Some Remarks on Capacitary Integrals and Measure Theory

TL;DR

This paper explores Choquet integrals with minimal assumptions, establishing key properties like sublinearity and convergence theorems, thereby advancing measure theory for non-additive set functions.

Contribution

It proves equivalence of sublinearity and strong subadditivity without regularity assumptions and extends classical measure convergence theorems to non-additive integrals.

Findings

Sublinearity is equivalent to strong subadditivity for Choquet integrals.

Standard measure convergence theorems hold for non-additive integrals under minimal assumptions.

Results generalize classical measure theory to non-additive set functions.

Abstract

We present results for Choquet integrals with minimal assumptions on the monotone set function through which they are defined. They include the equivalence of sublinearity and strong subadditivity independent of regularity assumptions on the capacity, as well as various forms of standard measure theoretic convergence theorems for these non-additive integrals, e.g. Fatou's lemma and Lebesgue's dominated convergence theorem.

Equations

$$\displaystyle H\Bigl{(}\bigcup\limits_{n=0}^{\infty}E_{n}\Bigr{)}$$

$$\displaystyle H\Bigl{(}\bigcap\limits_{n=0}^{\infty}K_{n}\Bigr{)}=\lim_{n\to\infty}H(K_{n});$$

$$\displaystyle H\Bigl{(}\bigcup\limits_{n=0}^{\infty}E_{n}\Bigr{)}=\lim_{n\to\infty}H(E_{n}).$$

$$\int f\mathrm{d}H:={\int }_0^{\infty }H(\{f>t\})\mathrm{d}t,$$

$$t\in (0,\infty )⟼H(\{f>t\}).$$

$$H(E\cap F)+H(E\cup F)\le H(E)+H(F).$$

$$\int \underset{n\to \infty }{\mathrm{liminf}}{f}_n\mathrm{d}H\le \underset{n\to \infty }{\mathrm{liminf}}\int f_n\mathrm{d}H,$$

$$\int \sum _{n=0}^{\infty }{f}_n\mathrm{d}H\le \sum _{n=0}^{\infty }\int f_n\mathrm{d}H,$$

$$\int \mathcal{M}f\mathrm{d}H\le \underset{n\to \infty }{\mathrm{liminf}}\int \mathcal{M}{f}_n\mathrm{d}H,$$

$$\mathcal{M}f(x):=\underset{r>0}{\sup }\frac{1}{r^d}{\int }_{B_r(x)}|f(y)|\mathrm{d}y$$

$$H(A):=\{\begin{array}{ll}0& \text{if A is finite,}\\ 1& \text{if A is infinite.}\end{array}$$

$${\tilde{\mathcal{H}}}_{\infty }^{\beta }(E):=\inf \left\{\sum _{n=0}^{\infty }\ell (Q_n{)}^{\beta }:Q_n\text{ is a dyadic cube and }E\subset \bigcup _{n=0}^{\infty }{Q}_n\right\},$$

$$f_n\to f\text{q.u.}$$

$$H(E\cup F)\le H(E)+H(F).$$

$$\int f\mathrm{d}H\le \underset{n\to \infty }{\mathrm{liminf}}\int f_n\mathrm{d}H.$$

$$|f_n-f|\le \eta \text{in Rd∖E.}$$

$$f\le f_n+|f_n-f|\le f_n+\eta \text{in Rd∖E.}$$

$$\{f>t+\eta \}\setminus E\subset \{f_n>t\}$$

$$\{f>t+\eta \}\subset \{f_n>t\}\cup E.$$

$$H(\{f>t+\eta \})\le H(\{f_n>t\})+H(E).$$

$$\begin{array}{rl}{\int }_{\eta }^{k+\eta }H(\{f>s\})\mathrm{d}s& ={\int }_0^{k}H(\{f>t+\eta \})\mathrm{d}t\\ & \le {\int }_0^{k}H(\{f_n>t\})\mathrm{d}t+H(E)k\le \int f_n\mathrm{d}H+ϵk.\end{array}$$

$${\int }_{\eta }^{k+\eta }H(\{f>s\})\mathrm{d}s\le \underset{n\to \infty }{\mathrm{liminf}}\int f_n\mathrm{d}H+ϵk.$$

$${\int }_{\eta }^{k+\eta }H(\{f>s\})\mathrm{d}s\le \underset{n\to \infty }{\mathrm{liminf}}\int f_n\mathrm{d}H.$$

$$\int f\mathrm{d}H={\int }_0^{\infty }H(\{f>s\})\mathrm{d}s\le \underset{n\to \infty }{\mathrm{liminf}}\int f_n\mathrm{d}H.\text{qed}$$

$$\underset{n\to \infty }{\lim }\int |f_n-f|\mathrm{d}H=0.$$

$$\int |f|\mathrm{d}H\le \underset{n\to \infty }{\mathrm{liminf}}\int |f_n|\mathrm{d}H\le \int F\mathrm{d}H<\infty .$$

$$\{|f_n-f|>t\}\subset E,$$

$$H(\{|f_n-f|>t\})\le H(E)\le ϵ.$$

$$\{|f_n-f|>t\}\subset \{F+|f|\ge t\}.$$

$$\begin{array}{rl}\int |f_n-f|\mathrm{d}H& ={\int }_0^{\eta }+{\int }_{\eta }^k+{\int }_k^{\infty }H(\{|f_n-f|>t\})\mathrm{d}t\\ & \le {\int }_0^{\eta }H(\{F+|f|>t\})\mathrm{d}t+kϵ+{\int }_k^{\infty }H(\{F+|f|>t\})\mathrm{d}t.\end{array}$$

Full text

D. Spector is supported by the Taiwan Ministry of Science and Technology under research grant number 110-2115-M-003-020-MY3 and the Taiwan Ministry of Education under the Yushan Fellow Program.

1. 1 Introduction
2. 2 Quasi-uniform convergence
3. 3 Approximation of quasicontinuous functions
4. 4 Sublinearity of the Choquet integral
5. 5 Lower semicontinuity of the Choquet integral
6. 6 The space $L^{1}(H)$

1. Introduction

The oeuvre of David R. Adams has had a profound influence on the study of Sobolev inequalities, including results early in his career on trace inequalities [A1, A7, A11], numerous papers over the years concerning potentials [A4, A5, A8, A9, A12, A13, A19-a], and of special interest in this paper, his body of work on capacities and Choquet integrals [A2, A3, A10, A14, A16, A17, A18, A19, A23-1998, A24, A25, A26]. That one should be interested in the study of Choquet integration is clear from the consideration of strong forms of the Sobolev inequality, namely V. Maz’ya’s capacitary inequalities [Mazya1, Mazya1a, Mazya2] and their various extensions [A14, Dahlberg:1979, A19, C6, Ponce-Spector:2018, Ponce-Spector:2020], as it is precisely in these improvements to typical Lebesgue or Lorentz inequalities that these integrals make an appearance. These inequalities give usual compactness results, though are strong enough even to provide information about fine properties of functions, and therefore motivate the need for as robust as possible of a framework of Choquet integration which contains these main capacitary inequalities as examples. The work we reference of D. R. Adams provides a number of results in this direction, most notably his survey [A23-1998].

