norm | · |. Specifically, the diameter of a set E ⊂ ℝ^{m} is defined as

d(E) = sup{|x − y| : x, y ∈ E} .

For r > 0, we let B(r) = (−r, r)^{m} and B[r] = [−r, r]^{m}, and if x ∈ ℝ^{m}, then

B(x, r) = {y ∈ ℝ^{m} : |x − y| < r} and B[x, r] = {y ∈ ℝ^{m} : |x − y| ≤ r} .

By x · y we denote the usual inner product of x, y ∈ ℝ^{m}, and let ∥x∥ = \sqrt{x · x}.

A cell is a compact nondegenerate subinterval of ℝ^{m}, and a figure is a finite (possibly empty) union of cells. The family of all figures is denoted by F. The closure, interior, and boundary of a set E ⊂ ℝ^{m} are denoted by cl E, int E, and ∂E, respectively. If A and B are sets, then A △ B denotes their symmetric difference.

When no attributes are added, a function is always real-valued. If f is a function defined on a set E ⊂ ℝ^{m}, the sets

null f = {x ∈ E : f(x) = 0} and supp f = cl {x ∈ E : f(x) ≠ 0}

are called, respectively, the null set and support of f. By χ_{E} we denote the indicator function of a set E ⊂ ℝ^{m}.

The Lebesgue measure in ℝ^{m} is denoted by L^{m}; however, for E ⊂ ℝ^{m}, we write |E| instead of L^{m}(E). A set E ⊂ ℝ^{m} with |E| = 0 is called negligible. Functions f and g defined on a set E ⊂ ℝ^{m} are called equivalent if they differ only on a negligible subset of E. Subsets A and B of ℝ^{m} are called equivalent whenever their indicator functions are equivalent, i.e., whenever |A △ B| = 0. As usual, we shall tacitly identify a set E ⊂ ℝ^{m}, or a function f defined on E, with the equivalence class determined by E, or f, respectively. Unless specified otherwise, the words “measure” and “measurable” as well as the expressions “almost all,” “almost everywhere,” and “absolutely continuous” always refer to the Lebesgue measure L^{m}.

For a measurable set E ⊂ ℝ^{m} and 0 ≤ α ≤ 1, let

E(α) = \{x ∈ ℝ^{m} : lim_{r→0+} \frac{|E ∩ B[x, r]|}{|B[x, r]|} = α\} .

The sets int_{∗}E = E(1), cl_{∗}E = ℝ^{m} − E(0), and ∂_{∗}E = ℝ^{m} − [E(0) ∪ E(1)] are called the essential interior, essential closure, and essential boundary of E, respectively. We say E is essentially closed whenever cl_{∗}E is closed.

If s ≥ 0 and k > s is an integer, we denote by H^{s} the s-dimensional Hausdorff measure in ℝ^{k}; since k is always determined by the context, this notation leads to no confusion. A set S ⊂ ℝ^{m} of σ-finite measure H^{m−1} is called thin. Clearly, each thin set is negligible but not vice versa.

In our exposition we shall use extensively the functions of bounded variation, called BV functions, and the sets of finite perimeter, called BV sets. For their definitions and properties we refer to [6, Chapter 5].

The linear space of all bounded BV functions in ℝ^{m} with compact support is denoted by BV_{c}^{∞}, and the linear space of all locally bounded locally BV functions in ℝ^{m} is denoted by BV_{loc}^{∞}, i.e.,

BV_{c}^{∞} = BV (ℝ^{m}) ∩ L_{c}^{∞}(ℝ^{m}) and BV_{loc}^{∞} = BV_{loc}(ℝ^{m}) ∩ L_{loc}^{∞}(ℝ^{m}) .

If f ∈ BV (ℝ^{m}), we denote by Df and ∥Df∥ the distributional gradient and the variational measure of f, respectively; the number ∥f∥ = ∥Df∥(ℝ^{m}) is called the total variation of f. For n = 1, 2, …, topologize

BV_{n} = {f ∈ BV_{c}^{∞} : supp f ⊂ B[n] and ∥f∥ + ∥f∥_{∞} ≤ n + 1}

by the L^{1}(ℝ^{m}) norm ∥ · ∥_{1}. According to [6, Section 5.2, Theorems 1 and 4], each BV_{n} is a compact set. The largest topology τ in BV_{c}^{∞} for which all inclusion maps BV_{n} ↪ BV_{c}^{∞} are continuous is a nonmetrizable locally convex Hausdorff topology that is sequential [5, Section 1.7] and sequentially complete. A sequence {f_{i}} in (BV_{c}^{∞}, τ) converges to zero if and only if lim ∥f_{i}∥_{1} = 0, sup(∥f_{i}∥ + ∥f_{i}∥_{∞}) < ∞, and supp f_{i} ⊂ K for a compact set K ⊂ ℝ^{m} and i = 1, 2, . . ..

The family of all bounded BV subsets of ℝ^{m} is denoted by BV. If E ∈ BV, we write ∥∂E∥ instead of ∥Dχ_{E}∥, and call ∥E∥ = ∥∂E∥(ℝ^{m}) the perimeter of E. The regularity of a set E ∈ BV is the number

r(E) = \begin{cases}\frac{|E|}{d(E)∥E∥} & if d(E)∥E∥ > 0, \\ 0 & otherwise, \end{cases}

which refines the usual concept of regularity employed in connection with Vitali’s covering theorem [20, Chapter 4, Section 2].

According to [6, Section 5.11, Theorem 1], a bounded measurable set E is a BV set if and only if H^{m−1}(∂_{∗}E) < ∞, in which case ∥E∥ = H^{m−1}(∂_{∗}E). If E is a BV set, then ∂_{∗}E is H^{m−1}-measurable, and there is an H^{m−1}-measurable unit vector field ν_{E} defined H^{m−1}-almost everywhere on ∂_{∗}E such that the Gauss-Green formula

∫_{E} div v dL^{m} = ∫_{∂_{∗}E} v · ν_{E} dH^{m−1}

holds for each v ∈ C_{c}^{1}(ℝ^{m}; ℝ^{m}) [6, Section 5.8, Theorem 1].

Identifying each set E ∈ BV with its indicator function χ_{E}, the family BV is a closed subspace of (BV_{c}^{∞}, τ). It follows the topology τ restricted to BV, still denoted by τ, is a sequential topology induced by a sequentially complete uniformity. For n = 1, 2, …, the topology τ restricted to

BV_{n} = BV ∩ BV_{n} = {A ∈ BV : A ⊂ B[n] and ∥A∥ ≤ n}

is compact and induced by the metric (B, C) ↦ |B △ C|. A sequence {A_{i}} in (BV, τ) converges to a set A ∈ BV if and only if lim |A_{i}△A| = 0, sup ∥A_{i}∥ < ∞, and A_{i} ⊂ K for a compact set K ⊂ ℝ^{m} and i = 1, 2, . . .. For a set A ∈ BV, let

BV^{∞}(A) = {f ∈ BV_{c}^{∞} : ℝ^{m} − A ⊂ null f} ,
BV(A) = BV ∩ BV^{∞}(A) = {B ∈ BV : B ⊂ A} .

3 Charges

A charge is an additive τ-continuous function defined on BV. Explicitly, a function F : BV → ℝ is a charge if it satisfies the following two conditions.

Additivity: F (A ∪ B) = F (A) + F (B) for each pair of disjoint bounded BV sets A and B.

Continuity: Given ε > 0, there is an η > 0 such that |F (C)| < ε for each BV set C ⊂ B(1/ε) with ∥C∥ < 1/ε and |C| < η.

Observe an additive function F : BV → ℝ is a charge whenever lim F (A_{i}) = 0 for each sequence {A_{i}} in BV that τ-converges to the empty set. By continuity, a charge vanishes on all bounded negligible sets; in particular, the value of a charge on a set E ∈ BV depends only on the equivalence class of E. If F is a charge and A is a locally BV set, define a charge F A by letting

(F A)(B) = F (A ∩ B)

for each B ∈ BV. We say F is a charge in A whenever F = F A. The linear spaces of all charges and of all charges in a locally BV set A are denoted by CH and CH(A), respectively. The following are canonical examples of charges.

(1) If f ∈ L_{loc}^{1}(ℝ^{m}) and F (A) = ∫_{A} f dL^{m} for each bounded BV set A, then F is a charge by the absolute continuity of the Lebesgue integral. It is called the indefinite Lebesgue integral of f, denoted by ∫ f dL^{m}.

(2) If v ∈ C(ℝ^{m}; ℝ^{m}) and F (A) = ∫_{∂_{∗}A} v · ν_{A} dH^{m−1} for each bounded BV set A, then F is a charge by [15, Proposition 11.2.8]. It is called the flux of v, denoted by ∫ v · ν dH^{m−1}.

According to DeGiorgi’s approximation theorem [2, Proposition 1.1], the family F of all figures is sequentially dense in (BV, τ), and hence (BV, τ) is the sequential completion of (F, τ). As each additive τ-continuous function on F is uniformly continuous, it has a unique extension to a charge. The next proposition, proved in [12, Proposition 1], provides a useful characterization of charges.

Proposition 3.1. An additive function F on BV is a charge if and only if given ε > 0, there is a θ > 0 such that

|F (B)| < θ|B| + ε(∥B∥ + 1)

for each BV set B ⊂ B(1/ε).

For a charge F and n = 1, 2, …, let

||||||F ||||||_{n} = sup{|F (B)| : B ∈ BV_{n}} .

As BV is the union of an increasing sequence {BV_{n}} of compact sets, {|||||| · ||||||_{n}} is an increasing sequence of seminorms in CH which defines a metrizable locally convex topology σ in CH. Since σ is the topology of uniform convergence on the families BV_{n}, it is easy to verify (CH, σ) is a Fréchet space (cf. the natural topology of [4, Definition 4.2]).

Lemma 3.2. Each C ∈ BV is the union of disjoint BV sets C_{1}, …, C_{p} such that p ≤ max{1, 2∥C∥} and ∥C_{i}∥ < 1 + 4d(C)^{m−1} for i = 1, …, p.

Proof. We employ an idea used in the proof of [12, Lemma 1]. Avoiding a triviality, assume γ = 2d(C)^{m−1} > 0, and for every t ∈ ℝ, let

C_{t}^{−} = C ∩ ((−∞, t) × ℝ^{m−1}) and C_{t}^{+} = C ∩ ([t, +∞) × ℝ^{m−1}) .

Then C is the disjoint union of the sets C_{t}^{±}, and

∥C_{t}^{−}∥ + ∥C_{t}^{+}∥ ≤ ∥C∥ + γ .

The function t ↦ ∥C_{t}^{−}∥ increases from 0 to ∥C∥, and

lim_{r→t+} ∥C_{r}^{−}∥ − lim_{r→t−} ∥C_{r}^{−}∥ ≤ \frac{γ}{2} .

Thus there is an s ∈ ℝ with

\frac{1}{2}∥C∥ ≤ ∥C_{s}^{−}∥ ≤ \frac{1}{2}∥C∥ + γ ,

which implies ∥C_{s}^{±}∥ ≤ \frac{1}{2}∥C∥ + γ. Now let n ≥ 0 be the least integer with ∥C∥ ≤ 2^{n}, and observe 2^{n} ≤ max{1, 2∥C∥}. Proceeding inductively, construct disjoint BV sets C_{1}, …, C_{2^{n}} so that C = ⋃_{k=1}^{2^{n}} C_{k} and

∥C_{i}∥ ≤ 2^{−n}∥C∥ + γ ∑_{k=0}^{n−1} 2^{−k} < 1 + 2γ

for i = 1, …, 2^{n}.

Corollary 3.3. Let C ∈ BV(B[k]) for an integer k ≥ 1 + 4d(C)^{m−1}, and assume ∥C∥ ≥ k. If F is a charge, then |F (C)| ≤ 2∥C∥ · ||||||F ||||||_{k}.

Proof. By Lemma 3.2, the set C is the union of disjoint BV sets C_{1}, …, C_{p} such that p ≤ 2∥C∥ and ∥C_{i}∥ < k for i = 1, …, p. Thus each C_{i} belongs to BV_{k}, and we have

|F (C)| ≤ ∑_{i=1}^{p}|F (C_{i})| ≤ 2∥C∥ · ||||||F ||||||_{k} .

