MULTIPLIERS FOR GENERALIZED RIEMANN INTEGRALS

Z. BUCZOLICH, T. DE PAUW, AND W.F. PFEFFER

Abstract. La présente note traite d’une intégrale de Riemann généralisée dans l’espace euclidien ℝ^{m}. Les résultats suivants sont annoncés. (1) Une fonction g est un multiplicateur (c’est-à-dire que fg est intégrable dès que f l’est) si et seulement si g est bornée et à variation bornée. (2) Le dual de l’espace des fonctions intégrables dans un ensemble A à variation bornée est isomorphe à l’espace des fonctions bornées à variation bornée s’annulant hors de A. En outre, chaque élément du dual est représenté par une formule intégrale usuelle. (3) Une formule d’intégration par parties est valide sur des domaines lipschitziens et pour des champs continus qui sont ponctuellement lipschitziens partout sauf éventuellement sur un ensemble de mesure de Hausdorff H^{m−1} σ-finie.

In dimension one, the problem of recovering derivative by integration has been solved a long time ago by Denjoy and Perron [8, 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 [6, 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 below, and as the continuous integral in [5, 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.

1. Preliminaries

The set of all real numbers is denoted by ℝ, and the ambient space of this note is ℝ^{m} where m ≥ 1 is a fixed integer. By x · y we denote the usual inner product of x, y ∈ ℝ^{m}, and let |x| = \sqrt{x · x}. The closed ball of radius r > 0 about the origin of ℝ^{m} is denoted by B_{r}. The symbols cl E, ∂E, and d(A) denote, respectively, the closure, boundary, and diameter 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). The essential closure cl_{∗}E and essential boundary ∂_{∗}E of a measurable set E ⊂ ℝ^{m} is, respectively, the set of all nondispersion points of E and all nondispersion points of E that are not density points of E. We say E is essentially closed whenever cl_{∗}E is a closed set. A thin set is a set S ⊂ ℝ^{m} whose (m − 1)-dimensional Hausdorff measure H^{m−1} is σ-finite.

The first author was supported in part by the Hungarian National Foundation of Scientific Research, Grant No. T019476 and FKFP 0189/1997.

The linear space of all locally bounded locally BV functions in ℝ^{m} is denoted by BV_{loc}^{∞}, and the linear space of all bounded BV functions in ℝ^{m} with compact support is denoted by BV_{c}^{∞}. If g ∈ BV_{c}^{∞}, we denote by Dg and ∥g∥ the distributional gradient and total variation of g, respectively. For n = 1, 2, …, topologize the family

BV_{n} = {g ∈ BV_{c}^{∞} : supp g ⊂ B_{n} and ∥g∥ + |g|_{∞} ≤ n + 1}

by the L^{1} norm |·|_{1}. Each BV_{n} is a compact set, and 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 and sequentially complete. We view the family BV of all bounded BV subsets of ℝ^{m} as a closed subspace of (BV_{c}^{∞}, τ). The perimeter and unit exterior normal of E ∈ BV are denoted by ∥E∥ and ν_{E}, respectively. Moreover, we let

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

2. Charges

A charge is an additive τ-continuous function defined on BV . 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 [6, Proposition 11.2.8]. It is called the flux of v, denoted by ∫ v · ν dH^{m−1}.

For a charge F and n = 1, 2, …, let

∥F ∥_{n} = sup{|F (B)| : B ∈ BV ∩ BV_{n}} ,

and observe {∥·∥_{n}} is an increasing sequence of seminorms, which defines a Fréchet topology σ in CH. If A ∈ BV , then ∥F ∥_{n} are equivalent Banach norms on CH(A) for all sufficiently large n.

Proposition 2.1. Let F be a charge, and let E be a bounded BV subset of ℝ^{m+1}. For L^{1}-almost all t ∈ ℝ, the section E^{t} = {x ∈ ℝ^{m} : (x, t) ∈ E} belongs to BV , and the function t ↦ F (E^{t}) 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).

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 the uniform convergence on each BV_{n}. For a nonnegative function g ∈ BV_{c}^{∞},

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

is a bounded BV subset of ℝ^{m+1}. In view of Proposition 2.1, 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}^{∞}.

Proposition 2.2. The pairing (F, g) ↦ ⟨F, g⟩ is a nondegenerate separately continuous bilinear functional on (CH × BV_{c}^{∞}, σ × τ).

Theorem 2.3. 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 ↦ ⟨F, ·⟩.

3. The R-integral

A charge F is derivable at x ∈ ℝ^{m} if lim[F (B_{n})/|B_{n}|] ≠ ±∞ exists for each {B_{n}} in BV with lim d(B_{n} ∪ {x}) = 0 and inf r(B_{n} ∪ {x}) > 0. If all these limits exist, they have the same value called the derivate of F at x, denoted by F′(x).

A charge F is called 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. It is well-known each AC charge F is derivable almost everywhere with F′ ∈ L_{loc}^{1}(ℝ^{m}) and ∫ F′ dL^{m} = F.

