Properties of the Riemann-Lebesgue integrability in the non-additive case

TL;DR

This paper explores the properties of Riemann-Lebesgue integrability for vector functions with respect to non-additive set functions, establishing classical properties and relationships among different integrability concepts.

Contribution

It extends classical integral properties to the non-additive case and clarifies the relationships among Riemann-Lebesgue, Birkhoff simple, and Gould integrabilities.

Findings

Classical integral properties are valid in the non-additive setting.

Continuity properties of the integral are characterized.

Relationships among various integrability notions are established.

Abstract

We study Riemann-Lebesgue integrability of a vector function relative to an arbitrary non-negative set function. We obtain some classical integral properties. Results regarding the continuity properties of the integral and relationships among Riemann-Lebesgue, Birkhoff simple and Gould integrabilities are also established.

Equations

$$\Vert \sum _{n=0}^{+\infty }f(t_n)m(A_n)-J\Vert \le \epsilon .$$

$$\Vert \sigma (f,\Pi )-\sigma (f,{\Pi }^{\prime })\Vert \le \Vert \sigma (f,\Pi )-J\Vert +\Vert J-\sigma (f,{\Pi }^{\prime })\Vert \le 2\epsilon .$$

$${\alpha }_1:=\sum _{I\in {\Pi }_A}f(t_I)m(I),{\alpha }_2:=\sum _{I\in {\Pi }_A^{\prime }}f(t_I^{\prime })m(I),\beta :=\sum _{I\in \Pi ,I\subset A^c}f({\tau }_I)m(I),$$

$$2\epsilon \ge \Vert {\alpha }_1+\beta -({\alpha }_2+\beta )\Vert =\Vert {\alpha }_1-{\alpha }_2\Vert .$$

$$\Vert \sum _{n=0}^{+\infty }f(s_n)m(B_n)-(|RL|){\int }_{A}fdm\Vert <\epsilon .$$

$$\Vert \sum _{n=0}^{+\infty }f{\chi }_A(u_n)m(A_n)-(|RL|){\int }_{A}fdm\Vert =$$

$$=$$

$$\Vert \sum _{n=0}^{+\infty }f{\chi }_A(t_n)m(C_n)-(|RL|){\int }_{T}f{\chi }_{A}dm\Vert <\epsilon .$$

$$\Vert \sum _{n=0}^{+\infty }f(t_n)m(D_n)-(|RL|){\int }_{T}f{\chi }_{A}dm\Vert =$$

$$=$$

$$(|RL|){\int }_{A}fdm=(|RL|){\int }_{T}f{\chi }_{A}dm.$$

$$\Vert (|RL|){\int }_{T}fdm\Vert \le \underset{t\in T}{\sup }\Vert f(t)\Vert \cdot \overline{m}(T).$$

$$\sum _{k=0}^n\Vert f(t_k)m(A_k)\Vert \le \underset{t\in T}{\sup }\Vert f(t)\Vert \cdot \overline{m}(T),$$

$$\Vert (|RL|){\int }_{T}fdm\Vert \le \underset{t\in T}{\sup }\Vert f(t)\Vert \cdot \overline{m}(T).$$

$$\Vert \sum _{n=0}^{+\infty }f(t_n)m(D_n)+\sum _{n=0}^{+\infty }f(s_n)m(E_n)\Vert =\Vert \sum _{n=0}^{+\infty }f(s_n)m(E_n)\Vert \le$$

$$\le \sum _{n=0}^{+\infty }\Vert f(s_n)\Vert m(E_n)\le M\cdot \overline{m}(B_{\epsilon })<\epsilon ,$$

$$(|RL|){\int }_{T}fd(\alpha m)=\alpha (|RL|){\int }_{T}fdm.$$

$$(|RL|){\int }_T(f\pm g)dm=(|RL|){\int }_{T}fdm\pm (|RL|){\int }_{T}gdm.$$

$$\Vert \sum _{n=0}^{+\infty }f(t_n)m(A_n)-(|RL|){\int }_{T}fdm\Vert <\frac{\epsilon }{2};$$

$$\Vert \sum _{n=0}^{+\infty }g(t_n)m(A_n)-(|RL|){\int }_{T}gdm\Vert <\frac{\epsilon }{2}.$$

$$\Vert \sum _{n=0}^{+\infty }(f+g)(t_n)m(C_n)-\left((|RL|){\int }_{T}fdm+(|RL|){\int }_{T}gdm\right)\Vert \le$$

$$\le$$

$$I_f(A)=(|RL|){\int }_{A}fdm\left(I_f(A)=(RL){\int }_{A}fdm\right).$$

$$(|RL|){\int }_{T}fdm=(|RL|){\int }_{T}gdm.$$

$$(|RL|){\int }_{T}fd(m_1+m_2)=(|RL|){\int }_{T}fd{m}_1+(|RL|){\int }_{T}fd{m}_2.$$