Proposition 3.4. Let A ∈ BV(B[k]) for an integer k ≥ 1 + 4d(A)^{m−1}. If n ≥ k is an integer and F is a charge in A, then

||||||F ||||||_{n} ≤ 2(∥A∥ + n)||||||F ||||||_{k} .

The seminorm |||||| · ||||||_{k} is a norm in CH(A) that induces the subspace topology in CH(A) ⊂ (CH, σ). In particular, (CH(A), |||||| · ||||||_{k}) is a Banach space.

Proof. Given a set C ∈ BV_{n}, we distinguish two cases. Either ∥A ∩ C∥ ≤ k which yields |F (C)| = |F (A ∩ C)| ≤ ||||||F ||||||_{k}, or ∥A ∩ C∥ > k and then

|F (C)| = |F (A ∩ C)| ≤ 2∥A ∩ C∥ · ||||||F ||||||_{k} ≤ 2(∥A∥ + n)||||||F ||||||_{k}

by Corollary 3.3. In either case ||||||F ||||||_{n} ≤ 2(∥A∥ + n)||||||F ||||||_{k}, and we conclude |||||| · ||||||_{k} and |||||| · ||||||_{n} are equivalent norms in CH(A).

If E is a bounded BV subset of ℝ^{m+1}, it follows from [10, Theorems 20 and 33] the set

E^{t} = {x ∈ ℝ^{m} : (x, t) ∈ E}

belongs to BV for L^{1}-almost all t ∈ ℝ.

Proposition 3.5. Let F be a charge, and let E be a bounded BV subset of ℝ^{m+1}. Then the function t ↦ F (E^{t}), defined for L^{1}-almost all t ∈ ℝ, belongs to L^{1}(ℝ). Letting

(F × L^{1})(E) = ∫_{ℝ} F (E^{t}) dt

for each bounded BV set E ⊂ ℝ^{m+1} defines a charge F × L^{1} in ℝ^{m+1}. The map F ↦ F × L^{1} is a continuous linear map from (CH(ℝ^{m}), σ) to (CH(ℝ^{m+1}), σ).

Proof. If A ⊂ ℝ^{m+1} is an (m + 1)-dimensional figure, then

F̂(A) = ∫_{ℝ} F (A^{t}) dt

is well defined, since t ↦ F (A^{t}) is an L^{1}-measurable simple function vanishing outside a compact set. Choose an ε > 0 and a < b. By Proposition 3.1, there is a θ > 0 such that

|F (B)| < θ|B| + ε(∥B∥ + 1)

for each BV set B ⊂ [a, b]^{m}. Thus if A is an (m + 1)-dimensional subfigure of [a, b]^{m+1}, then

\begin{aligned}|F̂(A)| & ≤ ∫_{a}^{b}|F (A^{t})| dt ≤ θ ∫_{a}^{b} |A^{t}| dt + ε ∫_{a}^{b}(∥A^{t}∥ + 1) dt \\ & ≤ θL^{m+1}(A) + ε[H^{m}(∂A) + (b − a)] . \end{aligned}  (3.1)

We infer F̂ has a unique extension to a charge in ℝ^{m+1}, denoted by F × L^{1}.

Select a BV set E in ℝ^{m+1} with cl E ⊂ (a, b)^{m+1}. By DeGiorgi’s approximation theorem [2, Proposition 1.1], there is a sequence {A_{i}} of (m+1)-dimensional figures contained in [a, b]^{m+1} such that

lim L^{m+1}(A_{i} △ E) = 0 and sup H^{m}(∂A_{i}) < ∞ .  (3.2)

If f_{i} : t ↦ F (A_{i}^{t}), then

|f_{i}(t) − f_{j}(t)| ≤ |F (A_{i}^{t} − A_{j}^{t})| + |F (A_{j}^{t} − A_{i}^{t})|

for all t ∈ ℝ, and inequality (3.1) implies

\begin{matrix}∫_{ℝ} |f_{i}(t) & − f_{j}(t)| dt = ∫_{a}^{b} |f_{i}(t) − f_{j}(t)| dt ≤ \\ θL^{m+1}(A_{i} △ A_{j}) + 2ε[H^{m+1}(∂A_{i}) + H^{m+1}(∂A_{j}) + (b − a)] . \end{matrix} (3.3)

As ε is arbitrary, relations (3.2) and (3.3) imply {f_{i}} is a Cauchy sequence in L^{1}(ℝ), and hence it converges in L^{1}(ℝ) to an f ∈ L^{1}(ℝ). It follows {f_{i}} has a subsequence, still denoted by {f_{i}}, such that lim f_{i}(t) = f(t) for all t ∈ ℝ − N where L^{1}(N) = 0. In view of the Fubini and dominated convergence theorems, we may assume lim |A_{i}^{t} △ E^{t}| = 0 for each t ∈ ℝ − N. Fatou’s lemma yields

∫_{ℝ} lim inf ∥A_{i}^{t}∥ dt ≤ lim inf ∫_{ℝ} ∥A_{i}^{t}∥ dt ≤ sup H^{m+1}(∂A_{i}) < ∞ ,

so we may further assume lim inf ∥A_{i}^{t}∥ < ∞ for every t ∈ ℝ − N. Select a t ∈ ℝ − N, and find a subsequence {A_{i_{k}}} of {A_{i}} with sup ∥A_{i_{k}}^{t}∥ < ∞. Since F is a charge,

f(t) = lim f_{i_{k}}(t) = lim F (A_{i_{k}}^{t}) = F (E^{t}) ,

and we conclude f(t) = F (E^{t}) for all t ∈ ℝ − N. As {f_{i}} converges to f in L^{1}(ℝ), we obtain

(F × L^{1})(E) = lim F̂(A_{i}) = lim ∫_{ℝ} F (A_{i}^{t}) dt = ∫_{ℝ} F (E^{t}) dt .

Now given an integer n ≥ 1, let k = max{n, 1 + 4(2n)^{m−1}}. If E ⊂ [−n, n]^{m+1} is a BV set in ℝ^{m+1} and ∥E∥ ≤ n, let

J_{−} = {t ∈ [−n, n] : ∥E^{t}∥ ≤ k} and J_{+} = {t ∈ [−n, n] : ∥E^{t}∥ > k} .

The first part of the proof and Corollary 3.3 imply

|(F × L^{1})(E)| ≤ ∫_{J_{−}} |F (E^{t})| dt + ∫_{J_{+}} |F (E^{t})| dt
≤ L^{1}(J_{−})||||||F ||||||_{k} + 2||||||F ||||||_{k} ∫_{J_{+}} ∥E^{t}∥ dt
≤ (L^{1}(J_{−}) + 2∥E∥)||||||F ||||||_{k} ≤ 4n||||||F ||||||_{k} ,

and hence ||||||F × L^{1}||||||_{n} ≤ 4n||||||F ||||||_{k}. As the map F ↦ F × L^{1} is clearly linear, it is also continuous.

4 Duality

Denote by CH^{∗} and (BV_{c}^{∞})^{∗} the dual spaces of (CH, σ) and (BV_{c}^{∞}, τ), respectively. We endow (BV_{c}^{∞})^{∗} with a Fréchet topology ν induced by seminorms

||||||T ||||||_{n} = sup{|T (f)| : f ∈ BV_{n}}

where T ∈ (BV_{c}^{∞})^{∗} and n = 1, 2 … (cf. the topology σ in CH defined in Section 3).

The subgraph of a nonnegative function g defined on ℝ^{m} is the set

Σ_{g} = {(x, t) ∈ ℝ^{m} × ℝ : 0 < t < g(x)} .

It follows from the coarea theorem [6, Section 5.5, Theorem 1] that the subgraph Σ_{g} of a nonnegative function g ∈ BV_{c}^{∞} is a bounded BV subset of ℝ^{m+1}. In view of Proposition 3.5, the number

⟨F, g⟩ = (F × L^{1})(Σ_{g^{+}}) − (F × L^{1})(Σ_{g^{−}})

is defined for each F ∈ CH and each g ∈ BV_{c}^{∞}, and we introduce the following functionals:

Θ(F) : BV_{c}^{∞} → ℝ : g ↦ ⟨F, g⟩ ,
Φ(g) : CH → ℝ : F ↦ ⟨F, g⟩ .

Proposition 4.1. The pairing (F, g) ↦ ⟨F, g⟩ is a nondegenerate separately continuous bilinear functional on (CH × BV_{c}^{∞}, σ × τ). In particular, for each F ∈ CH and each g ∈ BV_{c}^{∞}, the functionals Θ(F) and Φ(g) belong to (BV_{c}^{∞})^{∗} and CH^{∗}, respectively.

Proof. If ⟨F, g⟩ = 0 for all g ∈ BV_{c}^{∞}, then F = 0. On the other hand, if g is a nonzero element of BV_{c}^{∞}, we may assume by symmetry that L^{m+1}(Σ_{g^{+}}) > 0. Let E_{+} = {x ∈ ℝ^{m} : g(x) > 0} and observe F = L^{m} E_{+} is a charge with ⟨F, g⟩ = L^{m+1}(Σ_{g^{+}}). It follows the pairing (F, g) ↦ ⟨F, g⟩ is nondegenerate. According to Proposition 3.5, the linear functional Φ(g) belongs to CH^{∗}.

Let {g_{i}} be a sequence in BV_{c}^{∞} that τ-converges to a g ∈ BV_{c}^{∞}(ℝ^{m}). Then

lim L^{m+1}(Σ_{g_{i}^{±}} △ Σ_{g^{±}}) ≤ lim ∥g_{i} − g∥_{1} = 0 .

Moreover, sup(∥g_{i}∥+∥g_{i}∥_{∞}) = c < ∞, and there is a bounded open set U ⊂ ℝ^{m} such that supp g_{i} ⊂ U for i = 1, 2, . . .. Thus for each i,

∥Σ_{g_{i}^{±}}∥ ≤ 2|U| + ∥g_{i}∥ ≤ 2|U| + c

and Σ_{g_{i}^{±}} ⊂ U × [0, c]. We infer the sequences {Σ_{g_{i}^{±}}} τ-converge to Σ_{g^{±}}. Consequently lim⟨F, g_{i}⟩ = ⟨F, g⟩ for each charge F, and as τ is a sequential topology, Θ(F) is a continuous functional on (BV_{c}^{∞}, τ). It is a consequence of the coarea theorem [6, Section 5.5, Theorem 1] the linear space S_{BV} of all simple functions in BV_{c}^{∞} is dense in (BV_{c}^{∞}, τ). A standard argument reveals if g ∈ S_{BV} has the form g = ∑_{i=1}^{p} c_{i}E_{i} where c_{1}, …, c_{p} are real numbers and E_{1}, …, E_{p} are bounded BV sets, then

Θ(F)(g) = ⟨F, g⟩ = ∑_{i=1}^{p} c_{i}F (E_{i}) .

It follows Θ(F) is a linear functional on S_{BV}, and hence on BV_{c}^{∞}. Alternatively, by [16, Theorem 3.4], the functional Θ(F) is linear on the vector space of all Lipschitz functions g ∈ BV_{c}^{∞}, which is also dense in (BV_{c}^{∞}, τ). We conclude Θ(F) belongs to (BV_{c}^{∞})^{∗}.

Remark 4.2. As (CH, σ) is a Fréchet space, an application of the Banach-Steinhaus theorem [19, Theorem 2.17] improves the separate continuity established in Proposition 4.1 as follows: lim⟨F_{i}, g_{i}⟩ = ⟨F, g⟩ whenever {(F_{i}, g_{i})} is a sequence in CH × BV_{c}^{∞} that (σ × τ)-converges to (F, g).

Theorem 4.3. For T ∈ (BV_{c}^{∞})^{∗} and E ∈ BV, let Λ(T)(E) = T (χ_{E}). Then Λ(T) is a charge, and the map

Λ : ((BV_{c}^{∞})^{∗}, ν) → (CH, σ) : T ↦ Λ(T)

is a linear homeomorphism with Λ^{−1} : F ↦ Θ(F).

Proof. Since Λ(T) is the composition of T and the inclusion map BV ↪ BV_{c}^{∞}, it is easy to see that Λ(T) is a charge, and ||||||Λ(T)||||||_{n} ≤ ||||||T ||||||_{n} for n = 1, 2, . . .. Thus Λ, which is clearly linear, is also continuous and injective. Let

Θ : CH → (BV_{c}^{∞})^{∗} : F ↦ Θ(F) ,

and observe that for each charge F and each bounded BV set E,

(Λ ∘ Θ)(F)(E) = Θ(F)(χ_{E}) = (F × L^{1})(Σ_{χ_{E}}) = F (E) .