The starting place of D. R. Adams is the treatise of G. Choquet [Choquet:1954], who developed a theory of integration with respect to monotone, countably subadditive set functions with additional regularity assumptions: We say that $H:\mathcal{P}(\mathbb{R}^{d})\to[0,\infty]$, defined on the class $\mathcal{P}(\mathbb{R}^{d})$ of all subsets of $\mathbb{R}^{d}$, is a capacity in the sense of Choquet whenever it satisfies the conditions

empty set:

$H(\emptyset)=0$;

monotonicity:

If $E\subset F\subset\mathbb{R}^{d}$, then $H(E)\leq H(F)$;

countable subadditivity:

For every sequence of sets $E_{n}\subset\mathbb{R}^{d}$,

[TABLE]

outer regularity:

For every non-increasing sequence of compact subsets $K_{n}\subset\mathbb{R}^{d}$,

[TABLE]

inner regularity:

For every nondecreasing sequence of sets $E_{n}\subset\mathbb{R}^{d}$,

[TABLE]

Given a set function $H:\mathcal{P}(\mathbb{R}^{d})\to[0,\infty]$ that merely satisfies monotonicity: , one can define the Choquet integral with respect to $H$ of any function $f:\mathbb{R}^{d}\to[0,\infty]$ as

[TABLE]

where the right-hand side is understood as the Lebesgue integral of the non-increasing function

[TABLE]

Such an integral has a number of desirable properties, for example (1.1) is positively $1$-homogeneous and monotone. Moreover, one may replace the sets $\{f>t\}$ with $\{f\geq t\}$ and obtain the same value for the integral. However, one consequence of the choice to integrate outside the framework of Measure Theory is that this integral need not be linear, and in fact may not even be sublinear. Indeed, G. Choquet [Choquet:1954]*54.2 on p. 289 established a necessary and sufficient condition on $H$ that the integral be sublinear:

Theorem 1.1**.**

Let $H$ be a capacity in the sense of Choquet. Then, the Choquet integral (1.1) is sublinear if and only if $H$ is strongly subadditive.

Here we recall the notion of

strong subadditivity:

For every sets $E,F\subset\mathbb{R}^{d}$,

[TABLE]

The sublinearity of the integral implies one has a triangle inequality, from which Hölder’s and Minkowski’s inequalities follow from usual convexity arguments. These inequalities in turn serve as a basis for the study of a family of Banach spaces of functions $L^{p}(H)$, those suitably regular functions whose $p$th power has finite Choquet integral. Here typical questions have concerned the boundedness of maximal functions [A19, C8, STW], characterizations of the topological duals [A19, C10], and interpolation theory [CCM, CJ, CMS]. The assumption one has a capacity in the sense of Choquet ensures that even without the full strength of results from Measure Theory one has a number of useful tools, e.g. Fatou’s lemma (which follows from monotonicity: and inner regularity: , see [CMS]*Theorem 1 on pp. 98–99):

[TABLE]

for every sequence of functions $f_{n}:\mathbb{R}^{d}\to[0,\infty]$. As a result the Choquet integral built on a strongly subadditive capacity in the sense of Choquet enjoys countable sublinearity: For every sequence of functions $f_{n}:\mathbb{R}^{d}\to[0,\infty]$ one has

[TABLE]

and a Fatou-type lemma that is often appealed to (see e.g. the argument on p. 123 of [A19]: If $f_{n}\to f$ locally in $L^{1}(\mathbb{R}^{d})$ (or pointwise almost everywhere), then

[TABLE]

where

[TABLE]

is the Hardy-Littlewood maximal function. These results are a small sample of the theory of Choquet integration developed and recorded for capacities in the sense of Choquet, and we refer the reader to [A19, Choquet:1954, CMS] for further details.

Unfortunately, in practice the capacities that arise in various inequalities [Mazya1, Mazya1a, Mazya2, A14, Dahlberg:1979, A19, C6] may fail to satisfy inner regularity: or outer regularity: , a notable example being the Hausdorff content or its dyadic version, see [HH23] and Example 1.4 below. It is natural then that one address the necessity of these regularity assumptions in the resulting theory of Choquet integration. This program was initiated by Adams, who typically did not require that the set functions under consideration be capacities in the sense of Choquet, often with the initial assumptions of only empty set: , monotonicity: , and countable subadditivity: . He referred to these objects as capacities in the sense of N. Meyers, though the reader may also recognize these are the defining properties of an outer measure. To these he then added a continuity assumption, outer regularity: or inner regularity: , either of which is sufficient to obtain Choquet’s characterization of sublinearity of the integral (see Anger’s paper [Anger:1977] for the proof assuming outer regularity: or Saito, Tanaka, and Watanabe’s paper [STW]*Proposition 3.2 for the proof assuming inner regularity: ). The first observation of this paper is that neither assumption is necessary, that one has the following characterization independent of regularity assumptions.

Theorem 1.2**.**

Suppose that $H$ satisfies monotonicity: . Then, the Choquet integral (1.1) is sublinear if and only if $H$ is strongly subadditive.

Here is a simple example for which Theorem 1.2 applies but not Theorem 1.1:

Example 1.3**.**

Let $H$ be the set function defined by

[TABLE]

Then, $H$ is strongly subadditive and satisfies empty set: and monotonicity: , but does not satisfy countable subadditivity: , outer regularity: or inner regularity: .

A second pertinent example is the dyadic Hausdorff content for which Yang and Yuan [Yang-Yuan:2008] observed the following

Example 1.4**.**

Let $0<\beta<d$ and let $H=\tilde{\mathcal{H}}^{\beta}_{\infty}$ be the set function defined by

[TABLE]

where $\ell(Q_{n})$ denotes the side-length of $Q_{n}$. Then, $H$ is strongly subadditive and satisfies empty set: , monotonicity: , countable subadditivity: , and inner regularity: , but not outer regularity: for $\beta\leq d-1$.

The initial impetus for this work was the question of the validity of Theorem 1.2, though after obtaining a proof we discovered in our broader literature review that this was known to the community of non-additive Measure Theory [Denneberg]*Chapter 6. As it seems to have not been referenced in the results after Adams, we give the proof below for the convenience of the reader, the idea of which is as follows: First, one proves an algebraic result, which amounts to sublinearity for finite sums of characteristic functions (see e.g., the argument at the top of p. 249 of [Anger:1977], the argument on pp. 766–768 of [STW], or Proposition 4.2 below); Second, one argues the general case by approximation. In the papers [Anger:1977, STW] this is performed invoking either outer regularity: or inner regularity: to justify the limit, though as we show below it can be done by using only monotonicity: and properties of the Lebesgue integral in (1.1).

That the Choquet integral is sublinear without any regularity assumptions is perhaps surprising to the community working on capacitiary inequalities in the spirit of Adams, e.g. [Cap1, Cap2, C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, STW, Ponce-Spector:2018, Ponce-Spector:2020, Spector:PM, Chen-Spector], and suggests that it should be interesting to understand what other aspects of the theory of Choquet integration relies on these regularity assumptions and in what areas it can be dispensed. In this paper we take up this question as pertains to analogues of measure theoretic results, in particular Fatou’s lemma and Lebesgue’s dominated convergence theorem, as well as functional analysis results concerning spaces of functions with finite capacitary integral. Our results show that while one does not need regularity of the capacity, in one way or another regularity must make an appearance in order to obtain such results. In particular, if one assumes regularity on any of the capacity, the mode of convergence, or the functions involved, then it is possible to obtain analogues of (1.2) and (1.3), see Figure 1.