$$\Vert (|RL|){\int }_{T}fdm-(|RL|){\int }_{T}gdm\Vert \le \underset{t\in T}{\sup }\Vert f(t)-g(t)\Vert \cdot \overline{m}(T).$$

$$\left|{\int }_{T}fdm-\sum _{n=0}^{+\infty }f(t_n)m(C_n)\right|<\frac{\epsilon }{3};$$

$$\left|{\int }_{T}gdm-\sum _{n=0}^{+\infty }g(t_n)m(C_n)\right|<\frac{\epsilon }{3}.$$

$${\int }_{T}fdm-{\int }_{T}gdm={\int }_{T}fdm-\sum _{n=0}^{+\infty }f(t_n)m(C_n)+$$

$$+$$

$$<$$

$$\sum _{i=1}^n\Vert I_f(A_i)\Vert \le M\cdot \sum _{i=1}^n\overline{m}(A_i)\le M\cdot \overline{m}(T).$$

$$\left|I_f(A)-\sum _{n=0}^{+\infty }f(t_n)m(A_n)\right|<\frac{\epsilon }{2}.$$

Full text

**Key words Riemann-Lebesgue integral, Birkhoff simple integral, Gould integral, Non-negative set function, Monotone measure.

AMS class: 28B20, 28C15, 49J53**

1. Introduction

Riemann and Lebesgue integrals admit different extensions, some of them to the case of finitely additive, non-additive or set valued measures. Although (countable) additivity is one of the most important notion in measure theory, it can be useless in many problems, for example modeling different real aspects in data mining, computer science, economy, psychology, game theory, fuzzy logic, decision making, subjective evaluation. Non-additive measures have been used with different names: J.von Neumann and O. Morgenstern called them cooperative games in Economics; M. Sugeno called a non additive set function a fuzzy measure and proposed an integral with respect to a fuzzy measure useful for systems science. In [44] a broad overview of non-additive measures and their applications is presented. So, in general, non-additive measures are used when models based on the additive ones are not appropriate and that motivated us to study some integral properties in the non-additive case.

In the framework of an extension of the Riemann and Lebesgue integrals, a way of defining a new integral is to generalize the Riemann sums as in [1, 9, 16, 21, 22, 23, 24, 25, 18, 26, 27, 30, 12, 31, 32, 35, 36, 37, 41, 42, 43, 3]. One of these extensions has been defined by Birkhoff [1], using countable sums for vector functions with respect to complete finite measures. Then this integral has been intensively studied and generalized in [2, 5, 4, 6, 7, 10, 13, 14, 15, 8, 19, 20, 28, 29, 33, 34, 38, 39]. It is well-known that the Birkhoff integral and the (generalized) McShane integral lie strictly between the Bochner and Pettis integrals. The notions of Bochner and Birkhoff integrabilities are equivalent for bounded single-valued functions with values in a separable Banach space. Also, if the range of the Banach-valued function is separable, then Pettis, Birkhoff and McShane integrabilities coincide (see also [17]). The situation changes deeply in the non separable case, namely: every Birkhoff integrable function is McShane integrable and every McShane integrable function is Pettis integrable, but none of the reverse implications holds in general (see [40]).

Another generalization belongs to Kadets and Tseytlin [26], who introduced two integrals, called absolute Riemann-Lebesgue ($|RL|$) and unconditional Riemann-Lebesgue ($RL$), for functions with values in a Banach space, relative to a countably additive measure. In [26], it is proved that if $(T,\mathcal{A},m)$ is a finite measure space, then the following implications hold:

$f$ is Bochner integrable $\Rightarrow$ $f$ is $|RL|$-integrable $\Rightarrow$ $f$ is $RL$-integrable $\Rightarrow$ $f$ is Pettis integrable.

If $X$ is a separable Banach space, then $|RL|$-integrability coincides with Bochner integrability and $RL$-integrability coincides with Pettis integrability. Also, according to [31], if $(T,\mathcal{A},m)$ is a $\sigma$-finite measure space, then the Birkhoff integrability [1] is equivalent to the $RL$-integrability.

In this paper, we study the Riemann-Lebesgue integral with respect to an arbitrary set function, not necessarily countably additive. The paper is organized in four sections: Section 1 is devoted to the introduction; after, in Section 2, some basic concepts and results are presented,while in Section 3 the Riemann-Lebesgue integrability relative to an arbitrary set function is presented togheter with some classical (Theorem 3.5) and continuity properties (Theorem 3.12) of the integral. Finally Section 4 contains some comparative results among this Riemann-Lebesgue integral and some other integrals known in the literature as the Birkhoff simple and the Gould integrabilities. In particular we show that the Gould and the Riemann-Lebesgue integrabilities coincide when a bounded function is integrated with respect to a $\sigma$-additive (or monotone and $\sigma$-subadditive) measure of finite variation, see in this regard Proposition 4.5 and Theorem 4.7; lastly an example (Example 4.6) is given in order to show that the equivalence, in general, does not hold.