Hence Λ ∘ Θ is the identity map on CH, which implies Λ is surjective. By the closed graph theorem [19, Theorem 2.15], Λ is a linear homeomorphism and Λ^{−1} = Θ.

Remark 4.4. Let A be a bounded BV set. Considering BV^{∞}(A) and CH(A) instead of BV_{c}^{∞} and CH, respectively, we obtain results completely analogous to Proposition 4.1 and Theorem 4.3. In particular, it follows from Proposition 3.4 the topology in BV^{∞}(A)^{∗} is induced by a Banach norm.

Remark 4.5. If D(ℝ^{m}) is the space of Schwartz’s test functions equipped with its usual direct limit topology [19, Definition 6.3], then the inclusion map D(ℝ^{m}) ↪ BV_{c}^{∞} is continuous. Thus given a charge F, the restriction T of Θ(F) to D(ℝ^{m}) is a distribution, and it is easy to see the order of T is one.

A charge F is derivable at x ∈ ℝ^{m} if a finite limit

lim \frac{F (B_{n})}{|B_{n}|}

exists for each sequence {B_{n}} of figures such that

lim d(B_{n} ∪ {x}) = 0 and inf r(B_{n} ∪ {x}) > 0 .  (4.1)

If all these limits exist, they have the same value called the derivate of F at x, denoted by F′(x). Convenient necessary and sufficient conditions for a charge to be derivable almost everywhere in a set E ⊂ ℝ^{m} can be found in [2, Theorem 3.3 and Corollary 4.9].

Remark 4.6. It follows from [2, Lemma 3.1] that if a charge F is derivable at x, then lim[F (B_{n})/|B_{n}|] = F′(x) exists for each sequence {B_{n}} of bounded BV sets satisfying condition (4.1). This is an important fact we shall often use.

A charge F is called absolutely continuous (abbreviated as AC) if for each ε > 0 there is a δ > 0 such that |F (E)| < ε whenever E ⊂ B(1/ε) is a BV set with |E| < δ. The linear spaces of all AC charges and all AC charges in a locally BV set A are denoted by CH_{AC} and CH_{AC}(A), respectively. It follows from [15, Theorem 9.3.7] that each F ∈ CH_{AC} is derivable almost everywhere with F′ ∈ L_{loc}^{1}(ℝ^{m}) and ∫ F′ dL^{m} = F.

Lemma 4.7. Let B be a BV set, y ∈ ℝ^{m}, and B + y = {x + y : x ∈ B}. Then

|(B + y) △ B| ≤ ∥y∥ · ∥B∥ .

Proof. If g ∈ BV (ℝ^{m}) ∩ C^{1}(ℝ^{m}), then

g(x − y) − g(x) = ∫_{0}^{1} \frac{d}{dt}g(x − ty) dt = − ∫_{0}^{1} y · ∇g(x − ty) dt

for all x ∈ ℝ^{m}, and Schwartz’s inequality yields

∫_{ℝ^{m}} |g(x − y) − g(x)| dx ≤ ∫_{ℝ^{m}} (∫_{0}^{1} ∥y∥ · ∥∇g(x − ty)∥ dt) dx

= ∥y∥ ∫_{0}^{1} (∫_{ℝ^{m}} ∥∇g(x − ty)∥ dx) dt = ∥y∥ · ∥g∥ .

By the approximation theorem [6, Section 5.2.2, Theorem 2], there is a sequence {g_{i}} in BV (ℝ^{m}) ∩ C^{1}(ℝ^{m}) with lim ∥g_{i} − χ_{B}∥_{1} = 0 and lim ∥g_{i}∥ = ∥B∥. Since

|(B + y) △ B| = ∫_{ℝ^{m}} |χ_{B}(x − y) − χ_{B}(x)| dx ≤
∫_{ℝ^{m}} |χ_{B}(x − y) − g_{i}(x − y)| dx + ∫_{ℝ^{m}} |g_{i}(x − y) − g_{i}(x)| dx +
∫_{ℝ^{m}} |g_{i}(x) − χ_{B}(x)| dx ≤ 2∥g_{i} − χ_{B}∥_{1} + ∥y∥ · ∥g_{i}∥

for i = 1, 2, …, the lemma follows.

Proposition 4.8. The linear space CH_{AC} is dense in (CH, σ).

Proof. Let F be a charge, and let φ ∈ C_{c}^{1}(ℝ^{m}) be nonnegative. For a bounded BV set B, let

F_{φ}(B) = Θ(F)(φ ∗ χ_{B}) = ⟨F, φ ∗ χ_{B}⟩

where ∗ denotes the convolution operation. We shall need the following inequalities obtained by standard manipulations of convolutions:

\begin{matrix}∥φ ∗ χ_{B}∥_{∞} ≤ ∥φ∥_{1} , \\ ∥φ ∗ χ_{B}∥_{1} ≤ |B| · ∥φ∥_{1} , \\ ∥φ ∗ χ_{B}∥ ≤ min\{∥B∥ · ∥φ∥_{1}, ∥φ∥ · |B|\} . \end{matrix}  (4.2)

Given ε > 0, find a θ > 0 according to Proposition 3.1, and select a BV set B ⊂ B(1/ε). Letting b = ∥φ ∗ χ_{B}∥_{∞}, Proposition 3.1, Fubini’s theorem, coarea theorem, and inequalities (4.2) yield

|F_{φ}(B)| = |(F × L^{1})(Σ_{φ∗χ_{B}})| ≤ ∫_{0}^{b}|F (Σ_{φ∗χ_{B}}^{t})| dt
≤ θ ∫_{0}^{b} |Σ_{φ∗χ_{B}}^{t}| dt + ε ∫_{0}^{b}(∥Σ_{φ∗χ_{B}}^{t}∥ + 1) dt
= θ∥φ ∗ χ_{B}∥_{1} + ε(∥φ ∗ χ_{B}∥ + b)
≤ θ|B| · ∥φ∥_{1} + ε(∥φ∥ · |B| + ∥φ∥_{1}) ,

and we conclude F_{φ} is an AC charge.

For i = 1, 2, …, let φ_{i} = η_{1/i} be the standard mollifier [6, Section 4.2.1], and write F_{i} instead of F_{φ_{i}}. We establish the proposition by showing the sequence {F_{i}} of AC charges converges to F uniformly on each BV_{n}. To this end, select an integer n ≥ 1 and a B ∈ BV_{n}. Clearly supp (χ_{B} − φ_{i} ∗ χ_{B}) ⊂ B[n + 1], and since ∥φ_{i}∥_{1} = 1, the inequalities (4.2) imply

∥φ ∗ χ_{B}∥_{∞} ≤ 1 and ∥φ_{i} ∗ χ_{B}∥ ≤ ∥B∥ ≤ n .

Moreover, by Fubini’s theorem and Lemma 4.7,

∥χ_{B} − φ_{i} ∗ χ_{B}∥_{1} = ∫_{ℝ^{m}} |χ_{B}(x) ∫_{ℝ^{m}} φ_{i}(y) dy − ∫_{ℝ^{m}} χ_{B}(x − y)φ_{i}(y) dy| dx

≤ ∫_{ℝ^{m}} φ_{i}(y) (∫_{ℝ^{m}} |χ_{B}(x) − χ_{B}(x − y)| dx) dy

= ∫_{∥y∥≤ \frac{1}{i}} φ_{i}(y)|(B + y) △ B| dy

≤ ∥B∥ ∫_{∥y∥≤ \frac{1}{i}} ∥y∥φ_{i}(y) dy ≤ \frac{1}{i}∥B∥ ≤ \frac{n}{i} .

Thus {φ_{i} ∗ χ_{B}} is a sequence in BV_{c}^{∞} which τ-converges to χ_{B} uniformly with respect to B ∈ BV_{n}. Since the linear functional Θ(F) defined on BV_{c}^{∞} is τ-continuous (Proposition 4.1), we obtain

F (B) = Θ(F)(χ_{B}) = lim Θ(F)(φ_{i} ∗ χ_{B}) = lim F_{i}(B)

where the convergence is uniform with respect to B ∈ BV_{n}.

Corollary 4.9. For each locally BV set A, the linear space CH_{AC}(A) is dense in (CH(A), σ).

Proof. Let F be a charge in A. By Proposition 4.8, there is a sequence {F_{i}} in CH_{AC} that σ-converges to F. Clearly, {F_{i} A} is a sequence in CH_{AC}(A) that also σ-converges to F.

Endow L_{loc}^{1}(ℝ^{m}) with the Fréchet topology ρ induced by seminorms

∥f∥_{1,n} = ∫_{B[n]}|f(x)| dx

defined for each f ∈ L_{loc}^{1}(ℝ^{m}) and n = 1, 2, . . .. If F is an AC charge, then

||||||F ||||||_{n} = sup {|∫_{B} F′(x) dx| : B ∈ BV_{n}} ≤ ∫_{B[n]}|F′(x)| dx = ∥F′∥_{1,n}

and consequently, the linear bijection

Υ : (L_{loc}^{1}(ℝ^{m}), ρ) → (CH_{AC}, σ) : f ↦ ∫ f dL^{m}

is continuous. Note the inverse map Υ^{−1} : F ↦ F′ is not continuous.

Lemma 4.10. If T is a continuous linear functional on (L_{loc}^{1}(ℝ^{m}), ρ), then there is a g ∈ L_{c}^{∞}(ℝ^{m}) such that for each f ∈ L_{loc}^{1}(ℝ^{m}),

fg ∈ L^{1}(ℝ^{m}) and T (f) = ∫_{ℝ^{m}} f(x)g(x) dx .

Proof. For n = 1, 2, …, the subspace L_{n} = {fχ_{B[n]} : f ∈ L_{loc}^{1}(ℝ^{m})} of (L_{loc}^{1}(ℝ^{m}), ρ) is isomorphic to L^{1}(B[n]). According to [18, Theorem 6.16], there is a g_{n} ∈ L^{∞}(B[n]) such that

T (f) = ∫_{B[n]} f(x)g_{n}(x) dx

for each f ∈ L_{n}. Since

∫_{B} g_{n}(x) dx = T (χ_{B}) = ∫_{B} g_{n+1}(x) dx

for every measurable set B ⊂ B[n], the restriction of g_{n+1} to B[n] equals g_{n} almost everywhere. We infer there is a function g defined on ℝ^{m} whose restriction to each B[n] equals g_{n} almost everywhere; in particular g is measurable.

Select an f ∈ L_{loc}^{1}(ℝ^{m}) and observe the sequence {fχ_{B[n]}} ρ-converges to f. As |f| sign g belongs to L_{loc}^{1}(ℝ^{m}), we obtain

T (|f| sign g) = lim T [(|f| sign g)χ_{B[n]}]
= lim ∫_{B[n]}|f(x)g(x)| dx = ∫_{ℝ^{m}} |f(x)g(x)| dx .

Hence fg belongs to L^{1}(ℝ^{m}), which implies

T (f) = lim T (fχ_{B[n]}) = lim ∫_{B[n]} f(x)g(x) dx = ∫_{ℝ^{m}} f(x)g(x) dx .

Now it is easy to show by contradiction the equivalence class of g contains a bounded function with compact support.

Theorem 4.11. Let X be a linear subspace of CH, and let X^{∗} denote the dual space of (X, σ) equipped with the w^{∗} topology. If CH_{AC} ⊂ X, then the map

Φ : (BV_{c}^{∞}, τ) → X^{∗} : g ↦ Φ(g)

is a continuous linear bijection.