The plan of the paper is to make precise the sketch presented in Figure 1, as well as to meticulously develop the assumptions that lead to various results of the measure theoretic and functional analytic aspects of Choquet integration. Our results are roughly organized in terms of increasing assumptions on the capacity as one proceeds through the paper. With this framework, in Section 2, we rely on a strong form of pointwise convergence with respect to $H$, namely quasi-uniform convergence, which is sufficient to obtain versions of Fatou’s lemma and Lebesgue’s dominated convergence theorem for Choquet integrals with minimal assumptions on the capacity. Conversely, we show that convergence of a sequence of functions with respect to the Choquet integral implies that the sequence has this strong convergence property, the idea of which follows the typical completeness argument for the functional space $L^{1}(H)$. In Section 3, we recall the notion of quasicontinuity. When one imposes certain additional conditions on $H$, we show that quasicontinuous functions admit approximation with respect to the Choquet integral by functions in $C_{c}(\mathbb{R}^{d})$, the class of continuous functions that are compactly supported in $\mathbb{R}^{d}$. In Section 4, we prove Theorem 1.2 on the sublinearity of the Choquet integral. In Section 5 we show how one can relax the notion of convergence provided one works within the class of quasicontinuous functions. This relies on a classical application of the Hahn-Banach Theorem that realizes the Choquet integral as a supremum over Lebesgue integrals with respect to locally finite measures underneath the capacity. In Section 6, we introduce the Banach space $L^{1}(H)$ as the set of equivalence classes of quasicontinuous functions for which the Choquet integral is finite. The results in the preceding sections are then shown to imply standard results concerning this space such as completeness, density of $C_{c}(\mathbb{R}^{d})$, after which we discuss Fatou’s lemma in this context, namely we provide closure properties that guarantee that the limit of a sequence of functions remains in $L^{1}(H)$. For ease of reference we provide proofs of all of our assertions. This paper is an homage to David R. Adams’ work, for which we are extremely grateful, and is dedicated to his memory.

2. Quasi-uniform convergence

In this section, we present some convergence properties for the Choquet integral with respect to a monotone set function $H:\mathcal{P}(\mathbb{R}^{d})\to[0,\infty]$ under quasi-uniform convergence of the integrand.

Definition 2.1**.**

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of functions $f_{n}:\mathbb{R}^{d}\to[-\infty,\infty]$. We say that $(f_{n})_{n\in\mathbb{N}}$ converges quasi-uniformly to $f:\mathbb{R}^{d}\to[-\infty,\infty]$ whenever, for each $\epsilon>0$, there exists $E\subset\mathbb{R}^{d}$ such that $H(E)\leq\epsilon$ and $(f_{n})_{n\in\mathbb{N}}$ is finite and converges uniformly to $f$ in $\mathbb{R}^{d}\setminus E$. We denote

[TABLE]

For simplicity, we omit the dependence of $H$. In the case where $H$ is a measure, almost everywhere pointwise convergence implies quasi-uniform convergence on sets of finite measure. However, this is not true for capacities in general:

Example 2.2**.**

Take a capacity $H$ such that $H(\partial B_{r})\geq\eta$ for every $1<r<2$ and some fixed $\eta>0$, where $B_{r}\vcentcolon=B_{r}(0)$ is the ball of radius $r$ centered at [math]. Such is the case with the Hausdorff content $\mathcal{H}^{\beta}_{\infty}$ with $\beta\leq d-1$ or the Sobolev capacity $\operatorname{Cap}_{1,p}$ for $1\leq p<d$, see [AH, P] for their definitions and further properties. If $(f_{n})_{n\in\mathbb{N}_{*}}$ is any sequence of functions in $\mathbb{R}^{d}$ such that $f_{n}\geq 1$ on $\partial B_{1+2/n}$ with $f_{n}$ supported in $B_{1+3/n}\setminus B_{1+1/n}$, then $f_{n}\to 0$ pointwise in $\mathbb{R}^{d}$, but not quasi-uniformly since $H(\partial B_{1+2/n})\geq\eta$ for every $n\in\mathbb{N}_{*}$.

In this section we rely mostly on subadditivity involving finitely many sets:

finite subadditivity:

For every $E,F\subset\mathbb{R}^{d}$,

[TABLE]

The following version of Fatou’s lemma for quasi-uniform convergence then holds:

Proposition 2.3**.**

Suppose that $H$ satisfies monotonicity: and finite subadditivity: . If $(f_{n})_{n\in\mathbb{N}}$ is a sequence of nonnegative functions in $\mathbb{R}^{d}$ such that $f_{n}\to f$ q.u., then

[TABLE]

Proof.

Given $\epsilon>0$, there exists $E\subset\mathbb{R}^{d}$ such that $H(E)\leq\epsilon$ and $f_{n}\to f$ uniformly in $\mathbb{R}^{d}\setminus E$. Given $\eta>0$, take $N\in\mathbb{N}$ such that, for $n\geq N$,

[TABLE]

We then have

[TABLE]

Hence, for every $t>0$,

[TABLE]

which implies that

[TABLE]

By monotonicity and finite subadditivity of $H$, we get

[TABLE]

Fix $k\in\mathbb{N}$. Integrating with respect to $t$ over the interval $(0,k)$,

[TABLE]

This estimate holds for every $n\geq N$. Letting $n\to\infty$, we get

[TABLE]

As $\epsilon\to 0$, we deduce that

[TABLE]

To conclude we let $k\to\infty$ and $\eta\to 0$. We then get by Fatou’s lemma for the Lebesgue measure,

[TABLE]

We now show the following version of the Dominated Convergence Theorem:

Proposition 2.4**.**

Suppose that $H$ satisfies monotonicity: and finite subadditivity: . If $(f_{n})_{n\in\mathbb{N}}$ is a sequence of real-valued functions in $\mathbb{R}^{d}$ such that $f_{n}\to f$ q.u. and if there exists $F:\mathbb{R}^{d}\to[0,\infty]$ such that $\int F\,\mathrm{d}H<\infty$ and $|f_{n}|\leq F$ in $\mathbb{R}^{d}$ for every $n\in\mathbb{N}$, then $\int|f|\,\mathrm{d}H<\infty$ and

[TABLE]

Proof.

Since $|f_{n}|\to|f|$ q.u. and $|f_{n}|\leq F$ in $\mathbb{R}^{d}$, by Proposition 2.3 and by monotonicity of the Choquet integral we have

[TABLE]

Given $\epsilon>0$, take $E\subset\mathbb{R}^{d}$ with $H(E)\leq\epsilon$ such that $(f_{n})_{n\in\mathbb{N}}$ converges uniformly to $f$ in $\mathbb{R}^{d}\setminus E$. Then, given $\eta>0$, let $N\in\mathbb{N}$ be such that, for every $n\geq N$, $|f_{n}-f|\leq\eta$ in $\mathbb{R}^{d}\setminus E$. Thus, for $t>\eta$ and $n\geq N$, we have

[TABLE]

whence, by monotonicity of $H$,

[TABLE]

Also, for any $t>0$,

[TABLE]

It follows from (2.1) and (2.2) that, for any $k>\eta$ and $n\geq N$,

[TABLE]

As $n\to\infty$,

[TABLE]

By finite subadditivity of $H$,

[TABLE]

Since both $F$ and $|f|$ have finite Choquet integrals, we have the conclusion by letting $\epsilon,\eta\to 0$ and then $k\to\infty$ in (2.3). ∎

The proof of the partial converse of the Dominated Convergence Theorem involves countable subadditivity:

Proposition 2.5**.**

Suppose that $H$ satisfies monotonicity: and countable subadditivity: . Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of real-valued functions in $\mathbb{R}^{d}$ such that

[TABLE]

where $f:\mathbb{R}^{d}\to\mathbb{R}$ satisfies $\int|f|\,\mathrm{d}H<\infty$. Then, $f_{n}\to f$ q.u. and there exists $F:\mathbb{R}^{d}\to[0,\infty]$ such that $\int F\,\mathrm{d}H<\infty$ and $|f_{n}|\leq F$ in $\mathbb{R}^{d}$ for every $n\in\mathbb{N}$.

Proof.