2. Preliminaries

Let $T$ be a nonempty at least countable set, $\mathcal{P}(T)$ the family of all subsets of $T$ and $\mathcal{A}$ a $\sigma$-algebra of subsets of $T.$ Let $\mathbb{N}^{*}=\{1,2,3,\ldots\}.$

Definition 2.1**.**

• (2.1.i)

A finite (countable, resp.) partition of $T$ is a finite (countable, resp.) family of nonempty sets $P=\{A_{i}\}_{i=1,\ldots,n}$ ($\{A_{n}\}_{n\in\mathbb{N}}$, resp.) $\subset\mathcal{A}$ such that $A_{i}\cap A_{j}=\emptyset,i\neq j$ and $\bigcup\limits_{i=1}^{n}A_{i}=T$ ($\bigcup\limits_{n\in\mathbb{N}}A_{n}=T$, resp.).

• (2.1.ii)

If $P$ and $P^{\prime}$ are two partitions of $T$, then $P^{\prime}$ is said to be finer than $P$, denoted by $P\leq P^{\prime}$ (or $P^{\prime}\geq P)$, if every set of $P^{\prime}$ is included in some set of $P$.

• (2.1.iii)

The common refinement of two finite or countable partitions $P=\{A_{i}\}$ and $P^{\prime}=\{B_{j}\}$ is the partition $P\wedge P^{\prime}=\{A_{i}\cap B_{j}\}$.

We denote by $\mathcal{P}$ the class of all the partitions of $T$ and if $\mathrm{A\in}$ $\mathcal{A}$ is fixed, by $\mathcal{P}_{\mathrm{A}}$  we denote the class of all the partitions of $\mathrm{A.}$

Definition 2.2**.**

Let $m:\mathcal{A}\rightarrow[0,+\infty)$ be a non-negative function, with $m(\emptyset)=0.$ $m$ is said to be:

• (2.2.i)

monotone if $m(A)\leq m(B)$, $\forall A,B\in\mathcal{A}$, with $A\subseteq B$;

• (2.2.ii)

subadditive if $m(A\cup B)\leq m(A)+m(B),$ for every $A,B\in\mathcal{A}$, with $A\cap B=\emptyset;$

• (2.2.iii)

*a submeasure * (in the sense of Drewnowski) if $m$ is monotone and subadditive;

• (2.2.iv)

$\sigma$-subadditive if $m(A)\leq\sum\limits_{n=0}^{+\infty}m(A_{n}),$ for every sequence of (pairwise disjoint) sets* $(A_{n})_{n\in\mathbb{N}}\subset$* $\mathcal{A}$*, *with $A=\bigcup\limits_{n=0}^{+\infty}A_{n}$.

Let $(X,\|\cdot\|)$ be a Banach space and $m:\mathcal{A}\rightarrow X$ be a vector set function, with $m(\emptyset)=0.$

Definition 2.3**.**

A vector set function $m$ is said to be:

• (2.3.i)

finitely additive if $m(A\cup B)=m(A)+m(B),$ for every disjoint sets $A,B\in\mathcal{A};$

• (2.3.ii)

$\sigma$-additive if $m(\bigcup\limits_{n=0}^{+\infty}A_{n})=\sum\limits_{n=0}^{+\infty}m(A_{n})$ , for every sequence of pairwise disjoint sets $(A_{n})_{n\in\mathbb{N}}\subset\mathcal{A}$;

• (2.3.iii)

order-continuous (shortly, o-continuous) if $\lim\limits_{n\rightarrow+\infty}m(A_{n})=0,$ for every decreasing sequence of sets $(A_{n})_{n\in\mathbb{N}}\subset\mathcal{A}$, with $\bigcap\limits_{n=0}^{+\infty}A_{n}=\emptyset$ (denoted by $A_{n}\searrow\emptyset$);

• (2.3.iv)

*exhaustive *if $\lim\limits_{n\rightarrow+\infty}m(A_{n})=0,$ for every sequence of pairwise disjoint sets $(A_{n})_{n\in\mathbb{N}}\subset\mathcal{A}.$

• (2.3.v)

null-additive if, for every $A,B\in\mathcal{A}$, $m(A\cup B)=m(A)$ when $m(B)=0$.

Moreover $m$ satisfies property $(\sigma)$ if the ideal of $m$-zero sets is stable under countable unions, namely for every sequence $(A_{k})_{k}$ such that $m(A_{k})=0$ for each $k\in\mathbb{N}$, then also $m(\bigcup_{k}A_{k})=0$.

Definition 2.4**.**

The variation $\overline{m}$ of $m$ is the set function $\overline{m}:\mathcal{P}(T)\rightarrow[0,+\infty]$ defined by $\overline{m}(E)=\sup\{\sum\limits_{i=1}^{n}\|m(A_{i})\|\}$, for every $E\in\mathcal{P}(T)$, where the supremum is extended over all finite families of pairwise disjoint sets $\{A_{i}\}_{i=1}^{n}\subset\mathcal{A}$, with $A_{i}\subseteq E$, for every $i=1,\ldots,n$.