Proof. In view of Proposition 4.1 and the definition of the w^{∗}-topology in X^{∗} [19, Section 3.14], we only need to show Φ is surjective. To this end, select a T ∈ X^{∗}, and observe T ∘ Υ is a continuous linear functional on L_{loc}^{1}(ℝ^{m}). By Lemma 4.10, there is a g ∈ L_{c}^{∞}(ℝ^{m}) such that F′g belongs to L^{1}(ℝ^{m}) and

T (F) = (T ∘ Υ)(F′) = ∫_{ℝ^{m}} F′(x)g(x) dx

for every AC charge F. Employing the σ-continuity of T, find positive integers n and k so that |T (F)| ≤ k||||||F ||||||_{n} for all F ∈ X. If h ∈ C_{c}^{1}(ℝ^{m}; ℝ^{m}), then H = ∫ div h dL^{m} is an AC charge. By the Gauss-Green theorem,

|H(B)| = |∫_{∂_{∗}B} h · ν_{B} dH^{m−1}| ≤ ∥B∥ · ∥h∥_{∞}

for each BV set B, and hence ||||||H||||||_{n} ≤ n∥h∥_{∞}. Thus

∫_{ℝ^{m}} g(x)div h(x) dx = T (H) ≤ k||||||H||||||_{n} ≤ kn∥h∥_{∞} ,

which implies g ∈ BV^{∞}(A). Now Fubini’s theorem yields

T (F) = ∫_{ℝ^{m}} (F′(x) ∫_{0}^{g^{+}(x)} dt) dx − ∫_{ℝ^{m}} (F′(x) ∫_{0}^{g^{−}(x)} dt) dx

= ∫_{ℝ} (∫_{{g^{+}(x)>t}} F′(x) dx) dt − ∫_{ℝ} (∫_{{g^{−}(x)>t}} F′(x) dx) dt

= ∫_{ℝ} F (Σ_{g^{+}}^{t}) dt − ∫_{ℝ} F (Σ_{g^{−}}^{t}) dt
= (F × L^{1})(Σ_{g^{+}}) − (F × L^{1})(Σ_{g^{−}}) = ⟨F, g⟩ = Φ(g)(F)

for each F ∈ CH_{AC}. According to Corollary 4.9 and Proposition 4.1, the equality T (F) = Φ(g)(F) extends to all F ∈ X, and the surjectivity of Φ is established.

Question 4.12. Is there a simple description of the topology σ^{∗} in CH^{∗} for which Φ : (BV_{c}^{∞}, τ) → (CH^{∗}, σ^{∗}) is a homeomorphism?

Remark 4.13. Let A be a bounded BV set. In view of Corollary 4.9, a result completely analogous to Theorem 4.11 holds when the spaces CH and BV_{c}^{∞} are replaced by CH(A) and BV^{∞}(A), respectively (cf. Remark 4.2).

5 AC_{∗} charges

We show that a weaker notion of absolute continuity leads to a generalized Riemann integral akin to that of [15, Chapter 12].

A partition is a collection (possibly empty) P = {(A_{1}, x_{1}), …, (A_{p}, x_{p})} where A_{1}, …, A_{p} are disjoint bounded BV sets, and x_{i} ∈ ℝ^{m} for i = 1, …, p. Given ε > 0 and a nonnegative function δ on E ⊂ ℝ^{m}, the partition P is called

F-partition if A_{1}, …, A_{p} are figures,

ε-regular if r(A_{i} ∪ {x_{i}}) > ε for i = 1, …, p,

δ-fine if x_{i} ∈ E and d(A_{i} ∪ {x_{i}}) < δ(x_{i}) for i = 1, …, p.

Note if P is δ-fine, then each x_{i} belongs to E − null δ. A nonnegative function δ defined on a set E ⊂ ℝ^{m} is called a gage on E whenever null δ is a thin set.

The variation of a charge F on a set E ⊂ ℝ^{m} is the number

V_{∗}F (E) = sup_{ε>0} inf_{δ} sup_{P} ∑_{i=1}^{p}|F (A_{i})|

where δ is a gage on E, and P = {(A_{1}, x_{1}), …, (A_{p}, x_{p})} is an ε-regular δ-fine F-partition. According to [2, Lemmas 4.6 and 4.4], the extended real-valued function V_{∗}F : E ↦ V_{∗}F (E) is a Borel regular measure in ℝ^{m}, which coincides with the classical variation of F on every figure. In particular, V_{∗}F is Radon [6, Section 1.1] whenever it is locally finite. Our first proposition, proved in [3, Proposition 3.3], characterizes AC charges in terms of variations.

Proposition 5.1. A charge F is AC if and only if V_{∗}F is absolutely continuous and locally finite.

It turns out many important properties of AC charges remain valid when V_{∗}F is merely absolutely continuous. Specifically, we have a proposition, which follows from [9, Lemma 6] and [3, Corollary 4.3 and Theorem 4.4].

Proposition 5.2. Let F be a charge. If V_{∗}F is absolutely continuous, then the following is true.

(1) F is derivable almost everywhere, and F ≥ 0 whenever F′ ≥ 0 almost everywhere.

(2) There are closed sets C_{i} ⊂ ℝ^{m} with V_{∗}F (C_{i}) < ∞ for i = 1, 2, … and such that ℝ^{m} − ⋃_{i=1}^{∞} C_{i} is a thin set. In particular, V_{∗}F is σ-finite.

Definition 5.3. A charge F is called AC_{∗} whenever the measure V_{∗}F is absolutely continuous.

The linear space of all AC_{∗} charges or all AC_{∗} charges in a bounded BV set A is denoted by CH_{AC_{∗}} or CH_{AC_{∗}}(A), respectively. We wish to alert the reader our concept of AC_{∗} differs substantially from that of [20, Chapter 8, Section 8].

Remark 5.4. While neither of the spaces (CH_{AC_{∗}}, σ) and (CH_{AC_{∗}}(A), σ) is complete, an argument similar to that employed in [4, Theorem 4.1] shows the space (CH_{AC_{∗}}(A), σ) is barrelled.

In view of Proposition 5.2,(1), we can present a descriptive definition of the generalized Riemann integral (abbreviated as R-integral), which is completely analogous to the usual descriptive definition of the Lebesgue integral.

Definition 5.5. A function f defined on ℝ^{m} is locally R-integrable whenever there is an AC_{∗} charge F such that F′(x) = f(x) for almost all x ∈ ℝ^{m}. The charge F, uniquely determined by f, is called the indefinite R-integral of f. If A is a bounded BV set, the number F (A) is called the R-integral of f over A.

The linear space of all locally R-integrable functions is denoted by R_{loc}(ℝ^{m}). Clearly L_{loc}^{1}(ℝ^{m}) ⊂ R_{loc}(ℝ^{m}), and the indefinite R-integral F of f ∈ L_{loc}^{1}(ℝ^{m}) coincides with the indefinite Lebesgue integral of f. Thus without danger of confusion, we can denote F by ∫ f dL^{m}, and F (A) by ∫_{A} f dL^{m} for each bounded BV set A. An argument similar to [20, Chapter 4, Theorem 4.2] shows each f ∈ R_{loc}(ℝ^{m}) is measurable. It follows a nonnegative function f on ℝ^{m} belongs to R_{loc}(ℝ^{m}) if and only if it belongs to L_{loc}^{1}(ℝ^{m}) [13, Lemma 5.13]. The name “generalized Riemann integral” is justified by the next theorem, proved in [3, Theorem 4.6].

Theorem 5.6. A function f defined on ℝ^{m} is locally R-integrable if and only if there is a charge F satisfying the following condition: given ε > 0, we can find a gage δ on B(1/ε) so that

∑_{i=1}^{p}|f(x_{i})|A_{i}| − F (A_{i})| < ε

for each ε-regular δ-fine F-partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})}. In this case F is the indefinite R-integral of f.

To stress further the proximity of the R-integral and Lebesgue integral, we state a theorem due to McShane [11, Chapter 4, Definition 1-1 and Corollary 6-5]. It shows the regularity of partitions and use of gages are critical components of Theorem 5.6.

Theorem 5.7. A function f defined on ℝ^{m} is locally Lebesgue integrable if and only if there is a charge F satisfying the following condition: given ε > 0, we can find a positive function δ on B(1/ε) so that

∑_{i=1}^{p}|f(x_{i})|A_{i}| − F (A_{i})| < ε

for each δ-fine F-partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})}. In this case F is the indefinite Lebesgue integral of f.

The following proposition, which simplifies many technical manipulations, is a consequence of [17, Theorem 3.3], [13, Proposition 7.7], and Remark 4.6.

Proposition 5.8. If F is a charge and E ⊂ ℝ^{m}, then

V_{∗}F (E) = sup_{ε>0} inf_{δ} sup_{P} ∑_{i=1}^{p}|F (A_{i})|

where δ is a gage on E, and P = {(A_{1}, x_{1}), …, (A_{p}, x_{p})} is any ε-regular δ-fine partition. Similarly, Theorems 5.6 and 5.7 remain valid when F-partitions are replaced by partitions.

One of the principal reasons for introducing the R-integral is the divergence theorem for discontinuously differentiable vector fields. Let E ⊂ ℝ^{m} and let v : E → ℝ^{m} be a vector field on E. We say v is Lipschitz at x ∈ E whenever there are positive numbers c and r such that

|v(y) − v(x)| ≤ c|y − x|

for each y ∈ E ∩ B(x, r). If E is an open set and v is Lipschitz at each point of a set C ⊂ E, then v is differentiable almost everywhere in C by Stepanoff’s theorem [7, Theorem 3.1.9].

Lemma 5.9. Let F be the flux of a v ∈ C(ℝ^{m}; ℝ^{m}). For each x ∈ ℝ^{m}, the following is true.

(1) If v is Lipschitz at x, then given ε > 0, we can find positive numbers c and δ so that |F (B)| ≤ c|B| for each BV set B with r(B ∪ {x}) > ε and d(B ∪ {x}) < δ.

(2) If v is differentiable at x, then F is derivable at x and F′(x) = div v(x).

The proof of Lemma 5.9 is not difficult. The reader can find it in [15, Lemma 11.7.4].

Lemma 5.10. Let F be the flux of a v ∈ C(ℝ^{m}; ℝ^{m}). If v is Lipschitz at each point of a set E ⊂ ℝ^{m}, then (V_{∗}F) E is absolutely continuous.

Proof. Choose a negligible set N ⊂ E and an ε > 0. Given x ∈ N, use Lemma 5.9,(1) to find positive numbers c_{x} and δ_{x} so that |F (B)| ≤ c_{x}|B| for each BV set B with r(B ∪ {x}) > ε and d(B ∪ {x}) < δ_{x}. For k = 1, 2, …, let

N_{k} = {x ∈ N : k − 1 ≤ c_{x} < k} ,

and let U_{k} ⊂ ℝ^{m} be an open set with N_{k} ⊂ U_{k} and |U_{k}| < ε2^{−k}/k. Making δ_{x} smaller, we may assume B[x, δ_{x}] ⊂ U_{k} whenever x ∈ N_{k}. Observe δ : x ↦ δ_{x} is a positive function on N, and select an ε-regular δ-fine partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})}. Then

∑_{i=1}^{p}|F (A_{i})| ≤ ∑_{i=1}^{p} c_{x_{i}}|A_{i}| < ∑_{k=1}^{∞} ∑_{x_{i}∈N_{k}} k|A_{i}|

≤ ∑_{k=1}^{∞} k|U_{k}| < ∑_{k=1}^{∞} ε2^{−k} = ε ,

and from the arbitrariness of ε, we conclude V_{∗}F (N) = 0.

Theorem 5.11. Let v ∈ C(ℝ^{m}; ℝ^{m}), and suppose there is a thin set S such that v is Lipschitz at each point of ℝ^{m} − S. Then div v is locally R-integrable, and

∫ div v dL^{m} = ∫ v · ν dH^{m−1} .

Proof. Let F be the flux of v. Since V_{∗}F (S) = 0, Lemma 5.10 implies F is AC_{∗}. An application of Stepanoff’s theorem and Lemma 5.9,(2) completes the argument.

6 Multipliers

A multiplier for a family X of functions defined on a set E is a function g defined on E such that fg ∈ X for each f ∈ X. The collection of all multipliers for X is denoted by MX. If X is a linear space containing constant functions, then MX is a subalgebra of X — for instance, ML_{loc}^{1}(ℝ^{m}) = L^{∞}(ℝ^{m}). In this section we show MR_{loc}(ℝ^{m}) = BV_{loc}^{∞}.

Lemma 6.1. MR_{loc}(ℝ^{m}) ⊂ BV_{loc}^{∞}.

Proof. Since the one-dimensional R-integral is a restriction of the Denjoy-Perron integral [8, Chapter 11], for m = 1 the lemma follows from [21]. Thus we assume m ≥ 2, and select an f ∈ MR_{loc}(ℝ^{m}). According to the definition of a multiplier, f and f^{2} belong to R_{loc}(ℝ^{m}). Thus f is measurable and f^{2} is in L_{loc}^{1}(ℝ^{m}). In particular, f ∈ L_{loc}^{1}(ℝ^{m}).