By monotonicity of $H$ one has an analogue of Chebyshev’s inequality,

[TABLE]

For each $k\in\mathbb{N}$, denoting

[TABLE]

then, by countable subadditivity of $H$ and (2.4),

[TABLE]

Since the sequence $(1/2^{n})_{n\in\mathbb{N}}$ is summable, by the Weierstrass M-test the series $\sum\limits_{n=k}^{\infty}{|f_{n}-f|}$ converges uniformly in $\mathbb{R}^{d}\setminus A_{k}$ for every $k\in\mathbb{N}$, and then so does the sequence $(f_{n})_{n\in\mathbb{N}}$. We deduce from (2.5) that $(f_{n})_{n\in\mathbb{N}}$ converges quasi-uniformly to $f$ in $\mathbb{R}^{d}$.

To conclude, it suffices to verify that $F\vcentcolon=|f|+\sum\limits_{n=0}^{\infty}{|f_{n}-f|}$ has finite Choquet integral. We first observe that, by quasi-sublinearity of the Choquet integral, one has that for every two functions $g$ and $h$,

[TABLE]

Iterating this inequality, for every $j\in\mathbb{N}$ one gets

[TABLE]

By quasi-sublinearity of the Choquet integral and quasi-uniform convergence of the series $\sum\limits_{n=0}^{\infty}{|f_{n}-f|}$, we deduce from Proposition 2.3 that

[TABLE]

The need for countable subadditivity: in the statement of Proposition 2.5 can be seen in the following

Example 2.6**.**

Take a sequence $(a_{n})_{n\in\mathbb{N}}$ of distinct points in $\mathbb{R}^{d}$ and, for each $n\in\mathbb{N}$, let $f_{n}(a_{n})=1$ and $f_{n}(x)=0$ for $x\neq a_{n}$. If $H$ is the capacity given by Example 1.3, then $\int|f_{n}-0|\,\mathrm{d}H=H(\{a_{n}\})=0$ for each $n$ but $(f_{n})_{n\in\mathbb{N}}$ does not converge q.u. to [math] since, for every $j\in\mathbb{N}$, we have H\bigl{(}\bigcup\limits_{n\geq j}\{a_{n}\}\bigr{)}=1.

3. Approximation of quasicontinuous functions

We investigate in this section the question of approximation of quasicontinuous functions with finite Choquet integral by sequences of continuous functions.

Definition 3.1**.**

A function $f:\mathbb{R}^{d}\to[-\infty,\infty]$ is quasicontinuous whenever, for each $\epsilon>0$, there exists an open set $\omega\subset\mathbb{R}^{d}$ such that $H(\omega)\leq\epsilon$ and $f|_{\mathbb{R}^{d}\setminus\omega}$ is finite and continuous in $\mathbb{R}^{d}\setminus\omega$.

For later use, we observe that, for every $t\in\mathbb{R}$,

[TABLE]

even though $\{f>t\}$ itself need not be open. Indeed, by continuity of $f|_{\mathbb{R}^{d}\setminus\omega}$ the set $\{f>t\}\setminus\omega$ is open in $\mathbb{R}^{d}\setminus\omega$ with respect to the relative topology. Hence, there exists an open set $U$ in $\mathbb{R}^{d}$ such that $\{f>t\}\setminus\omega=U\setminus\omega$ and then

[TABLE]

is open in $\mathbb{R}^{d}$ as claimed.

We begin by approximating quasicontinuous functions by sequences of bounded continuous functions:

Proposition 3.2**.**

Suppose that $H$ satisfies monotonicity: and finite subadditivity: . If $f:\mathbb{R}^{d}\to[-\infty,\infty]$ is quasicontinuous and $\int|f|\,\mathrm{d}H<\infty$, then there exists a sequence $(f_{n})_{n\in\mathbb{N}}$ of bounded continuous functions in $\mathbb{R}^{d}$ such that

[TABLE]

Proof.

Suppose that $f$ is quasicontinuous and has finite Choquet integral. We compose $f$ with the truncation function $T_{k}:[-\infty,\infty]\to\mathbb{R}$ at height $k>0$ defined by

[TABLE]

Then, the composition $T_{k}(f)$ satisfies

[TABLE]

Since $\int|f|\,\mathrm{d}H<\infty$, the integral in the right-hand side tends to zero as $k\to\infty$. Next, by quasicontinuity of $f$, for every $\epsilon>0$ we may find an open set $\omega\subset\mathbb{R}^{d}$ such that $H(\omega)\leq\epsilon$ and $f|_{\mathbb{R}^{d}\setminus\omega}$ is continuous on the closed set $\mathbb{R}^{d}\setminus\omega$. Then, by composition, $T_{k}(f)|_{\mathbb{R}^{d}\setminus\omega}$ is also continuous. Thus, the Tietze Extention Theorem allows us to extend $T_{k}(f)|_{\mathbb{R}^{d}\setminus\omega}$ as a bounded continuous function $g_{k,\epsilon}:\mathbb{R}^{d}\to\mathbb{R}$ with $|g_{k,\epsilon}|\leq k$. We can therefore estimate using the quasi-sublinearity (2.6) of the Choquet integral,

[TABLE]

Taking $n\in\mathbb{N}$, we can find $k=k_{n}>0$ such that the second term is bounded by $1/(n+1)$. For the first term, since $|g_{k_{n},\epsilon}-T_{k_{n}}(f)|$ is bounded by $2k_{n}$ and vanishes on $\mathbb{R}^{d}\setminus\omega$, we have

[TABLE]

Thus,

[TABLE]

Choosing $\epsilon=\epsilon_{n}>0$ so that $4k_{n}\epsilon_{n}\leq 1/(n+1)$, the right-hand side is less than or equal to $3/(n+1)$ and we have the conclusion with $f_{n}\vcentcolon=g_{k_{n},\epsilon_{n}}$. ∎

To obtain the approximation by continuous functions with compact support, one needs a vanishing property of the capacity at infinity:

evanescence:

For every open subset $U\subset\mathbb{R}^{d}$ with $H(U)<\infty$ and every closed subset $F\subset U$,

[TABLE]

Under this additional assumption, we prove

Proposition 3.3**.**

Suppose that $H$ satisfies monotonicity: , finite subadditivity: , and evanescence: . If $f:\mathbb{R}^{d}\to[-\infty,\infty]$ is quasicontinuous and $\int|f|\,\mathrm{d}H<\infty$, then there exists a sequence $(\varphi_{n})_{n\in\mathbb{N}}$ in $C_{c}(\mathbb{R}^{d})$ such that

[TABLE]

Proof.

By Proposition 3.2 and quasi-sublinearity (2.6) of the Choquet integral, we may assume that $f$ is bounded and continuous. Given $r>0$, take $\psi_{r}\in C_{c}(\mathbb{R}^{d})$ such that $0\leq\psi_{r}\leq 1$ in $\mathbb{R}^{d}$ and $\psi_{r}=1$ in $B_{r}$. Since $|f\psi_{r}-f|\leq|f|$ in $\mathbb{R}^{d}$ and $|f\psi_{r}-f|=0$ in $B_{r}$, for every $t>0$ we have

[TABLE]

By boundedness of $f$, there exists $M\geq 0$ such that $|f|\leq M$. For any $0<\eta\leq M$ we then have

[TABLE]

Using the monotonicity of $H$ and (3.3), we estimate

[TABLE]

and

[TABLE]

Thus,

[TABLE]

Observe that $\{|f|\geq\eta\}$ is closed and contained in the open set $\{|f|>\eta\}$, which satisfies $H(\{|f|>\eta\})<\infty$ by the Chebyshev inequality. Thus, by evanescence: , the second term in the right-hand side of (3.4) converges to zero as $r\to\infty$. Since $\int|f|\,\mathrm{d}H<\infty$, the first term in the right-hand side of (3.4) also tends to zero as $\eta\to 0$. Therefore, if we let $r\to\infty$ and then $\eta\to 0$ in (3.4), we obtain

[TABLE]

which implies the desired conclusion since $f\psi_{r}\in C_{c}(\mathbb{R}^{d})$. ∎

The assumption evanescence: is necessary for the density of functions with compact support. Indeed,

Proposition 3.4**.**

Suppose that $H$ satisfies monotonicity: . If every quasicontinuous function $f:\mathbb{R}^{d}\to\mathbb{R}$ with $\int|f|\,\mathrm{d}H<\infty$ can be approximated in terms of the Choquet integral by a sequence in $C_{c}(\mathbb{R}^{d})$, then $H$ satisfies evanescence: .