$m$ is said to be of finite variation (on $\mathcal{A}$) if $\overline{m}(T)<+\infty$.

$\widetilde{m}$ $:\mathcal{P}(T)\rightarrow[0,+\infty]$ is defined for every $A\subseteq T$, by $\widetilde{m}(A)=\inf\{\overline{m}(B);A\subseteq B,B\in\mathcal{A}\}$.

Remark 2.5**.**

$\overline{m}$ is monotone and super-additive on $\mathcal{P}(T)$, that is $\overline{m}(\bigcup_{i\in I}A_{i})\geq\sum_{i\in I}\overline{m}(A_{i}),$ for every finite or countable partition $\{A_{i}\}_{i\in I}$ of $T$.

If $m:\mathcal{A}\rightarrow[0,+\infty)$ is finitely additive, then $\overline{m}(A)=m(A),$ for every $A\in\mathcal{A}.$

If $m:\mathcal{A}\rightarrow[0,+\infty)$ is subadditive ($\sigma$-subadditive, respectively), then $\overline{m}$ is finitely additive ($\sigma$-additive, respectively).

For all unexplained definitions, see for example [10, 11].

3. Riemann-Lebesgue integrability with respect to an arbitrary set function

In this section, we study Riemann-Lebesgue integrability of vector functions with respect to an arbitrary non-negative set function, and point out some classic properties and continuity properties of the integral. In what follows, $m:\mathcal{A}\to[0,+\infty)$ is a non-negative set function.

As in [26, Definition 4.5] (for scalar functions) and [33, Definition 7] and [26], (for vector functions), we introduce the following definition:

Definition 3.1**.**

A function $f:T\to X$ is called absolutely (unconditionally respectively) Riemann-Lebesgue ($|RL|$) ($RL$ respectively) $m$-integrable (on $T$) if there exists $b\in X$ such that for every $\varepsilon>0$, there exists a countable partition $P_{\varepsilon}$ of $T$, so that for every other countable partition $P=\{A_{n}\}_{n\in\mathbb{N}}$ of $T$ with $P\geq P_{\varepsilon}$, $f$ is bounded on every $A_{n}$, with $m(A_{n})>0$ and for every $t_{n}\in A_{n}$, $n\in\mathbb{N}$, the series $\sum_{n=0}^{+\infty}f(t_{n})m(A_{n})$ is absolutely (unconditionally respectively) convergent and $\|\sum_{n=0}^{+\infty}f(t_{n})m(A_{n})-b\|<\varepsilon.$

The vector $b$ (necessarily unique) is called the absolute (unconditional) Riemann-Lebesgue $m$-integral of $f$ on $T$ and it is denoted by ${\scriptstyle(|RL|)}\displaystyle{\int_{T}}fdm$ (${\scriptstyle(RL)}\displaystyle{\int_{T}}fdm$ respectively).

The $|RL|$ ($RL$ respectively) definitions of the integrability on a subset $A\in\mathcal{A}$ are defined in the classical manner. Moreover, the next characterization holds:

Theorem 3.2**.**

For a vector function $f:T\to X$, the following properties hold:

• 3.2.a)

If $f$ is $|RL|$ $m$-integrable on $T$, then $f$ is $|RL|$ $m$-integrable on every $A\in\mathcal{A}$;

• 3.2.b)

$f$* is $|RL|$ $m$-integrable on every $A\in\mathcal{A}$ if and only if $f\chi_{A}$ is $|RL|$ $m$-integrable on $T$. In this case, ${\scriptstyle(|RL|)}\displaystyle{\int}_{A}fdm={\scriptstyle(|RL|)}\displaystyle{\int}_{T}f\chi_{A}dm.$*

(The same holds for $RL$-integrability).

Proof.

As soon as $f$ is $|RL|$ (RL) integrable in $T$, then it is integrable as well on every subset $A\in\mathcal{A}$. Indeed, fix any $A\in\mathcal{A}$ and denote by $J$ the integral of $f$; then, fixed any $\varepsilon>0$, there exists a partition $P_{\varepsilon}$ of $T$, such that, for every finer partition $P^{\prime}$, $P^{\prime}=\{A_{n}\}_{n\in\mathbb{N}}$ it is

[TABLE]

Now, denote by $P_{0}$ any partition finer than $P$ and also finer than $\{A,T\setminus A\}$, and by $P_{A}$ the partition of $A$ consisting of all the elements of $P_{0}$ that are contained in $A$. Next, let $\Pi_{A}$ and $\Pi^{\prime}_{A}$ denote two partitions of $A$ finer than $P_{A}$, and extend them with a common partition of $T\setminus A$ (also with the same tags) in such a way that the two resulting partitions, denoted by $\Pi$ and $\Pi^{\prime}$, are both finer than $P$. So, if we denote by $\sigma(f,P):=\sum_{n=0}^{+\infty}f(t_{n})m(A_{n}),\,\{A_{n}\in\Pi\}$,

[TABLE]

Now, setting:

[TABLE]

(with the obvious meaning of the symbols), one has

[TABLE]

By the arbitrariness of $\Pi_{A}$ and $\Pi^{\prime}_{A}$, this means that the sums $\sigma(f,\Pi_{A})$ satisfy a Cauchy principle in $X$, and so the first claim follows by completeness.