Suppose f ∉ L_{loc}^{∞}(ℝ^{m}). By symmetry, we may assume there is a cell C with f^{+} ∉ L^{∞}(C). Each C_{n} = {x ∈ C : f(x) ≥ 2^{n}} is a measurable set of finite positive measure, and

g = ∑_{n=1}^{∞} \frac{1}{2^{n}|C_{n}|}χ_{C_{n}}

is a function from L^{1}(ℝ^{m}). If E = {x ∈ ℝ^{m} : f(x) ≥ 0}, then gχ_{E} belongs to L^{1}(ℝ^{m}), and hence to R_{loc}(ℝ^{m}). Since f ∈ MR_{loc}(ℝ^{m}), the function f^{+}g = (fχ_{E})g = f(gχ_{E}) is in R_{loc}(ℝ^{m}) and, being nonnegative, in L_{loc}^{1}(ℝ^{m}). On the other hand, as

|C_{n}| ≤ 2^{−n} ∫_{ℝ^{m}} |f| dL^{m}  and  ∫_{C_{n}} f^{+}g ≥ 1

for n = 1, 2, …, the function f^{+}g is not in L^{1}(C), a contradiction.

Next suppose f ∉ BV_{loc}(ℝ^{m}), and select an open cube U = (a, b)^{m} so that f ∉ BV (U). For a < t < b, let

U_{t−} = (a, t) × (a, b)^{m−1} and U_{t+} = (t, b) × (a, b)^{m−1} .

There is a v_{1} ∈ C^{1}(ℝ^{m}, ℝ^{m}) with ∥v_{1}∥_{∞} ≤ 1 and ∫_{U} f div v_{1} > 2 · 2^{2}. Find a t ∈ (a, b) so that

∫_{U_{t−}} f div v_{1} = ∫_{U_{t+}} f div v_{1} = \frac{1}{2} ∫_{U} f div v_{1} > 2^{2} ,

and denote by V_{1} the element of {U_{t−}, U_{t+}} in which f is not BV; such a V_{1} exists by the extension property of BV functions [6, Section 5.4]. The other element of {U_{t−}, U_{t+}} is denoted by U_{1} = (r_{1}, s_{1}) × (a, b)^{m−1}. Applying the previous argument to V_{1}, we obtain disjoint subsets U_{2} = (r_{2}, s_{2}) × (a, b)^{m−1} and V_{2} = (t, t′) × (a, b)^{m−1} of V_{1} such that f is not BV in V_{2} and

∫_{U_{2}} f div v_{2} > 2^{4}

for a v_{2} ∈ C^{1}(ℝ^{m}, ℝ^{m}) with ∥v_{2}∥_{∞} ≤ 1. Proceeding inductively, we construct disjoint open intervals U_{k} = (r_{k}, s_{k}) × (a, b)^{m−1} and v_{k} ∈ C^{1}(ℝ^{m}, ℝ^{m}) with

∥v_{k}∥_{∞} ≤ 1 and ∫_{U_{k}} f div v_{k} > 2^{2k} for k = 1, 2, ….

From the construction it is clear the set {r_{k}, s_{k} : k = 1, 2, … } has at most two cluster points α and β. As m ≥ 2, we can define disjoint open intervals

U_{k,i} = (r_{k}, s_{k}) × (a + (i − 1)\frac{b − a}{2^{k}}, a + i\frac{b − a}{2^{k}}) × (a, b)^{m−2} ,

i = 1, …, 2^{k}, which cover U_{k} almost entirely. Thus for each k there is an i_{k} with ∫_{U_{k,i_{k}}} f div v_{k} > 2^{k}. Now U_{k,i_{k}} contains a cell A_{k} such that

∫_{A_{k}} f div v_{k} > 2^{k} ,

and by mollification, we may assume v_{k} vanishes outside U_{k,i_{k}}. The vector field v = ∑_{k=1}^{∞} 2^{−k}v_{k} is continuous everywhere, and it is differentiable at each x ∈ ℝ^{m} that is not in the thin set {α, β}×(a, b)^{m−1}. According to Theorem 5.11, div v belongs to R_{loc}(ℝ^{m}) and, as f ∈ MR_{loc}(ℝ^{m}), so does f div v. Denote by F the indefinite R-integral of f div v. Since

∑_{k=1}^{∞} ∥A_{k}∥ ≤ ∑_{k=1}^{∞} ∥U_{k,i_{k}}∥ ≤ ∥U∥ ,

the union A = ⋃_{k=1}^{∞} A_{k} is a bounded BV set. A contradiction follows:

F (A) = lim F ( ⋃_{k=1}^{n} A_{k}) = ∑_{k=1}^{∞} F (A_{k})

= ∑_{k=1}^{∞} ∫_{A_{k}} f div v = ∑_{k=1}^{∞} 2^{−k} ∫_{A_{k}} f div v_{k} = ∞ .

Let F be a charge, and let g ∈ BV_{loc}^{∞}. For each bounded BV set B, the function gχ_{B} belongs to BV_{c}^{∞}, and we let

G : B ↦ ⟨F, gχ_{B}⟩

where ⟨·, ·⟩ is the pairing introduced at the beginning of Section 4. Using Proposition 4.1, it is easy to see G is a charge; we say G is induced by F via g.

Lemma 6.2. Let F be an AC_{∗} charge, and let g ∈ BV_{loc}^{∞} satisfy 0 ≤ g ≤ 1. If G is a charge induced by F via g, then G is AC_{∗}.

Proof. Denote by I the interval [0, 1]. For a bounded BV set B and t ∈ I, let

B^{t} = {x ∈ B : g(x) > t} ,

and observe that G(B) = ∫_{I} F (B^{t}) dt. It suffices to show V_{∗}G vanishes on each bounded negligible set. Choose such a set N, and an ε > 0 so that N ⊂ B(1/ε). By Proposition 3.1, there is a θ > 0 such that

|F (B)| < θ|B| + ε(∥B∥ + 1)  (6.1)

for every BV set B ⊂ B(1/ε). As F is AC_{∗}, there is a gage δ on N such that

∑_{i=1}^{p}|F (A_{i})| < ε

for each (ε/6)-regular δ-fine partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})} (Proposition 5.8). Find an open set U ⊂ B(1/ε) with N ⊂ U and |U| < ε/θ. Making δ smaller, we may assume δ(x) ≤ 1 and B[x, δ(x)] ⊂ U for every x ∈ N.

Select an ε-regular δ-fine F-partition {(B_{1}, x_{1}), …, (B_{q}, x_{q})}, and note each B_{i} ⊂ U by the choice of δ. If U_{i} = int B_{i}, then

∥∂B_{i}^{t}∥(U_{i}) ≤ ∥B_{i}^{t}∥ ≤ ∥∂B_{i}^{t}∥(U_{i}) + ∥B_{i}∥  (6.2)

for L^{1}-almost all t ∈ I and i = 1, …, q. Now let S_{i} = {t ∈ I : ∥B_{i}^{t}∥ ≤ 2∥B_{i}∥}, T_{i} = I − S_{i}, and distinguish two cases.

(1) Let S′_{i} = {t ∈ S_{i} : |B_{i}| ≤ 2|B_{i}^{t}|}, S′′_{i} = S_{i} − S′_{i}, and

A_{i,t} = \begin{cases}B_{i}^{t} & if t ∈ S′_{i}, \\ B_{i} − B_{i}^{t} & if t ∈ S′′_{i}. \end{cases}

As ∥A_{i,t}∥ ≤ ∥B_{i}∥ + ∥B_{i}^{t}∥ ≤ 3∥B_{i}∥ and |B_{i}| ≤ 2|A_{i,t}|, we obtain

r(A_{i,t}, x_{i}) ≥ \frac{1}{2} · \frac{|B_{i}|}{d(B_{i} ∪ \{x_{i}\})∥B_{i}∥} · \frac{∥B_{i}∥}{∥A_{i,t}∥} > \frac{ε}{6}

for each t ∈ S_{i}.

(2) By inequality (6.2),

∥B_{i}^{t}∥ ≤ ∥∂B_{i}^{t}∥(U_{i}) + ∥B_{i}∥ < ∥∂B_{i}^{t}∥(U_{i}) + \frac{1}{2}∥B_{i}^{t}∥ ,

and hence ∥B_{i}^{t}∥ < 2∥∂B_{i}^{t}∥(U_{i}) for almost all t ∈ T_{i}.

Using cases (1) and (2), we estimate

∑_{i=1}^{q}|G(B_{i})| = ∑_{i=1}^{q} |∫_{I} F (B_{i}^{t}) dt| ≤ ∑_{i=1}^{q} ∫_{S_{i}} |F (B_{i}^{t})| dt + ∑_{i=1}^{q} |∫_{T_{i}} F (B_{i}^{t}) dt|

by estimating separately each sum on the right. According to case (1), the collection P_{t} = {(A_{i,t}, x_{i}) : t ∈ S_{i}} is an (ε/6)-regular δ-fine partition for each t ∈ I. Thus

∑_{i=1}^{q} ∫_{S_{i}} |F (B_{i}^{t})| dt = ∑_{i=1}^{q} (∫_{S′_{i}} |F (A_{i,t})| dt + ∫_{S′′_{i}} |F (B_{i} − A_{i,t})| dt)

≤ ∑_{i=1}^{q} (∫_{S_{i}} |F (A_{i,t})| dt + ∫_{S′′_{i}} |F (B_{i})| dt)

≤ ∫_{I} (∑_{i=1}^{q} χ_{S_{i}}(t)|F (A_{i,t})|) dt + ∫_{I} (∑_{i=1}^{q}|F (B_{i})|) dt

< ∫_{I} (∑_{t∈S_{i}}|F (A_{i,t})|) dt + ε < 2ε .

After a suitable reordering, we may assume there is a nonnegative integer p ≤ q such that ∫_{T_{i}} F (B_{i}^{t}) dt ≥ 0 for i ≤ p, and ∫_{T_{i}} F (B_{i}^{t}) dt < 0 for i > p. Letting

B_{+} = ⋃{B_{i}^{t} : t ∈ T_{i} and i ≤ p} and B_{−} = ⋃{B_{i}^{t} : t ∈ T_{i} and i > p} ,

case (2), inequality (6.1), and the coarea theorem imply

∑_{i=1}^{q} |∫_{T_{i}} F (B_{i}^{t}) dt| = ∑_{i=1}^{p} ∫_{T_{i}} F (B_{i}^{t}) dt − ∑_{i=p+1}^{q} ∫_{T_{i}} F (B_{i}^{t}) dt

= ∫_{I} (∑_{i=1}^{p} χ_{T_{i}}(t)F (B_{i}^{t}) − ∑_{i=p+1}^{q} χ_{T_{i}}(t)F (B_{i}^{t})) dt

= ∫_{I} ( ∑_{{i≤p:t∈T_{i}}} F (B_{i}^{t}) − ∑_{{i>p:t∈T_{i}}} F (B_{i}^{t})) dt

= ∫_{I}[F (B_{+}) − F (B_{−})]dt ≤ ∫_{I}(|F (B_{+})| + |F (B_{−})|)dt

≤ ∫_{I}[θ(|B_{+}| + |B_{−}|) + ε(∥B_{+}∥ + ∥B_{−}∥) + 2ε]dt

≤ ∫_{I} θ|U| dt + ε ∫_{I} (∑_{t∈T_{i}} ∥B_{i}^{t}∥) dt + 2ε

< 3ε + 2ε ∫_{I} (∑_{t∈T_{i}} ∥∂B_{i}^{t}∥(U_{i})) dt

≤ 3ε + 2ε ∑_{i=1}^{q} ∫_{I} ∥∂B_{i}^{t}∥(U_{i}) dt

= 3ε + 2ε ∑_{i=1}^{p} ∥Dg∥(U_{i}) ≤ 3ε + 2ε∥Dg∥(U) .

Combining the previous inequalities yields

∑_{i=1}^{q}|G(B_{i})| < ε[5 + 2∥Dg∥(U)] ,

and the lemma follows.