A nonnegative function δ defined on a set E ⊂ ℝ^{m} is called a gage on E whenever its null set is thin. 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_{1}, …, x_{p} are points of ℝ^{m}. Given ε > 0 and a gage δ on E ⊂ ℝ^{m}, the partition P is called

• 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. The critical 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 partition. 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 compact interval. The following proposition was proved in [2, Proposition 3.3].

Proposition 3.1. A charge F is AC if and only if V_{∗}F is absolutely continuous and locally finite.

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. While neither of the spaces (CH_{AC_{∗}}, σ) and (CH_{AC_{∗}}(A), σ) is complete, both are barrelled [3, Theorem 4.1]. It turns out many important properties of AC charges remain valid for AC_{∗} charges.

Definition 3.2. 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 denote F by ∫ f dL^{m}, and F (A) by ∫_{A} f dL^{m} for each bounded BV set A. The next theorem, proved in [2, Theorem 4.6], shows the R-integral is akin to the generalized Riemann integrals of [1, Definition 5.1].

Theorem 3.3. 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 partition {(A_{1}, x_{1}), …, (A_{p}, x_{p})}. In this case F is the indefinite R-integral of f.

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 the family X is denoted by MX. It follows from [7, Proposition 3.1] that MR_{loc}(ℝ^{m}) ⊂ BV_{loc}^{∞}.

Theorem 3.4. 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}^{∞}.

Theorem 3.4 is proved by showing G : B ↦ ⟨F, gχ_{B}⟩ is an AC_{∗} charge such that G′ = F′g almost everywhere.

For a function f defined on a set E ⊂ ℝ^{m}, let \overline{f} be the extension of f to ℝ^{m} by zero. If E ∈ BV , we denote by BV^{∞}(E) the family of all functions g defined on E for which \overline{g} belongs to BV_{c}^{∞}.

Definition 3.5. A function f defined on a bounded BV set A is called R-integrable 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). The indefinite integral F of a function f ∈ R(A) is a charge in A, and the map

F ↦ F′ ↾ A : CH_{AC_{∗}}(A) → R(A)

is a linear bijection, which induces in R(A) a normed barrelled topology. If f ∈ R(A), we denote by ∫_{A} f dL^{m} the R-integral of f over A. The R-integral has the usual properties associated with the word “integral,” including the common convergence theorems. Moreover, the R-integral is invariant with respect to lipeomorphisms (i.e., bi-Lipschitzian maps). The term “locally R-integrable,” introduced in Definition 3.2, is now justified: 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.

Theorem 3.6. 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.

One of the principal reasons for introducing the R-integral is the divergence theorem for discontinuously differentiable vector fields. If E ⊂ ℝ^{m}, we say a vector field v : E → ℝ^{m} 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 with |y − x| < r. Let E ⊂ ℝ^{m} be a closed set, and let v ∈ C(E; ℝ^{m}) be Lipschitz at almost all x ∈ E. In view of Whitney’s and Stepanoff’s theorems, div v is uniquely defined almost everywhere in E.

Theorem 3.7. Let A be a bounded BV set that is essentially closed, let v ∈ C(cl_{∗}A; ℝ^{m}), and let S ⊂ cl_{∗}A be a thin set such that v is Lipschitz at each x ∈ cl_{∗}A − S. Then div v is R-integrable in A and

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

If Ω is a Lipschitz domain [4, Section 4.2.1] and f is a BV function in Ω, we can define the trace of f, which is a function in L^{1}(∂Ω, H^{m−1}), denoted by Tr f.

Theorem 3.8. 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 ∈ Ω − S. Then

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

for each g ∈ BV^{∞}(Ω).

4. The R_{∗}-integral

While the R-integral introduced in Sections 3 is an averaging process of appreciable appeal, it is not adequate for integration over arbitrary bounded BV sets: Theorem 3.7 is false when A is not essentially closed [5, Example 5.21]. Fortunately, a simple extension of the R-integral eliminates this shortcoming and retains all desirable properties of the R-integral.

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. The closure operator in the space (BV , τ) is denoted by cl_{τ}.

Definition 4.1. 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 [5, Sections 8 and 9] show the R_{∗}-integral has the usual “integral properties,” including the invariance with respect to lipeomorphic changes of coordinates. The validity of Theorems 3.6 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. As τ is a sequential topology, it is easy to see no new integral is produced by a further iteration of Definition 4.1.

Lemma 4.2. 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.

Under the assumptions of Lemma 4.2, we call the function f the divergence of v, still denoted by div v. The last claim of Lemma 4.2 shows this is consistent with our previous definition of div v introduced prior to Theorem 3.7.

Theorem 4.3. Let A be any bounded BV set, and let 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} ⊂ cl_{∗}E so that v ↾ cl E is Lipschitz at each x ∈ cl_{∗}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} .

References

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[3] T. De Pauw, Topologies for the space of BV -integrable functions in ℝ^{N}, J. Func. Anal., 144 (1997), 190–231.

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Department of Analysis, Eötvös Loránd University, Múzeum krt. 6–8, H–1088 Budapest, Hungary

   E-mail address: buczo@ludens.elte.hu

Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK

   E-mail address: thierry@math.ucl.ac.uk

Department of Mathematics, University of California, Davis, CA 95616, USA

   E-mail address: wfpfeffer@ucdavis.edu