Now, let us suppose that $f$ is $|RL|$ $m$-integrable on $A\in\mathcal{A}$. Then for every $\varepsilon>0$ there exists a partition $P_{A}^{\varepsilon}\in\mathcal{P}_{A}$ so that for every partition $P_{A}=\{B_{n}\}_{n\in\mathbb{N}}$ of $A$ with $P_{A}\geq P_{A}^{\varepsilon}$ and for every $s_{n}\in B_{n},n\in\mathbb{N}$, we have

[TABLE]

Let us consider $P_{\varepsilon}=P_{A}^{\varepsilon}\cup\{T\setminus A\}$ , which is a partition of $T$. If $P=\{A_{n}\}_{n\in\mathbb{N}}$ is a partition of $T$ with $P\geq P_{\varepsilon}$, then without any loss of generality we may write $P=\{C_{n},D_{n}\}_{n\in\mathbb{N}}$ with pairwise disjoint $C_{n},D_{n}$ such that $A=\displaystyle\cup_{n=0}^{+\infty}C_{n}$ and $\displaystyle\cup_{n=0}^{+\infty}D_{n}=T\setminus A.$ Now, for every $u_{n}\in A_{n},n\in\mathbb{N}$ we get by (3.1):

[TABLE]

where $t_{n}\in C_{n},s_{n}\in D_{n},$ for every $n\in\mathbb{N},$ which says that $f\chi_{A}$ is $|RL|$ $m$ -integrable on $T$ and $\displaystyle{{\scriptstyle(|RL|)}\int_{T}}f\chi_{A}dm={\scriptstyle(|RL|)}\int_{A}fdm.$

Finally, suppose that $f\chi_{A}$ is $|RL|$ $m$-integrable on $T$. Then for every $\varepsilon>0$ there exists $P_{\varepsilon}=\{B_{n}\}_{n\in\mathbb{N}}\in\mathcal{P}$ so that for every $P=\{C_{n}\}_{n\in\mathbb{N}}$ partition of $T$ with $P\geq P_{\varepsilon}$ and every $t_{n}\in C_{n},n\in\mathbb{N}$, we have

[TABLE]

Let us consider $P_{A}^{\varepsilon}=\{B_{n}\cap A\}_{n\in\mathbb{N}}$, which is a partition of $A$. Let $P_{A}=\{D_{n}\}_{n\in\mathbb{N}}$ be an arbitrary partition of $A$ with $P_{A}\geq P_{A}^{\varepsilon}$ and $P=P_{A}\cup\{T\setminus A\}.$ Then $P\in\mathcal{P}$ and $P\geq P_{\varepsilon}.$ Let us take $t_{n}\in D_{n},n\in\mathbb{N}$ and $s\in T\setminus A$. By (3.2) we obtain

[TABLE]

which assures that $f$ is $|RL|$ $m$-integrable on $A$ and

[TABLE]

∎

If $X$ is finite dimensional, then $|RL|$ $m$-integrability is equivalent to $RL\,m$-integrability. In this case, it is called $m$-integrability for short and the integral is denoted by $\displaystyle{\int_{T}fdm}$.

We observe also that the $RL$-integral is stronger than the Birkhoff integral (in the sense of Fremlin), and it has been examined, in the countably additive case, by Potyrala in [33].

Obviously, if $f$ is $|RL|$ $m$-integrable, then it is also RL $m$-integrable.

Proposition 3.3**.**

If $f:T\to X$ is bounded and $\overline{m}(T)<+\infty$, then $f$ is $|RL|$ $m$-integrable and

[TABLE]

Proof.

The series $\sum_{n=0}^{+\infty}f(t_{n})m(A_{n})$ is absolutely convergent. Indeed, for every $n\in\mathbb{N}$, we have

[TABLE]

which means that the series $\sum_{n=0}^{+\infty}f(t_{n})m(A_{n})$ is absolutely convergent. Then clearly, if $f$ is $|RL|$ $m$-integrable, we have

[TABLE]

∎

Theorem 3.4**.**

Let $\overline{m}(T)<+\infty$ and let $f:T\rightarrow X$ be bounded. If $f$ $=0$ $m$-a.e., then $f$ is $|RL|$ $m$-integrable and $\displaystyle{{\scriptstyle(|RL|)}\int_{T}}fdm=0$.

Proof.