Lemma 6.3. Let F be an AC_{∗} charge, let g ∈ BV_{loc}^{∞} satisfy 0 ≤ g ≤ 1, and let G be the charge induced by F via g. There is a set E ⊂ ℝ^{m} with |ℝ^{m} − E| = 0 and sequences {r_{x,n}} of positive reals such that the following conditions are met:

(1) F′(x) exists and lim r_{x,n} = 0 for every x ∈ E; (2) given ε > 0, we can find a gage δ on E ∩ B(1/ε) such that

∑_{i=1}^{p}|g(x_{i})F′(x_{i})|A_{i}| − G(A_{i})| < ε

whenever A_{i} = B[x_{i}, r_{x_{i},n_{i}}] and r_{x_{i},n_{i}} < δ(x_{i}) for i = 1, …, p.

Proof. The absolutely continuous part of the distributional gradient Dg of g has a density vector ∇g ∈ L_{loc}^{1}(ℝ^{m}) [6, Section 5.1]. Let E consist of all x ∈ ℝ^{m} at which F is derivable, and

\frac{1}{|B(x, s)|} ∫_{B(x,s)}|g(y) − g(x) − ∇g(x) · (y − x)| dy = o(s)

as s → 0. It follows from Proposition 5.2,(1), [6, Section 6.1, Theorem 1], and Hölder’s inequality that |ℝ^{m} − E| = 0. Fix an x ∈ E and let B_{s} = B[x, s]. For all sufficiently small s > 0,

∫_{B_{s}} |g(y) − g(x)| dy ≤ o(s^{m+1}) + ∫_{B_{s}} |∇g(x) · (y − x)| dy
≤ o(s^{m+1}) + s\sqrt{m}∥∇g(x)∥ · |B_{s}|
≤[1 + \sqrt{m}∥∇g(x)∥]2^{m}s^{m+1}

and hence, by the Fubini or coarea theorem,

∫_{0}^{s}(∫_{∂B_{r}} |g(y)−g(x)| dH^{m−1}(y)) dr ≤

(m + 1)[1 + \sqrt{m}∥∇g(x)∥] ∫_{0}^{s}(2r)^{m} dr .

Thus there is a sequence {r_{n}} of positive reals such that lim r_{n} = 0 and

\begin{aligned}∫_{B_{rn}} |g(y) − g(x)| dy & ≤ r_{n}[1 + \sqrt{m}∥∇g(x)∥] · |B_{r_{n}}| , \\ ∫_{∂B_{rn}} |g(y) − g(x)| dH^{m−1}(y) & ≤ (m + 1)[1 + \sqrt{m}∥∇g(x)∥] · |B_{r_{n}}| \end{aligned} (6.3)

for n = 1, 2, . . .. Let r_{x,n} = r_{n} and choose a positive ε < 1/(2m). To simplify the notation, let U = B(1/ε) and F̂ = F × L^{1}.

According to Propositions 3.5 and 3.1, there is a θ > 0 such that

|F̂(C)| < θ|C| + ε(∥C∥ + 1)  (6.4)

for every BV set C ⊂ U ×[0, 1]. Extend F′ to a function f defined on ℝ^{m}. Since both f and ∥∇g∥ belong to R_{loc}(ℝ^{m}), Theorem 5.6 implies there is a gage Δ on U such that

∑_{i=1}^{p}|f(x_{i})|A_{i}| − F (A_{i})| < ε ,

∑_{i=1}^{p} |∥∇g(x_{i})∥ · |A_{i}| − ∫_{A_{i}} ∥∇g(x)∥ dx| < 1

for each ε-regular Δ-fine F-partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})}. Let δ be the restriction of Δ/2 to E ∩ U. Making δ smaller, we may assume δ < ε/θ and B[x, δ(x)] ⊂ U for each x in E ∩ U − null δ. Thus for every ε-regular 2δ-fine F-partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})}, the following inequalities hold:

∑_{i=1}^{p}|F′(x_{i})|A_{i}| − F (A_{i})| < ε ,  (6.5)

\begin{aligned}∑_{i=1}^{p}∥∇g(x_{i})∥ · |A_{i}| & < 1 + ∑_{i=1}^{p} ∫_{A_{i}} ∥∇g(x)∥ dx \\ & ≤ 1 + ∑_{i=1}^{p} ∥Dg∥(A_{i}) ≤ 1 + ∥Dg∥(U) . \end{aligned}  (6.6)

If A_{i} = B[x_{i}, r_{x_{i},n_{i}}] where x_{i} ∈ E ∩ U and r_{x_{i},n_{i}} < δ(x_{i}), then {(A_{1}, x_{1}), …, (A_{p}, x_{p})} is an ε-regular 2δ-fine F-partition. The sets C_{i}^{1} = A_{i} × [0, g(x)] and C_{i}^{2} = Σ_{gχ_{A_{i}}} are contained in U × [0, 1] by the choice of δ. For j, k = 1, 2 and S ⊂ {1, …, p}, inequalities 6.3 and 6.6 yield

|⋃_{i∈S}(C_{i}^{j} − C_{i}^{k})| = ∑_{i∈S} |C_{i}^{j} − C_{i}^{k}| ≤ ∑_{i=1}^{p} ∫_{A_{i}} |g(y) − g(x_{i})| dy

≤ ∑_{i=1}^{p} r_{x_{i},n_{i}}[1 + \sqrt{m}∥∇g(x_{i})∥] · |A_{i}|

< \frac{ε}{θ} (|U| + \sqrt{m}[1 + ∥Dg∥(U)]) ,

∥⋃_{i∈S}(C_{i}^{j} − C_{i}^{k})∥ ≤ ∑_{i∈S} ∥C_{i}^{j} − C_{i}^{k}∥

≤ ∑_{i=1}^{p} [2|A_{i}| + ∥Dg∥(A_{i}) + ∫_{∂A_{i}} |g(y) − g(x_{i})| dH^{m−1}(y)]

< 2|U| + ∥Dg∥(U) + (m + 1) ∑_{i=1}^{p}[1 + \sqrt{m}∥∇g(x_{i})∥] · |A_{i}|

< (m + 3)(|U| + \sqrt{m}[1 + ∥Dg∥(U)]) .

From these inequalities and (6.4), we obtain

∑_{i∈S} F̂(C_{i}^{j} − C_{i}^{k}) = F̂ [⋃_{i∈S}(C_{i}^{j} − C_{i}^{k})] < εα ,

where

α = 1 + (m + 4)(|U| + \sqrt{m}[1 + ∥Dg∥(U)])

does not depend on ε. As S is arbitrary,

∑_{i=1}^{p}|F̂(C_{i}^{1}) − F̂(C_{i}^{2})| ≤ ∑_{i=1}^{p}[|F̂(C_{i}^{1} − C_{i}^{2})| + |F̂(C_{i}^{2} − C_{i}^{1})|] < 4αε .

Since F̂(C_{i}^{1}) = g(x_{i})F (A_{i}) and F̂(C_{i}^{2}) = G(A_{i}), the previous inequality together with (6.5) implies

∑_{i=1}^{p}|g(x_{i})F′(x_{i})|A_{i}| − G(A_{i})| ≤

∑_{i=1}^{p} g(x_{i})|F′(x_{i})|A_{i}| − F (A_{i})| + ∑_{i=1}^{p}|g(x_{i})F (A_{i}) − G(A_{i})| < ε(1 + 4α) ,

and the lemma is proved.

Theorem 6.4. Let F be the indefinite R-integral of an f ∈ R_{loc}(ℝ^{m}), and let g ∈ BV_{loc}^{∞}. Then fg ∈ R_{loc}(ℝ^{m}) and

∫_{B} fg dL^{m} = ⟨F, gχ_{B}⟩

for each bounded BV set B. In particular, MR_{loc}(ℝ^{m}) = BV_{loc}^{∞}.

Proof. With no loss of generality, suppose 0 ≤ g ≤ 1. Let E and the sequences {r_{x,n}} be as in Lemma 6.3, and let G be the charge induced by F via g. In view of Lemma 6.2 and Proposition 5.2,(1), making E smaller, we may assume G′(x) exists and F′(x) = f(x) for each x ∈ E. We complete the argument by showing G′(x) = f(x)g(x) for almost all x ∈ E.

For k = 1, 2, …, let N_{k} be the set of all x ∈ E ∩ B(k) such that

\frac{1}{k} < |lim \frac{G(B[x, r_{x,n}])}{|B[x, r_{x,n}]|} − f(x)g(x)| .  (6.7)

Fix an integer k ≥ 1, choose a positive ε < 1/k, and use Lemma 6.3 to find a gage δ on E ∩ B(1/ε) so that

∑_{i=1}^{p}|g(x_{i})f(x_{i})|A_{i}| − G(A_{i})| < \frac{ε}{k}

whenever A_{i} = B[x_{i}, r_{x_{i},n_{i}}] and r_{x_{i},n_{i}} < δ(x_{i}) for i = 1, …, p. In view of (6.7), the collection B of all cells B[x, r_{x,n}] such that x ∈ N_{k} −null δ, r_{x,n} < δ(x), and

\frac{1}{k} |B[x, r_{x,n}]| < |G(B[x, r_{x,n}]) − f(x)g(x)|B[x, r_{x,n}]||

covers N_{k} − null δ. According to [15, Lemma 9.2.1], there are disjoint cells B[x_{1}, r_{1}], B[x_{2}, r_{2}], … in B such that N_{k} − null δ ⊂ ⋃_{i} B[x_{i}, 5r_{i}]. Since

∑_{i}|B[x_{i}, r_{i}]| < k ∑_{i}|G(B[x_{i}, r_{i}]) − f(x_{i})g(x_{i})|B[x_{i}, r_{i}]|| ≤ ε ,

we have |N_{k}| = |N_{k} − null δ| < 5^{m}ε. By the arbitrariness of ε, the set N_{k} is negligible, and so is N = ⋃_{k=1}^{∞} N_{k}. As G′(x) = f(x)g(x) for each x ∈ E − N, our assertion is proved.

The following natural question remains unanswered.

Question 6.5. If F is an AC_{∗} charge, is the charge F × L^{1} also AC_{∗}?

7 The R-integral

If f is a function defined on a set E ⊂ ℝ^{m}, let

\overline{f}(x) = \begin{cases}f(x) & if x ∈ E, \\ 0 & if x ∈ ℝ^{m} − E. \end{cases}

Definition 7.1. A function f defined on a bounded BV set A is called Rintegrable in A whenever \overline{f} is locally R-integrable. The indefinite R-integral F of \overline{f} is called the indefinite R-integral of f, and the number F (A) is called the R-integral of f over A.

The linear space of all R-integrable functions in a bounded BV set A is denoted by R(A). If f ∈ R(A), we denote by ∫_{A} f dL^{m} the R-integral of f over A.

Proposition 7.2. A function f defined on a bounded BV set A is R-integrable in A whenever there is a g ∈ R_{loc}(ℝ^{m}) such that g ↾ A = f. If G is the indefinite integral of g, then G A is the indefinite integral of f. In particular,

∫_{A} f dL^{m} = ∫_{A} g dL^{m} .

Proof. By Theorem 6.4, the function \overline{f} = gχ_{A} is locally R-integrable and

∫_{B} \overline{f} dL^{m} = ∫_{B} gχ_{A} dL^{m} = ⟨G, χ_{A∩B}⟩ = (G A)(B)

for each bounded BV set B.

According to Proposition 7.2, the indefinite integral F of a function f in R(A) is a charge in A. Since the map

F ↦ F′ ↾ A : CH_{AC_{∗}}(A) → R(A)

is a linear bijection, Proposition 3.4 turns R(A) into a normed space, which is barrelled according to Remark 5.4.

Remark 7.3. As the boundaries of figures are thin sets, it follows from Theorem 5.6, [17, Lemma 3.2], and [15, Theorem 12.2.2], that on figures the Rintegral coincides with the F-integral of [15, Definition 12.2.1].

The term “locally R-integrable,” introduced in Definition 5.5, is justified by the next proposition.

Proposition 7.4. A function f defined on ℝ^{m} is locally R-integrable if and only if f ↾ A is R-integrable in A for each bounded BV set A.

Proof. As the converse follows from Proposition 7.2, assume f ↾ A is R-integra-ble in A for each bounded BV set A. For n = 1, 2, …, denote by F_{n} the indefinite R-integral of f ↾ B[n], and observe F_{n+1} B[n] = F_{n} by Proposition 7.2. Thus there is a charge F such that F B[n] = F_{n} for n = 1, 2, …, and it is easy to verify F′(x) = f(x) for almost all x ∈ ℝ^{m}.