Proof.

Given an open set $U\subset\mathbb{R}^{d}$ with $H(U)<\infty$ and a closed subset $F\subset U$, let $f:\mathbb{R}^{d}\to\mathbb{R}$ be a continuous function supported in $U$ such that $f=1$ on $F$ and $0\leq f\leq 1$ in $\mathbb{R}^{d}$. Then, by monotonicity of $H$,

[TABLE]

By assumption, there exists a sequence $(f_{n})_{n\in\mathbb{N}}$ in $C_{c}(\mathbb{R}^{d})$ such that $\int|f_{n}-f|\,\mathrm{d}H\to 0$. Given $k\in\mathbb{N}$ to be chosen below, let $R>0$ be such that $\operatorname{supp}{f_{k}}\subset B_{R}$. By monotonicity of $H$, we have

[TABLE]

Given $\epsilon>0$, take $k\in\mathbb{N}$ so that the integral in the left-hand side is less than $\epsilon$. Then, for every $r\geq R$, we have by monotonicity of $H$,

[TABLE]

4. Sublinearity of the Choquet integral

The proof of the sublinearity of the Choquet integral in the discrete case relies on the following minimization property:

Lemma 4.1**.**

Suppose that $H$ satisfies strong subadditivity: . Then, for every $n\in\mathbb{N}_{*}$ and $C_{1},\dots,C_{n}\subset\mathbb{R}^{d}$, there exist $D_{1},\dots,D_{n-1}\subset D_{n}$ such that

[TABLE]

From the identity between the characteristic functions, we have $\bigcup\limits_{i=1}^{n}{D_{i}}=\bigcup\limits_{i=1}^{n}{C_{i}}$ and, since $D_{n}$ contains all sets $D_{i}$, it then follows that $D_{n}=\bigcup\limits_{i=1}^{n}{C_{i}}$.

Proof of Lemma 4.1.

We proceed by induction on $n$. For $n=1$, it suffices to take $D_{1}=C_{1}$. We now suppose the statement is true for some $n\in\mathbb{N}_{*}$. Assume we are given sets $C_{1}$, …, $C_{n+1}\subset\mathbb{R}^{d}$, and apply the induction assumption to the first $n$ sets $C_{1}$, …, $C_{n}$ to get $\widetilde{D}_{1}$, …, $\widetilde{D}_{n-1}\subset\widetilde{D}_{n}$ that satisfy the conclusion. By strong subadditivity of $H$, we have

[TABLE]

Let $D_{i}\vcentcolon=\widetilde{D}_{i}$ for $i\in\{1,\ldots,n-1\}$, $D_{n}\vcentcolon=D_{n}\cap C_{n+1}$ and $D_{n+1}\vcentcolon=D_{n}\cup C_{n+1}$. Then,

[TABLE]

By an application of the induction hypothesis, we then have

[TABLE]

We also observe that

[TABLE]

Since $\sum\limits_{i=1}^{n}{\chi_{\widetilde{D}_{i}}}=\sum\limits_{i=1}^{n}{\chi_{C_{i}}}$, one sees that

[TABLE]

as claimed. Finally, it remains to verify that $D_{i}\subset D_{n+1}$ for $i\in\{1,\ldots,n\}$. This follows from the choice of $D_{n+1}=\widetilde{D}_{n}\cup C_{n+1}$. Indeed, the first $n-1$ sets $D_{1},\ldots,D_{n-1}$ are contained in $\widetilde{D}_{n}$, which is a subset of $D_{n+1}$. To conclude, one notes that $D_{n}=\widetilde{D}_{n}\cap C_{n+1}$, which is also contained in $D_{n+1}$. ∎

The following proposition is claimed true by an induction argument at the top of p. 249 of [Anger:1977], while a detailed proof can be found on pp. 766–768 of [STW]. We rely on a different organization of the proof based on Lemma 4.1.

Proposition 4.2**.**

Suppose that $H$ satisfies strong subadditivity: . Then, for every $C_{1},\dots,C_{n}\subset\mathbb{R}^{d}$, the nested family of sets $A_{1}\subset\ldots\subset A_{n}\subset\mathbb{R}^{d}$ such that

[TABLE]

satisfies

[TABLE]

For every $i\in\{1,\ldots,n\}$, the set $A_{i}$ is the collection of points that belong to at least $n-i+1$ sets among $C_{1},\dots,C_{n}$. Note that $A_{i}$ can be also identified as the superlevel set $\{a\geq i\}$ of the function $a\vcentcolon=\sum\limits_{i=1}^{n}{\chi_{C_{i}}}$.

Proof of Proposition 4.2.

We proceed by induction on the number of sets $C_{1},\dots,C_{n}$. The conclusion is trivially true for $n=1$ since in this case $A_{1}=C_{1}$. We now assume that $n\geq 2$ and that the statement holds for any family of $n-1$ sets. Then, given $C_{1},\ldots,C_{n}\subset\mathbb{R}^{d}$, take

[TABLE]

given by Lemma 4.1, so that

[TABLE]

By the induction assumption applied to $D_{1},\ldots,D_{n-1}$, we have a family of nested sets $A_{1}\subset\ldots\subset A_{n-1}\subset\mathbb{R}^{d}$ such that

[TABLE]

In particular, $A_{n-1}=\bigcup\limits_{i=1}^{n-1}{D_{i}}$. Define $A_{n}\vcentcolon=D_{n}$. From (4.1), we deduce that $A_{1}\subset\ldots\subset A_{n-1}\subset A_{n}$ are nested. Moreover, by (4.2) and (4.3), they satisfy

[TABLE]

and also

[TABLE]

By induction, the conclusion then follows. ∎

From this result, sublinearity of the Choquet integral where the functions in consideration have values in a discrete set follows easily:

Corollary 4.3**.**

Suppose that $H$ satisfies monotonicity: and strong subadditivity: . If $f,g:\mathbb{R}^{d}\to\mathbb{N}/k$ for some $k\in\mathbb{N}_{*}$, then

[TABLE]

Proof.

By positive $1$-homogeneity of the Choquet integral we may assume that $f$ and $g$ take values in $\mathbb{N}$. Let us first assume they are both bounded from above by $n$, in which case

[TABLE]

with analogous identities for $g$ and $f+g$; note that this last function is bounded from above by $2n$. Since

[TABLE]

and the sets $\{f+g\geq i\}$ for $i\in\{1,\ldots,2n\}$ are nested, we may apply Proposition 4.2 with sets $\{f\geq i\}$ and $\{g\geq i\}$ for $i\in\{1,\ldots,n\}$ to get

[TABLE]

From the identities verified by the Choquet integrals of $f+g$, $f$ and $g$, this inequality is precisely (4.4).