Since $f$ is bounded, then there exists $M\in[0,+\infty)$ such that $\|f(t)\|\leq M$, for every $t\in T$. If $M=0$, then the conclusion is obvious. Suppose $M>0.$ Let us denote by $A$ the set $A:=\{t\in T;f(t)\neq 0\}$. Since $f=0$ $m$-a.e., we have $\widetilde{m}(A)=0$. Then, for every $\varepsilon>0$, there exists $B_{\varepsilon}\in\mathcal{A}$ so that $A\subseteq B_{\varepsilon}$ and $\overline{m}(B_{\varepsilon})<\varepsilon/M.$ Let us take the partition $P_{\varepsilon}=\{C_{n}\}_{n\in\mathbb{N}}$ of $B_{\varepsilon}$, and let $C_{0}=T\setminus B_{\varepsilon}$ and add $C_{0}$ to $P_{\varepsilon}$.

Let us consider an arbitrary partition $P=\{A_{n}\}_{n\in\mathbb{N}}$ of $T$ such that $P\geq P_{\varepsilon}$.

Without any loss of generality, we suppose that $P=\{D_{n},E_{n}\}_{n\in\mathbb{N}}\subset\mathcal{A},$ with pairwise disjoint sets $D_{n},E_{n}$ such that $\bigcup_{n\in\mathbb{N}}D_{n}=C_{0}$ and $\bigcup_{n\in\mathbb{N}}E_{n}=B_{\varepsilon}.$ Let $t_{n}\in D_{n},s_{n}\in E_{n}$, for every $n\in\mathbb{N}.$ Then we can write

[TABLE]

which ensures that $f$ is $|RL|$ $m$-integrable and $\displaystyle{{\scriptstyle(|RL|)}\int_{T}}fdm=0.$ ∎

The following properties hold:

Theorem 3.5**.**

Let $f,g:T\rightarrow X$ be $|RL|$ $m$-integrable functions and $\alpha\in\mathbb{R}$. Then:

3.5**.a): **

$\alpha f$* is $|RL|$ $m$-integrable and $\displaystyle{{\scriptstyle(|RL|)}\int_{T}}\alpha fdm=\alpha\displaystyle{{\scriptstyle(|RL|)}\int_{T}}fdm.$*

3.5**.b): **

$f$* is $|RL|$ $\alpha m$-integrable (for $\alpha\in[0,+\infty))$ and*

[TABLE]

3.5**.c): **

$f\pm g$* is $|RL|$ $m$-integrable and*

[TABLE]

A similar result holds also for the $RL$-integrability.

Proof.

Statements 3.5.a), 3.5.b) are straightforward. Let us prove 3.5.c) for the sum. Let $\varepsilon>0$ be arbitrary. Since $f$ is $|RL|$ $m$-integrable, then there exists a countable partition $P_{1}\in\mathcal{P}$ so that for every other countable partition $P\in\mathcal{P},P=\{A_{n}\}_{n\in\mathbb{N}},$ with $P\geq P_{1}$ and every $t_{n}\in A_{n},n\in\mathbb{N}$, the series $\sum_{n=0}^{+\infty}f(t_{n})m(A_{n})$ and $\sum_{n=0}^{+\infty}g(t_{n})m(A_{n})$ are absolutely convergent and

[TABLE]

Then, for every countable partition $P=\{C_{n}\}_{n\in\mathbb{N}}\in\mathcal{P},$ with $P\geq P_{1}$ and every $t_{n}\in C_{n},n\in\mathbb{N}$, the series $\sum_{n=0}^{+\infty}(f+g)(t_{n})m(C_{n})$ is absolutely convergent and we get

[TABLE]

Hence $f+g$ is $|RL|$ $m$-integrable and (3.3) is satisfied. ∎

We point out that the previous result holds without any additivity condition on $m$. In particular, for every $f:T\rightarrow X$ that is $|RL|$ ($RL$) $m$-integrable on every set $A\in\mathcal{A}$, let $I_{f}:\mathcal{A}\rightarrow X$ defined by

[TABLE]

We have:

Corollary 3.6**.**

Let $f:T\rightarrow X$ be any $|RL|$ (resp. $RL$) $m$-integrable function. Then $I_{f}$ is finitely additive.

Proof.

It follows from Theorem 3.2 and Theorem 3.5. ∎

Corollary 3.7**.**

Suppose $\overline{m}(T)<+\infty$. Let $f,g:T\rightarrow X$ be vector functions with $\sup\limits_{t\in T}\|f(t)-g(t)\|<+\infty$, $f$ is $|RL|$ $m$-integrable and $f=g$ $m$-a.e.. Then $g$ is $|RL|$ $m$-integrable and

[TABLE]

Proof.

It is enough to observe that $g-f$ satisfies the hypotheses of Proposition 3.4, and so it is $|RL|$-integrable, with null integral. Then $g=(g-f)+f$ is integrable as well, and its integral coincides with ${\scriptstyle(|RL|)}\displaystyle{\int}_{T}fdm$ by Theorem 3.5. ∎

By a similar reasoning, as in Theorem 3.5.c), we get the following result:

Theorem 3.8**.**

Let $m_{i}:\mathcal{A}\rightarrow[0,+\infty)$ be a non-negative set function, $i=1,2$. Suppose $f:T\rightarrow X$ is both $|RL|$ (resp. RL) $m_{1}$-integrable and $|RL|$ (resp. RL) $m_{2}$-integrable. Then $f$ is $|RL|$ (resp. RL) $m_{1}+m_{2}$-integrable and

[TABLE]

Theorem 3.9**.**

Let $\overline{m}(T)<+\infty$. If $f,g:T\rightarrow X$ are $|RL|$ $m$-integrable functions, then

[TABLE]

Proof.