It follows from Proposition 7.2 the R-integral has the usual properties (such as linearity, nonnegativity, and additivity) associated with the word “integral.” Moreover, the common convergence theorems are also valid for the R-integral [13, Lemma 5.13]. By [1, Theorem 5.1], the R-integral is invariant with respect to lipeomorphisms (i.e., bi-Lipschitzian maps).

Theorem 7.5. Let f be an R-integrable function in a bounded BV set B ⊂ ℝ^{m}, and let Ψ : A → B be a lipeomorphism from a set A ⊂ ℝ^{m} onto B. Then A is a bounded BV set, and if J_{Ψ} is the Jacobian of Ψ, then (f ∘ Ψ)J_{Ψ} is R-integrable in A with

∫_{A}(f ∘ Ψ)J_{Ψ} dL^{m} = ∫_{B} f dL^{m} .

Throughout the remainder of this paper, given A ∈ BV, we shall identify the linear spaces BV^{∞}(A) and {g ↾ A : g ∈ BV^{∞}(A)}. In other words, we shall view BV^{∞}(A) as the linear space of all functions g defined on A for which \overline{g} ∈ BV_{c}^{∞}.

The next two propositions follow easily from Theorems 6.4 and 4.11, and Remarks 4.13 and 4.2.

Proposition 7.6. Let A be a bounded BV set. Then MR(A) = BV^{∞}(A), and for each g ∈ BV^{∞}(A), the map

T_{g} : f ↦ ∫_{A} fg dL^{m}

is a continuous linear functional on R(A). The map g ↦ T_{g} is a continuous linear bijection from (BV^{∞}(A), τ) onto the dual space of R(A) equipped with the w^{∗}-topology.

Proposition 7.7. Let A be a bounded BV set, and let k ≥ 1 + 4d(A)^{m−1} be an integer for which A ⊂ B[k]. Let f ∈ R(A) and g ∈ BV^{∞}(A), and let {f_{i}} and {g_{i}} be sequences in R(A) and BV^{∞}(A), respectively, such that

(i) lim(sup{|∫_{A∩B}(f_{i} − f) dL^{m}| : B ∈ BV_{k}}) = 0,

(ii) lim ∥g_{i} − g∥_{1} = 0 and sup(∥g_{i}∥ + ∥g_{i}∥_{∞}) < ∞.

Then lim ∫_{A} f_{i}g_{i} dL^{m} = ∫_{A} fg dL^{m}.

Proposition 7.6 inspires a question related to Question 4.12.

Question 7.8. What is the meaning of the topology σ^{∗} in the dual R(A)^{∗} of R(A) for which

g ↦ T_{g} : (BV^{∞}(A), τ) → (R(A)^{∗}, σ^{∗})

is a homeomorphism?

Observation 7.9. Let E ⊂ ℝ^{m} be a measurable set, x ∈ int_{∗}E, and let B_{j} = B[x, d_{j}] where {d_{j}} is a sequence of positive numbers converging to zero. If A_{j} ⊂ B_{j} are measurable sets and |A_{j}|/|B_{j}| ≥ α > 0 for j = 1, 2, …, then

lim \frac{|E ∩ A_{j}|}{|A_{j}|} = 1 .

Proof.

1 ≥ lim sup \frac{|E ∩ A_{j}|}{|A_{j}|} ≥ lim inf \frac{|E ∩ A_{j}|}{|A_{j}|} ≥ lim inf \frac{|E ∩ B_{j}| − |B_{j} − A_{j}|}{|A_{j}|}

= lim inf [1 − \frac{|B_{j}|}{|A_{j}|} · (1 − \frac{|E ∩ B_{j}|}{|B_{j}|})] ≥ 1 − \frac{1}{α} lim (1 − \frac{|E ∩ B_{j}|}{|B_{j}|}) = 1 .

Lemma 7.10. Let A ∈ BV and ε > η > 0. There is a gage Δ on cl_{∗}A such that given ε-regular Δ-fine partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})}, the partition

{(A ∩ A_{1}, x_{1}), …, (A ∩ A_{p}, x_{p})}

is η-regular.

Proof. Since the lemma is obvious for m = 1, assume m ≥ 2. By [6, Section 2.3, Theorem 1], there is an H^{m−1}-negligible set S ⊂ int_{∗}A such that

lim_{r→0+} \frac{H^{m−1}(B[x, r] ∩ ∂_{∗}A)}{r^{m−1}} = 0

for each x ∈ int_{∗}A − S. Fix an x ∈ int_{∗}A − S, and select BV sets C_{j} so that lim d(C_{j} ∪ {x}) = 0 and r(C_{j} ∪ {x}) ≥ ε for j = 1, 2, . . .. Letting d_{j} = d(C_{j} ∪ {x}) and B_{j} = B[x, d_{j}], observe C_{j} ⊂ B_{j} and

∥A ∩ C_{j}∥ ≤ ∥C_{j}∥ + H^{m−1}(B_{j} ∩ ∂_{∗}A) .

Since r(C_{j} ∪ {x}) ≥ ε, the isoperimetric inequality yields |C_{j}|/d_{j}^{m} ≥ κε^{m} where κ > 0 depends only on the dimension m [6, Section 5.6.2, Theorem 2]. Hence |C_{j}|/|B_{j}| ≥ κ(ε/2)^{m}, and lim(|A ∩ C_{j}|/|C_{j}|) = 1 by Observation 7.9. Now

\frac{d_{j}∥A ∩ C_{j}∥}{|C_{j}|} ≤ \frac{d_{j}∥C_{j}∥}{|C_{j}|} + \frac{d_{j}^{m}}{|C_{j}|} · \frac{H^{m−1}(B_{j} ∩ ∂_{∗}A)}{d_{j}^{m−1}}

≤ \frac{1}{ε} + \frac{1}{κε^{m}} · \frac{H^{m−1}(B_{j} ∩ ∂_{∗}A)}{d_{j}^{m−1}} ,

and we obtain

lim inf r[(A ∩ C_{j}) ∪ \{x\}] ≥ lim inf (\frac{|A ∩ C_{j}|}{|C_{j}|} · \frac{|C_{j}|}{d_{j}∥A ∩ C_{j}∥}) ≥ ε .

It follows there is a Δ_{x} > 0 such that r[(A ∩ C) ∪ {x}] > η for each BV set C with d(C ∪ {x}) < Δ_{x} and r(C ∪ {x}) ≥ ε. As S ∪ ∂_{∗}A is a thin set, extending the function x ↦ Δ_{x} to cl_{∗}A by zero produces the desired gage Δ.

Lemma 7.11. Let F be a charge, and let A be a bounded BV set. Then

[V_{∗}(F A)] cl_{∗}A ≤ (V_{∗}F) cl_{∗}A ,

and (F A)′(x) = F′(x) at almost all x ∈ A at which F′(x) exists.

Proof. Proceeding towards a contradiction, suppose there is an E ⊂ cl_{∗}A with V_{∗}(F A)(E) > V_{∗}F (E), and find an ε > 0 so that

inf_{δ} sup_{P} ∑_{i=1}^{p}|(F A)(A_{i})| > inf_{σ} sup_{Q} ∑_{j=1}^{q}|F (B_{j})|

where δ and σ are gages on E, P = {(A_{1}, x_{1}), …, (A_{p}, x_{p})} is an ε-regular δ-fine partition and Q = {(B_{1}, y_{1}), …, (B_{q}, y_{q})} is an (ε/2)-regular δ-fine partition. Let Δ be a gage on cl_{∗}A associated with η = ε/2 according to Lemma 7.10. There is a gage σ on E with σ ≤ (Δ ↾ E) and an ε-regular σ-fine partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})} such that

∑_{i=1}^{p}|F (A ∩ A_{i})| = ∑_{i=1}^{p}|(F A)(A_{i})| > sup_{Q} ∑_{j=1}^{q}|F (B_{j})|

where Q = {(B_{1}, y_{1}), …, (B_{q}, y_{q})} is an (ε/2)-regular σ-fine partition. In view of Lemma 7.10, this is a contradiction.

Suppose F is derivable at an x ∈ A ∩ (int_{∗}A − null Δ), and select a sequence {B_{j}} of BV sets such that lim d(B_{j} ∪ {x}) = 0 and inf r(B_{i} ∪ {x}) > ε. Lemma 7.10 implies r[(A ∩ B_{j}) ∪ {x}] > ε/2 for all sufficiently large j. Arguing as in the proof of Lemma 7.10, we show lim(|A ∩ B_{j}|/|B_{j}|) = 1, and hence

lim \frac{(F A)(B_{j})}{|B_{j}|} = lim (\frac{F (A ∩ B_{j})}{|A ∩ B_{j}|} · \frac{|A ∩ B_{j}|}{|B_{j}|}) = F′(x) .

Thus (F A)′(x) = F′(x), and it suffices to observe the sets A∩(int_{∗}A−null Δ) and A are equivalent.

Corollary 7.12. Let f be a function defined on an essentially closed set A ∈ BV. Then f ∈ R(A) whenever there is a charge F such that (V_{∗}F) cl_{∗}A is absolutely continuous, and F′(x) = f(x) for almost all x ∈ A. In this case F A is the indefinite R-integral of f.

Proof. Since cl_{∗}A is closed, Lemma 7.11 implies the charge F A is AC_{∗}, and (F A)′(x) = \overline{f}(x) for almost all x ∈ ℝ^{m}.

We omit the proof of the next lemma, which is similar to that of Whitney’s extension theorem presented in [22, Chapter 6, Theorem 3].

Lemma 7.13. Let E ⊂ ℝ^{m} be a closed set, and let v ∈ C(E; ℝ^{m}) be Lipschitz at each point of a set B ⊂ E. Then v can be extended to a w ∈ C(ℝ^{m}; ℝ^{m}) that is still Lipschitz at each x ∈ B.

Let A ∈ BV, and let v ∈ C(cl A; ℝ^{m}) be Lipschitz at almost all x ∈ A. In view of Lemma 7.13, v can be extended to a w ∈ C(ℝ^{m}; ℝ^{m}) that is still Lipschitz at almost all x ∈ A. By Stepanoff’s theorem, w is differentiable almost everywhere in A, and we define a function div v on A, called the divergence of v, by the formula

div v(x) = \begin{cases}div w(x) & if w is differentiable at x, \\ 0 & otherwise. \end{cases}

If F is the flux of w, Lemmas 5.9,(2) and 7.11 imply

div w(x) = F′(x) = (F A)′(x)

for almost all x ∈ A. Since F A depends only on v, and not on the extension w of v, so does the equivalence class of div v; the same conclusion follows also from [15, Lemma 10.5.5].

Theorem 7.14. Let A ∈ BV be essentially closed, let v ∈ C(cl_{∗}A; ℝ^{m}), and let S ⊂ int_{∗}A be a thin set such that v is Lipschitz at each x ∈ int_{∗}A − S. Then div v is R-integrable in A and

∫_{A} div v dL^{m} = ∫_{∂_{∗}A} v · ν_{A} dH^{m−1} .

Proof. Using Lemma 7.13, extend v to a w ∈ C(ℝ^{m}; ℝ^{m}) that is Lipschitz at each x ∈ int_{∗}A − S, and denote by F the flux of w. Since S ∪ ∂_{∗}A is a thin set, Lemmas 5.10 and 5.9,(2) imply (V_{∗}F) cl_{∗}A is absolutely continuous, and F′(x) = div w(x) for almost all x ∈ A. The theorem follows from Corollary 7.12 and the definition of div v.

Remark 7.15. [13, Example 5.21] shows Theorem 7.14 is false without assuming the set A is essentially closed.

Remark 7.16. [15, Example 11.1.2] shows the integrability of partial derivatives of all differentiable functions is incompatible with Fubini’s theorem. In particular, in view of Theorem 7.14, the Fubini theorem is false for the R-integral. This explains in part the complexity of determining the algebra MR_{loc}(ℝ^{m}). Indeed, Theorem 6.4 relies on the following transposition of integrals:

∫_{B} (∫_{0}^{g(x)} f(x) dt) dx = ∫_{B} f(x)g(x) dx = ∫_{0}^{∥g∥_{∞}} (∫_{g(x)>t} f(x) dx) dt

whenever B ∈ BV, f ∈ R_{loc}(ℝ^{m}), and g ∈ BV_{loc}^{∞} is nonnegative.