To handle the case where $f$ or $g$ is unbounded, we compose them with the truncation function $T_{n}$ at height $n\in\mathbb{N}$ defined by (3.2). As $T_{n}(f)$ and $T_{n}(g)$ are both bounded and have values in $\mathbb{N}$, we can apply the sublinearity of the Choquet integral we already proved in this setting to deduce that

[TABLE]

Since $T_{n}(f)\leq f$ and $T_{n}(g)\leq g$, by monotonicity of the Choquet integral we get

[TABLE]

We also note that $T_{n}(f+g)\leq T_{n}(f)+T_{n}(g)$, which again by monotonicity of the Choquet integral gives

[TABLE]

Since

[TABLE]

it follows from Fatou’s lemma for the Lebesgue integral that

[TABLE]

It now suffices to combine (4.5) and (4.6). ∎

Proof of Theorem 1.2.

Given $E,F\subset\mathbb{R}^{d}$, we have

[TABLE]

Thus, if the Choquet integral is sublinear, then we have

[TABLE]

Hence, $H$ is strongly subadditive.

To prove the converse, we now assume that $H$ is strongly subadditive. Denote by $\lfloor\alpha\rfloor$ the integer part of the real number $\alpha$. Let $f,g:\mathbb{R}^{d}\to[0,\infty]$ and $k\in\mathbb{N}_{*}$. An application of Corollary 4.3 to the $\mathbb{N}/k$-valued functions $\lfloor kf\rfloor/k$ and $\lfloor kg\rfloor/k$ gives

[TABLE]

Since $\lfloor kf\rfloor/k\leq f$ and $\lfloor kg\rfloor/k\leq g$, by monotonicity of the Choquet integral we then have

[TABLE]

Since $k\alpha-1\leq\lfloor k\alpha\rfloor$ for every $\alpha\geq 0$, we have

[TABLE]

which implies

[TABLE]

By definition and monotonicity of the Choquet integral,

[TABLE]

Hence, by (4.7) and (4.8),

[TABLE]

An application of Fatou’s lemma for the Lebesgue integral then yields

[TABLE]

As a consequence of Proposition 2.3, one gets countable sublinearity for the Choquet integral for series of functions that converge quasi-uniformly. Note that countable subadditivity of $H$ is not necessary in this case.

Corollary 4.4**.**

Suppose that $H$ satisfies monotonicity: and strong subadditivity: . If $(f_{n})_{n\in\mathbb{N}}$ is a sequence of real-valued functions in $\mathbb{R}^{d}$ such that \Bigl{(}\sum\limits_{n=0}^{k}f_{n}\Bigr{)}_{k\in\mathbb{N}} converges q.u. to $F:\mathbb{R}^{d}\to\mathbb{R}$, then

[TABLE]

Proof.

By monotonicity and sublinearity of the Choquet integral, for every $k\in\mathbb{N}$ we have

[TABLE]

By Proposition 2.3, we have the conclusion as $k\to\infty$. ∎

Remark 4.5**.**

The statements above and their proofs in this section apply without change to a set function $H$ defined on an algebra $\mathcal{X}$ associated to a set $X$, that is, $\mathcal{X}$ is a family of subsets of $X$ such that $\emptyset\in\mathcal{X}$ and

[TABLE]

More generally, one expects that results concerning the Choquet integral stated for capacities satisfying finite subadditivity: hold in the setting of an algebra, provided one works with the class of $H$-capacitable functions, i.e. those for which their upper-level sets are elements in this algebra.

5. Lower semicontinuity of the Choquet integral

To obtain a Fatou’s lemma without quasi-uniform convergence, we restrict ourselves to the setting of quasicontinuous functions. The main tool is based on the following standard consequence of the Hahn-Banach Theorem:

Proposition 5.1**.**

Suppose that $H$ satisfies monotonicity: , strong subadditivity: , and evanescence: . If $f:\mathbb{R}^{d}\to[0,\infty]$ is a quasicontinuous function such that

[TABLE]

then

[TABLE]

We handle separately the inequality “$\geq$” in the following

Lemma 5.2**.**

Suppose that $H$ satisfies monotonicity: and finite subadditivity: . If $f:\mathbb{R}^{d}\to[0,\infty]$ is quasicontinuous, then, for every locally finite nonnegative Borel measure $\mu$ in $\mathbb{R}^{d}$ such that $\mu\leq H$ on open subsets of $\mathbb{R}^{d}$, we have

[TABLE]

Proof of Lemma 5.2.

Since $f$ is quasicontinuous, for every $\epsilon>0$ there exists an open set $\omega\subset\mathbb{R}^{d}$ such that $H(\omega)\leq\epsilon$ and $f|_{\mathbb{R}^{d}\setminus\omega}$ is continuous. Then, by (3.1), the set $\{f>t\}\cup\omega$ is open in $\mathbb{R}^{d}$ for every $t>0$. Since $\mu$ is monotone, $\mu\leq H$ on open sets, and $H$ is finitely subadditive, we get

[TABLE]

Given $k>0$, we integrate both members with respect to $t\in(0,k)$ to get

[TABLE]

Letting $\epsilon\to 0$ and then $k\to\infty$, we have the conclusion using Cavalieri’s principle for the Lebesgue integral. ∎

Proof of Proposition 5.1.

Inequality “$\geq$” in the statement readily follows from Lemma 5.2 for any nonnegative quasicontinuous function. We thus focus on the proof of the reverse inequality “$\leq$”. We first show it for a nonnegative function $f\in C_{c}(\mathbb{R}^{d})$, assuming in addition that $H$ is finite. Since $H$ is strongly subadditive, the function

[TABLE]

is well-defined and sublinear. By the Hahn-Banach Theorem, the linear functional

[TABLE]

defined on the one dimensional vector subspace $\{tf:t\in\mathbb{R}\}$ has a linear extension $F:C_{c}(\mathbb{R}^{d})\to\mathbb{R}$ such that $F\leq P$ on $C_{c}(\mathbb{R}^{d})$.

Observe that if $\psi\in C_{c}(\mathbb{R}^{d})$ is nonpositive, we have $F(\psi)\leq P(\psi)=0$. Thus, $F$ is positive and then, for every compact subset $S\subset\mathbb{R}^{d}$, its restriction to $C(S)$ is a continuous linear functional. Hence, by the Riesz Representation Theorem, there exists a locally finite nonnegative Borel measure $\mu$ in $\mathbb{R}^{d}$ such that

[TABLE]

In particular, since $f$ is nonnegative,

[TABLE]

It remains to observe that $\mu\leq H$ on open subsets of $\mathbb{R}^{d}$. To this end, take a nonempty open set $U\subset\mathbb{R}^{d}$ and a compact subset $K\subset U$. There exists $\psi\in C_{c}(\mathbb{R}^{d})$ such that $0\leq\psi\leq 1$ in $\mathbb{R}^{d}$, $\psi=1$ on $K$ and $\operatorname{supp}{\psi}\subset U$. We then have by monotonicity of $\mu$ and $H$,

[TABLE]

Since this inequality holds for every compact subset $K\subset U$, by inner regularity of $\mu$ we conclude that $\mu(U)\leq H(U)$.