If $\sup_{t\in T}\|f(t)-g(t)\|=+\infty$, then the conclusion is obvious. If $\sup_{t\in T}\|f(t)-g(t)\|<+\infty$ the conclusion follows directly from Theorem 3.5 and Proposition 3.3, applied to $f-g$. ∎

Theorem 3.10**.**

If $f,g:T\rightarrow\mathbb{R}$ are $m$-integrable functions such that $f(t)\leq g(t),$ for every $\mathit{t\in T,}$ then $\displaystyle{\int_{T}}fdm\leq\displaystyle{\int_{T}}gdm$.

Proof.

Let $\varepsilon>0$ be arbitrary. Since $f,g$ are $m$-integrable, there exists $P_{0}\in\mathcal{P}$ such that for every $P=\{C_{n}\}_{n\in\mathbb{N}}\in\mathcal{P},P\geq P_{0}$ and every $t_{n}\in C_{n},n\in\mathbb{N}$, the series $\sum_{n=0}^{+\infty}f(t_{n})m(C_{n})$, $\sum_{n=0}^{+\infty}g(t_{n})m(C_{n})$ are absolutely convergent and

[TABLE]

Therefore

[TABLE]

since, by the hypothesis, $\sum_{n=0}^{+\infty}f(t_{n})m(C_{n})\leq\sum_{n=0}^{+\infty}g(t_{n})m(C_{n}).$

Consequently, $\displaystyle{\int_{T}}fdm-\displaystyle{\int_{T}}gdm\leq 0.$ ∎

We analogously obtain the following theorem.

Theorem 3.11**.**

Let $m_{1}$, $m_{2}:\mathcal{A}\rightarrow[0,+\infty)$ be set functions such that $m_{1}(A)\leq m_{2}(A)$, for every $A\in\mathcal{A}$ and $f:T\rightarrow[0,\infty)$ a simultaneously $m_{1}$-integrable and $m_{2}$-integrable function. Then $\displaystyle{\int_{T}}fdm_{1}\leq\displaystyle{\int_{T}}fdm_{2}$.

In the next theorem we point out some results concerning the properties of the set function $I_{f}$.

Theorem 3.12**.**

Let $f:T\rightarrow X$ be a vector function such that $f$ is $|RL|$ $m$-integrable on $T$. Then the following properties hold:

• 3.12.i)

If $f$ is bounded, and $\overline{m}(T)<+\infty$, then

• 3.12.i.a)

$I_{f}\ll\overline{m}$* (in the $\varepsilon$-$\delta$ sense) and $I_{f}$ is of finite variation.*

• 3.12.i.b)

If moreover $\overline{m}$ is o-continuous (exhaustive respectively), then $I_{f}$ is also o-continuous (exhaustive respectively).

• 3.12.ii)

If $f:T\rightarrow[0,\infty)$ and $m$ is scalar-valued and monotone, then the same holds for $I_{f}$.

Proof.

3.12.i.a) The absolute continuity of $I_{f}$ in the $\varepsilon$-$\delta$ sense follows from Proposition 3.3. In order to prove that $I_{f}$ is of finite variation let $\{A_{i}\}_{i=1,\ldots,n}\subset\mathcal{P}(T)$ be pairwise disjoint sets and $M=$ $\underset{t\in T}{\sup}\|f(t)\|.$ By Proposition 3.3 and Remark 2.5, it follows

[TABLE]

This implies $\overline{I_{f}}(T)\leq M$ $\overline{m}(T)$, which ensures that $I_{f}$ is of finite variation.

3.12.i.b) Let $M=\sup_{t\in T}\|f(t)\|.$ If $M=0$, then $f=0$, hence $I_{f}=0.$

Let us suppose that $M>0$. According to Proposition 3.3 we have $\|I_{f}(A)\|\leq M\cdot\overline{m}(A),$ for every $A\in\mathcal{A}.$ If $\overline{m}$ is o-continuous, it follows that $I_{f}$ is also o-continuous. The proof is similar when $\overline{m}$ is exhaustive.