A Lipschitz domain is a bounded open set Ω ⊂ ℝ^{m} whose boundary ∂Ω is Lipschitz [6, Section 4.2.1]; in particular, Ω is a BV set with int_{∗}Ω = Ω, cl_{∗}Ω = cl Ω, and ∂_{∗}Ω = ∂Ω. If f is a BV function in a Lipschitz domain Ω, then there is a function Tr f ∈ L^{1}(∂Ω, H^{m−1}), called the trace of f on ∂Ω, such that

Tr f(x) = lim_{r→0+} \frac{1}{|B[x, r]|} ∫_{Ω∩B[x,r]} f(y) dy

for H^{m−1}-almost all x ∈ ∂Ω [6, Section 5.3].

Theorem 7.17. Let Ω be a Lipschitz domain, v ∈ C(cl Ω; ℝ^{m}), and let S ⊂ Ω be a thin set such that v is Lipschitz at each x ∈ int Ω − S. Then

∫_{Ω} g div v dL^{m} = ∫_{∂Ω}(Tr g) v · ν_{Ω} dH^{m−1} − ∫_{Ω} v · Dg  (7.1)

for each g ∈ BV^{∞}(Ω).

Proof. There is a sequence {v_{i}} in C^{1}(ℝ^{m}; ℝ^{m}) that converges to v uniformly in cl Ω. By [6, Section 5.3, Theorem 1], equality (7.1) holds for each v_{i}, and the dominated convergence theorem yields

lim ∫_{∂Ω}(Tr g) v_{i} · ν_{Ω} dH^{m−1} = ∫_{∂Ω}(Tr g) v · ν_{Ω} dH^{m−1}

and lim ∫_{Ω} v_{i} · Dg = ∫_{Ω} v · Dg. According to Theorem 7.14,

|∫_{B}(div v_{i} − div v) dL^{m}| = |∫_{∂_{∗}B}(v_{i} − v) · ν_{B} dH^{m−1}|

≤ ∥B∥ sup_{x∈cl Ω}∥v_{i}(x) − v(x)∥

for every bounded BV set B. Thus lim ∫_{Ω} g div v_{i} dL^{m} = ∫_{Ω} g div v dL^{m} for each g ∈ BV^{∞}(Ω) by Proposition 7.7.

8 The R_{∗}-integral

The R-integral introduced in Sections 4 and 6 is an averaging process of appreciable appeal:

• Its definition naturally extends the descriptive definition of the Lebesgue integral (Definition 5.5).

• It has a simple Riemannian description (Theorem 5.6).

• It is invariant with respect to lipeomorphic changes of variables (Theorem 7.5).

• It has an interesting algebra of multipliers (Theorem 6.4).

• Continuous linear functionals on the space of R-integrable functions have the usual integral representation (Proposition 7.6).

• It provides the divergence theorem and integration by parts formula for all differentiable vector fields (Theorems 7.14 and 7.17).

Unfortunately, the R-integral is not adequate for integration over bounded BV sets which are not essentially closed (Remark 7.15). We show, however, that a simple extension of the R-integral inherits all desirable properties and eliminates the shortcomings.

If f is a function defined on a bounded BV set A and F is a charge, denote by R(f, F; A) the family of all B ∈ BV(A) for which f ↾ B belongs to R(B) and F B is the indefinite R-integral of f ↾ B.

Definition 8.1. A function f defined on a BV set A is called R_{∗}-integrable in A if there is a charge F in A such that A ∈ cl_{τ} R(f, F; A). The charge F, uniquely determined by f, is called the indefinite R_{∗}-integral of f, and the number F (A) is called the R_{∗}-integral of f over A.

The linear space of all R_{∗}-integrable functions in a bounded BV set A is denoted by R_{∗}(A). Clearly R(A) ⊂ R_{∗}(A), and arguments similar to those of [13, Sections 8 and 9] show the R_{∗}-integral has the usual “integral properties,” including the invariance with respect to lipeomorphic changes of coordinates (Theorem 5.11). The validity of Propositions 7.6 and 7.7 for the R_{∗}-integral is also easy to verify. With no danger of confusion, for f ∈ R_{∗}(A) we denote by ∫_{A} f dL^{m} the R_{∗}-integral of f over A.

Since each BV set in ℝ is equivalent to a figure, using arguments analogous to those employed in [15, Section 11.5], it is easy to see the R-integral and R_{∗}-integral coincide in dimension one. The next proposition shows that in any dimension, the R_{∗}-integral and R_{∗∗}-integral coincide — in other words, no new integral is produced by iterating Definition 8.1.

Proposition 8.2. Let f be a function defined on a bounded BV set A, and let F be a charge. If R_{∗}(f, F; A) is the family of all B ∈ BV(A) for which f ↾ B belongs to R_{∗}(B) and F B is the indefinite R_{∗}-integral of f ↾ B, then

cl_{τ} R_{∗}(f, F; A) = cl_{τ} R(f, F; A) .

Proof. Since

R_{∗}(f, F; A) ⊂ ⋃{cl_{τ} R(f, F; B) : B ∈ R_{∗}(f, F; A)}

⊂ cl_{τ} (⋃{R(f, F; B) : B ∈ R_{∗}(f, F; A)}) ⊂ cl_{τ} R(f, F; A) ,

we have cl_{τ} R_{∗}(f, F; A) ⊂ cl_{τ} R(f, F; A). As R(f, F; A) ⊂ R_{∗}(f, F; A), the proposition follows.

The reader should compare the following proposition with Corollary 7.12.

Proposition 8.3. Let f be a function defined on any set A ∈ BV. Then f ∈ R_{∗}(A) whenever there is a charge F such that (V_{∗}F) cl_{∗}A is absolutely continuous, and F′(x) = f(x) for almost all x ∈ A. In this case F A is the indefinite R_{∗}-integral of f.

Proof. As in dimension one the proposition coincides with Corollary 7.12, assume m ≥ 2. According to [23], there is a sequence {A_{i}} of essentially closed BV subsets of A such that lim ∥A − A_{i}∥ = 0. By Corollary 7.12, the charge F A_{i} is the indefinite R-integral of f ↾ A_{i}. Thus each A_{i} belongs to R(f, F A; A), and by the isoperimetric inequality, {A_{i}} τ-converges to A.

Lemma 8.4. Let A ∈ BV, and let E ⊂ BV(A) be such that A ∈ cl_{τ} E. There is a sequence {E_{i}} in E such that |A − ⋃_{i=1}^{∞} E_{i}| = 0.

Proof. Let {C_{t} : t ∈ T } be the collection of all countable subfamilies of E, and let {C_{i}} be a sequence in C = ⋃_{t∈T} cl_{τ} C_{t} which τ-converges to C. Given an integer i ≥ 1, find a t_{i} ∈ T with C_{i} ∈ cl_{τ} C_{t_{i}}, and observe ⋃_{i=1}^{∞} C_{t_{i}} = C_{s} for an s ∈ T. Since ⋃_{i=1}^{∞} cl_{τ} C_{t_{i}} ⊂ cl_{τ} C_{s}, the set C belongs to cl_{τ} C_{s}, and hence to cl_{τ} C. As τ is a sequential topology (see Section 2), the family C is τ-closed, and as E ⊂ C, the family C contains A. Thus A ∈ cl_{τ} C_{u} for a u ∈ T, and we can find a sequence {E_{i}} with C_{u} = {E_{1}, E_{2}, … }. If E = ⋃ C_{u}, then

B ={B ∈ BV(A) : |B − E|}

contains C_{u}. Employing again τ is a sequential topology, it is easy to verify B is τ-closed. We conclude A ∈ B, which proves the lemma.

Lemma 8.5. Let A ∈ BV and v ∈ C(cl A; ℝ^{m}). Suppose there is a family E ⊂ BV(A) such that for each E ∈ E, the restriction v ↾ cl E is Lipschitz at almost all x ∈ E. If A ∈ cl_{τ} E, then there is a function f, defined on A and unique up to equivalence, such that given E ∈ E,

f(x) = div (v ↾ cl E)(x)

for almost all x ∈ E. In particular, f(x) = div v(x) for almost all x ∈ A whenever A ∈ E.

Proof. Extend v to a w ∈ C(ℝ^{m}; ℝ^{m}), and denote by F the flux of w. Given E ∈ E, use Lemma 7.13 to extend v ↾ cl E to a w_{E} ∈ C(ℝ^{m}; ℝ^{m}) which is Lipschitz at almost all x ∈ E, and denote by F_{E} the flux of w_{E}. Clearly F E = F_{E} E, and we recall from the paragraph preceding Theorem 7.14 there is a negligible set N_{E} ⊂ E such that at each x ∈ E − N_{E}, the charge F E is derivable and

div (v ↾ cl E)(x) = (F E)′(x) .

For every pair B, C ∈ E,

(F B) C = F (B ∩ C) = (F C) B .

According to Lemma 7.11, we can find a negligible set N_{B,C} ⊂ B ∩ C so that at each x ∈ B ∩ C − N_{B,C}, the charge F (B ∩ C) is derivable and

(F B)′(x) = [(F B) C]′(x) = [(F C) B]′(x) = (F C)′(x) .

By Lemma 8.4, there is a sequence {E_{i}} in E such that N_{0} = A − ⋃_{i=1}^{∞} E_{i} is a negligible set. The set

N = N_{0} ∪ (⋃_{i=1}^{∞} N_{E_{i}}) ∪ ( ⋃_{i,j=1}^{∞} N_{E_{i},E_{j}} )

is negligible, and we define a function f on A by the formula

f(x) = \begin{cases}(F E_{i})′(x) & if x ∈ E_{i} − N, \\ 0 & if x ∈ N. \end{cases}

Our choice of N shows that f is well-defined, and that given E ∈ E,

f(x) = div (v ↾ cl E)(x)

for each x ∈ E − N ∪ N_{E}. Let g be a function on A such that given E ∈ E,

g(x) = div (v ↾ cl E)(x)

for all x ∈ E − Z_{E} where Z_{E} is a negligible subset of E. If Z = ⋃_{i=1}^{∞} Z_{E_{i}}, then g(x) = f(x) for each x ∈ A − N ∪ Z, which establishes the equivalence class of f is unique.

Under the assumptions of Lemma 8.5, we call the function f the divergence of v, still denoted div v. The last claim of Lemma 8.5 shows this is consistent with our previous definition of div v introduced prior to Theorem 7.14.

Theorem 8.6. Let A ∈ BV and v ∈ C(cl A; ℝ^{m}). Suppose there is a family E ⊂ BV(A) with the following property: given E ∈ E, we can find a thin set S_{E} ⊂ int_{∗}E so that v ↾ cl E is Lipschitz at each x ∈ int_{∗}E − S_{E}. If A ∈ cl_{τ} E, then div v is R_{∗}-integrable in A and

∫_{A} div v dL^{m} = ∫_{∂_{∗}A} v · ν_{A} dH^{m−1} .

Proof. Extend v to a w ∈ C(ℝ^{m}; ℝ^{m}), and denote by F the flux of w. Furthermore, for each E ∈ E, extend v ↾ cl E to a w_{E} ∈ C(ℝ^{m}; ℝ^{m}) that is Lipschitz at each x ∈ int_{∗}E − S_{E}, and denote by F_{E} the flux of w_{E}. These extensions exist according to Lemma 7.13, and clearly F E = F_{E} E. The definition of div v, together with Lemmas 5.9,(2) and 7.11, yields

div v(x) = div (v ↾ cl E)(x) = (F_{E} E)(x) = (F E)′(x)

for almost all x ∈ E. As S_{E} ∪ ∂_{∗}E is a thin set, (V_{∗}F_{E}) cl_{∗}E is absolutely continuous by Lemmas 5.10, and Lemma 7.11 implies

[V_{∗}(F E)] cl_{∗}E = [V_{∗}(F_{E} E)] cl_{∗}E

is absolutely continuous. Now it follows from Proposition 8.3 that (div v) ↾ E belongs to R_{∗}(E) and F E is the indefinite R_{∗}-integral of (div v) ↾ E. An application of Proposition 8.2 completes the proof.

Remark 8.7. By [2, Theorem 5.3] and [13, Proposition 7.7], each R-integrable function is variationally integrable according to [13, Definition 5.1]. On the other hand, Propositions 8.3 and 5.2,(2), Lemma 7.11, [2, Theorem 5.3], and [13, Proposition 7.7] imply each variationally integrable function is R_{∗}-integrable. Thus using Proposition 8.2, we conclude the R_{∗}-integral coincides with the continuous integral of [13, Definition 9.1].

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