We next prove the inequality “$\leq$” for a nonnegative function $f\in C_{c}(\mathbb{R}^{d})$ that satisfies (5.1). Take $\eta>0$ to be chosen below and consider the set function $H_{\eta}$ defined by contraction for every $A\subset\mathbb{R}^{d}$ as

[TABLE]

Observe that $H_{\eta}$ is also monotone and strongly subadditive. Moreover, for every $n\in\mathbb{N}$,

[TABLE]

We may apply the previous case with finite set function $H_{\eta}$ to get

[TABLE]

Combining (5.2) and (5.3), we then obtain

[TABLE]

As $\eta\to 0$, it follows from Fatou’s lemma for the Lebesgue integral that

[TABLE]

To conclude, we now prove (5.4) for an arbitrary nonnegative quasicontinuous function $f$ for which (5.1) holds. To this end, we apply Proposition 3.2 that ensures the existence of a sequence of nonnegative functions $(\varphi_{n})_{n\in\mathbb{N}}$ in $C_{c}(\mathbb{R}^{d})$ such that $\int|\varphi_{n}-f|\,\mathrm{d}H\to 0$. For every $n\in\mathbb{N}$, we have by sublinearity of the Choquet integral and by Lemma 5.2 applied to the quasicontinuous function $|\varphi_{n}-f|$,

[TABLE]

Taking the supremum on both sides with respect to $\mu\leq H$, we have by the first part of the proof,

[TABLE]

As $n\to\infty$, by convergence of the sequence $(\varphi_{n})_{n\in\mathbb{N}}$ with respect to the Choquet integral we get (5.4), which completes the proof. ∎

Assumption (5.1) can be removed under inner regularity of open sets with infinite capacity:

semifinite:

For every open set $U\subset\mathbb{R}^{d}$ such that $H(U)=\infty$ and every $M\geq 0$, there exists a subset $E\subset U$ with $M\leq H(E)<\infty$.

We then get the counterpart of Proposition 5.1 for all nonnegative quasicontinuous functions with infinite Choquet integral:

Proposition 5.3**.**

Suppose that $H$ satisfies monotonicity: , strong subadditivity: , evanescence: , and semifinite: . If $f:\mathbb{R}^{d}\to[0,\infty]$ is a quasicontinuous function such that

[TABLE]

then

[TABLE]

Proof.

From (5.5), we have $\int f\,\mathrm{d}H=\infty$. Next, by quasicontinuity of $f$, there exists an open set $\omega\subset\mathbb{R}^{d}$ such that $H(\omega)\leq 1$ and $f|_{\mathbb{R}^{d}\setminus\omega}$ is continuous. In particular, by (3.1), the set $\{f>t\}\cup\omega$ is open in $\mathbb{R}^{d}$. By monotonicity of $H$ and (5.5),

[TABLE]

Applying semifinite: , for every $n\in\mathbb{N}$ there exists a subset $E_{n}\subset\{f>t\}\cup\omega$ with $n\leq H(E_{n})<\infty$. Let $H_{n}$ be the set function defined for every $A\subset\mathbb{R}^{d}$ by $H_{n}(A)\vcentcolon=H(A\cap E_{n})$. Note that

[TABLE]

Since $E_{n}\subset\{f>t\}\cup\omega$, by monotonicity and finite subadditivity of $H$,

[TABLE]

Thus,

[TABLE]

Since $H_{n}$ is finite and $H_{n}\leq H$, by Proposition 5.1 we have

[TABLE]

Combining (5.6) and (5.7), we get

[TABLE]

This implies the conclusion since $n$ can be chosen arbitrarily large. ∎

From Propositions 5.1 and 5.3, we deduce the following analogue of Fatou’s lemma for quasicontinuous functions:

Corollary 5.4**.**

Suppose that $H$ satisfies monotonicity: , strong subadditivity: , evanescence: , and semifinite: . If $(f_{n})_{n\in\mathbb{N}}$ is a sequence of nonnegative quasicontinuous functions in $\mathbb{R}^{d}$, then, for every quasicontinuous function $f:\mathbb{R}^{d}\to[0,\infty]$ such that $f\leq\liminf\limits_{n\to\infty}{f_{n}}$ in $\mathbb{R}^{d}$, we have

[TABLE]

Proof.

We may assume that each $f_{n}$ has finite Choquet integral and that $\liminf\limits_{n\to\infty}{\int f_{n}\,\mathrm{d}H}<\infty$. For every nonnegative locally finite Borel measure $\mu$ in $\mathbb{R}^{d}$ such that $\mu\leq H$ on open subsets of $\mathbb{R}^{d}$, we have by monotonicity and Fatou’s lemma for the Lebesgue integral,

[TABLE]

Thus, by Lemma 5.2,

[TABLE]

Since $f$ is quasicontinuous and the right-hand side is finite, by Proposition 5.3 we must have $H(\{f>t\})<\infty$ for every $t>0$. The assumptions of Proposition 5.1 are then satisfied by $f$ and it thus suffices to take the supremum with respect to $\mu$ in the left-hand side of (5.8). ∎

The assumption semifinite: avoids a gap between sets of finite and infinite capacities, which is illustrated in the following example where the conclusion of Corollary 5.4 fails:

Example 5.5**.**

Let $H$ be defined for every nonempty subset $A\subset\mathbb{R}^{d}$ as

[TABLE]

For each $n\in\mathbb{N}$, take $f_{n}\in C_{c}(\mathbb{R}^{d})$ such that $0\leq f_{n}\leq 1$ in $\mathbb{R}^{d}$, $f_{n}=1$ in $B_{n}$ and $\operatorname{supp}{f_{n}}\subset B_{n+1}$. Then, $\int f_{n}\,\mathrm{d}H=1$ and $f_{n}\to 1$ pointwise in $\mathbb{R}^{d}$ as $n\to\infty$, but $\int 1\,\mathrm{d}H=\infty$.

One can fulfill assumption semifinite: by replacing $H$ with a more regular set function $\widetilde{H}:\mathcal{P}(\mathbb{R}^{d})\to[0,\infty]$ defined for every $A\subset\mathbb{R}^{d}$ by

[TABLE]

Observe that $\widetilde{H}$ is monotone regardless of $H$. Under monotonicity of $H$ itself, these set functions coincide on sets of finite $H$ capacity:

Proposition 5.6**.**

If $H$ satisfies monotonicity: , then

[TABLE]

Hence, $\widetilde{H}$ also satisfies semifinite: .

Proof.

By the monotonicity of $H$, the supremum in the definition of $\widetilde{H}(A)$ is achieved by the set $A$ itself whenever $H(A)<\infty$. That semifinite: holds, then follows from the fact that we may then reformulate the definition of $\widetilde{H}$ as

[TABLE]

Note that $\widetilde{H}$ inherits several properties of $H$:

Proposition 5.7**.**

Suppose that $H$ satisfies monotonicity: . If $H$ also satisfies any of the assumptions finite subadditivity: , countable subadditivity: , strong subadditivity: , or evanescence: , then so does $\widetilde{H}$.

Proof.

That $\widetilde{H}$ verifies evanescence: whenever $H$ does follows from the fact that $\widetilde{H}=H$ on sets where $H$ is finite. We now assume that $H$ satisfies countable subadditivity: . Given a sequence of sets $(E_{n})_{n\in\mathbb{N}}$ and $D\subset\bigcup\limits_{n=0}^{\infty}{E_{n}}$ with $H(D)<\infty$, by monotonicity of $H$ for every $n\in\mathbb{N}$ we have $H(E_{n}\cap D)<\infty$. Hence, by countable subadditivity of $H$ and definition of $\widetilde{H}$,

[TABLE]

It now suffices to take the supremum with respect to $D$ in the left-hand side. Similarly, one verifies that $\widetilde{H}$ satisfies finite subadditivity: whenever $H$ does.

We now assume that $H$ verifies strong subadditivity: . Given $E,F\subset\mathbb{R}^{d}$, take $C\subset E\cap F$ and $D\subset E\cup F$ with $H(C)<\infty$ and $H(D)<\infty$. By finite subadditivity of $H$, $H(C\cup D)<\infty$ and then, by monotonicity, $H$ is finite on every subset of $C\cup D$. Note that, since $C\subset E\cap F$ and $D\subset E\cup F$,

[TABLE]

By monotonicity and strong subadditivity of $H$, we then have

[TABLE]

Since $(C\cup D)\cap E$ and $(C\cup D)\cap F$ are subsets of $E$ and $F$ where $H$ is finite, we get by definition of $\widetilde{H}$,

[TABLE]

It now suffices to take the supremum in the left-hand side with respect to $C$ and $D$. ∎

6. The space $L^{1}(H)$

We assume that $H$ satisfies monotonicity: and strong subadditivity: . We introduce an equivalence relation $\sim$ among elements in the vector space of real-valued quasicontinuous functions in $\mathbb{R}^{d}$ by denoting $f\sim g$ whenever $f=g$ quasi-everywhere (q.e.), that is, there exists $E\subset\mathbb{R}^{d}$ such that $H(E)=0$ and $f=g$ in $\mathbb{R}^{d}\setminus E$. Then, $[f]$ is the equivalence class that contains $f$. We let

[TABLE]

We may naturally equip this set with addition and multiplication by scalar: For every quasicontinuous functions $f,g$ and $\lambda\in\mathbb{R}$, let

[TABLE]

Observe that the function

[TABLE]

is well-defined in $L^{1}(H)$ and, by sublinearity and $1$-homogeneity of the Choquet integral, is a norm in this space.

Proposition 6.1**.**

Suppose that $H$ satisfies monotonicity: , strong subadditivity: and countable subadditivity: . Then, $L^{1}(H)$ is a Banach space.

The proof is standard, e.g. [C5]*Proposition 2.1 and 2.2, though as stated in the introduction we include it for completeness.

Proof.

Let $([f_{n}])_{n\in\mathbb{N}}$ be a Cauchy sequence in $L^{1}(H)$, where each $f_{n}:\mathbb{R}^{d}\to\mathbb{R}$ is quasicontinuous. We may find positive integers $n_{1}<n_{2}<\ldots<n_{j}$ such that

[TABLE]

This implies

[TABLE]

and therefore

[TABLE]

Since $|f_{n_{j}}-f_{n_{j+1}}|$ is quasicontinuous, there exists an open set $\omega_{j}\subset\mathbb{R}^{d}$ such that $H(\omega_{j})\leq 1/2^{j}$ and $|f_{n_{j}}-f_{n_{j+1}}|$ is continuous in $\mathbb{R}^{d}\setminus\omega_{j}$. By (3.1), the set

[TABLE]

is open and, by finite subadditivity of $H$, $H(G_{j})\leq{1}/{2^{j-1}}$.

Let $F_{m}\vcentcolon=\bigcup\limits_{j=m}^{\infty}G_{j}$. For any $x\in\mathbb{R}^{d}\setminus F_{m}$ we have

[TABLE]

Therefore if for $x\in\mathbb{R}^{d}\setminus F_{m}$ one defines

[TABLE]

then by the Weierstrass $M$-test, $(f_{n_{j}})_{j\in\mathbb{N}}$ converges uniformly to $f$ in $\mathbb{R}^{d}\setminus F_{m}$, whence $f$ is continuous in $\mathbb{R}^{d}\setminus F_{m}$. As $(F_{m})_{m\geq 1}$ is non-increasing, the function $f$ is well-defined on $\bigcup\limits_{m\geq 1}{(\mathbb{R}^{d}\setminus F_{m})}$. Finally, let $f(x)=0$ for $x\in\bigcap\limits_{m\geq 1}{F_{m}}$. Since, by countable subadditivity of $H$,

[TABLE]

as $m\to\infty$, we deduce that $f$ is quasicontinuous and $f_{n_{j}}\to f$ q.u.

Since $([f_{n}])_{n\in\mathbb{N}}$ is a Cauchy sequence in $L^{1}(H)$, to prove its convergence to $[f]$ in $L^{1}(H)$ it suffices to prove that the subsequence $([f_{n_{j}}])_{j\geq 1}$ converges q.u. to $[f]$. For every $i\geq j\geq 1$, by (6.1) we have

[TABLE]

As $(|f_{n_{j}}-f_{n_{i}}|)_{i\geq 1}$ converges to $|f_{n_{j}}-f|$ when $i\to\infty$, it follows from Proposition 2.3 that

[TABLE]

from which the conclusion follows. ∎

Every function in $C_{c}(\mathbb{R}^{d})$ has finite Choquet integral provided that $H$ satisfies

locally finite:

$H(U)<+\infty$ for every bounded open set $U\subset\mathbb{R}^{d}$.

Then, the quotient space $C_{c}(\mathbb{R}^{d})/\!\sim$ is contained in $L^{1}(H)$. Moreover,

Corollary 6.2**.**

Suppose that $H$ satisfies monotonicity: , strong subadditivity: , countable subadditivity: , evanescence: , and locally finite: . Then, $L^{1}(H)$ is the completion of $C_{c}(\mathbb{R}^{d})/\!\sim\;$ with respect to the $L^{1}(H)$ norm.

Proof.

From Proposition 3.3 we have that $C_{c}(\mathbb{R}^{d})/\!\sim\;$ is dense in $L^{1}(H)$. By Proposition 6.1, $L^{1}(H)$ is complete. ∎

We conclude this section with a closure property for bounded sequences in $L^{1}(H)$. Firstly, concerning quasi-uniform convergence:

Corollary 6.3**.**

Suppose that $H$ satisfies monotonicity: and strong subadditivity: . If $([f_{n}])_{n\in\mathbb{N}}$ is a bounded sequence in $L^{1}(H)$ such that $(f_{n})_{n\in\mathbb{N}}$ converges q.u. to a quasicontinuous function $f$, then $[f]\in L^{1}(H)$ and

[TABLE]

Proof.

It suffices to apply Proposition 2.3, which implies that $\int|f|\,\mathrm{d}H<\infty$ and then $[f]\in L^{1}(H)$ by quasicontinuity of $f$. ∎

Observe that if $f,g:\mathbb{R}^{d}\to\mathbb{R}$ are such that $f=g$ q.e. and $f$ is quasicontinuous, it need not be true that $g$ is quasicontinuous. To make sure that such a property holds, one may require some regularity on sets where $H$ vanishes:

zero-capacity regularity:

For every $E\subset\mathbb{R}^{d}$ with $H(E)=0$ and every $\epsilon>0$, there exists an open set $\omega\supset E$ such that $H(\omega)\leq\epsilon$.

Corollary 6.4**.**

Suppose that $H$ satisfies monotonicity: , strong subadditivity: , evanescence: , semifinite: , and zero-capacity regularity: . If $([f_{n}])_{n\in\mathbb{N}}$ is a bounded sequence in $L^{1}(H)$ such that $(f_{n})_{n\in\mathbb{N}}$ converges q.e. to a quasicontinuous function $f$, then $[f]\in L^{1}(H)$ and

[TABLE]

Proof.

We observe that $(|f_{n}|)_{n\in\mathbb{N}}$ converges q.e. to $|f|$ and that $|f|$ is quasicontinuous. It then follows from zero-capacity regularity: that the function $\liminf\limits_{n\to\infty}{|f_{n}|}$, which equals $|f|$ q.e., is also quasicontinuous. Moreover, by Corollary 5.4, we deduce that

[TABLE]

from which the conclusion follows. ∎

Acknowledgments

D. Spector is supported by the National Science and Technology Council of Taiwan under research grant number 110-2115-M-003-020-MY3 and the Taiwan Ministry of Education under the Yushan Fellow Program. Part of this work was undertaken while D. Spector was visiting IRMP Institute of the Université catholique de Louvain. He would like to thank the IRMP Institute for its support and A. C. Ponce for his warm hospitality during the visit.