3.12.ii) Let be $A,B\in\mathcal{A}$ with $A\subseteq B$ and $\varepsilon>0$. Since $f$ is $m$-integrable on $A$, there exists a countable partition $P_{1}=\{C_{n}\}_{n\in\mathbb{N}}\in\mathcal{P}_{A}$ so that for every other countable partition $P=\{A_{n}\}_{n\in\mathbb{N}}\in\mathcal{P}_{A},P\geq P_{1}$ and every $t_{n}\in A_{n},n\in\mathbb{N}$, the series $\sum\limits_{n=0}^{+\infty}f(t_{n})m(A_{n})$ is absolutely convergent and

[TABLE]

Analogously, since $f$ is $m$-integrable on $B$, there exists a countable partition $P_{2}=\{D_{n}\}_{n\in\mathbb{N}}\in\mathcal{P}_{B}$ such that for every other countable partition $P=\{B_{n}\}_{n\in\mathbb{N}}\in\mathcal{P}_{B},P\geq P_{2}$ and every $t_{n}\in B_{n},n\in\mathbb{N}$, the series $\sum\limits_{n=0}^{+\infty}f(t_{n})m(B_{n})$ is absolutely convergent and

[TABLE]

Consider $\widetilde{P}_{1}=\{C_{n},B\setminus A\}_{n\in\mathbb{N}}$. Then $\widetilde{P}_{1}\in\mathcal{P}_{B}$ and $\widetilde{P}_{1}\wedge P_{2}\in\mathcal{P}_{B}.$ Let $P=\{E_{n}\}_{n\in\mathbb{N}}\in\mathcal{P}_{B}$ be an arbitrary countable partition, with $P\geq\widetilde{P}_{1}\wedge P_{2}$. We observe that $P^{{}^{\prime\prime}}=\{E_{n}\cap A\}_{n\in\mathbb{N}}$ is also a partition of $A$ and $P^{{}^{\prime\prime}}\geq P_{1}.$ If $t_{n}\in E_{n}\cap A,n\in\mathbb{N}$, by (3.5) and (3.6), we have $|I_{f}(B)-\sum\limits_{n=0}^{+\infty}f(t_{n})m(E_{n})|<\dfrac{\varepsilon}{2}$ and $|I_{f}(A)-\sum\limits_{n=0}^{+\infty}f(t_{n})m(E_{n}\cap A)|<\dfrac{\varepsilon}{2}$. Therefore

[TABLE]

From the hypothesis, it results $\sum\limits_{n=0}^{+\infty}f(t_{n})m(E_{n}\cap A)\leq\sum\limits_{n=0}^{+\infty}f(t_{n})m(E_{n}).$ Consequently, $I_{f}(A)-I_{f}(B)\leq\varepsilon$, for every $\varepsilon>0$, which implies $I_{f}(A)\leq I_{f}(B).$ ∎

4. Comparative results

In this section, comparison results among Riemann-Lebesgue, and Birkhoff simple [10] integrabilities are presented.

Definition 4.1**.**

([10, Definition 3.2]) A vector function $f:T\rightarrow X$ is called Birkhoff simple $m$-integrable (on $T$) if there exists $a\in X$ such that for every $\varepsilon>0$, there exists a countable partition $P_{\varepsilon}$ of $T$ so that for every other countable partition $P=\{A_{n}\}_{n\in\mathbb{N}}$ of $T$, with $P\geq P_{\varepsilon}$ and every $t_{n}\in A_{n},n\in\mathbb{N},$ it holds $\limsup_{n\rightarrow+\infty}\|\sum_{k=0}^{n}f(t_{k})m(A_{k})-a\|<\varepsilon.$

The vector $a$ is denoted by $(Bs)\displaystyle{\int_{T}}fdm$ and it is called the Birkhoff simple integral of $f$ (on $T$) with respect to $m$.

Theorem 4.2**.**

If $f:T\rightarrow X$ is RL $m$-integrable, then $f$ is also Birkhoff simple $m$-integrable and the two integrals coincide.

Proof.

Let $\varepsilon>0$ be arbitrary and let $P_{\varepsilon}$ the countable partition of $T$ that one given by Definition 3.1. Considering any countable partition $P=\{A_{n}\}_{n\in\mathbb{N}}$ of $T$, with $P\geq P_{\varepsilon}$ and $t_{n}\in A_{n},n\in\mathbb{N}$, by Definition 3.1, it results that the series $\sum\limits_{n=0}^{+\infty}f(t_{n})m(A_{n})$ is unconditionally convergent and

[TABLE]

which implies the Birkhoff simple $m$-integrability of $f$ and $(Bs)\displaystyle{\int_{T}}fdm={\scriptstyle(|RL|)}\displaystyle{\int_{T}}fdm.$ ∎

Next, we highlight some comparison results between Riemann-Lebesgue and Gould integrabilities. We first recall the Gould integrability for vector functions (called RLF-integrability in [33]).

Let $(X,\|\cdot\|)$ be a Banach space. If $f:T\rightarrow X$ is a vector function, we denote by $\sigma(P)$ the sum $\sigma(P):=\sum\limits_{i=1}^{n}f(t_{i})m(A_{i}),$ for every finite partition of $T$, $P=\{A_{i}\}_{i=1,\ldots,n}\in\mathcal{P}$ and every $t_{i}\in A_{i},i=1,\ldots,n.$

Definition 4.3**.**

([25]) A vector function $f:T\rightarrow X$ is called Gould $m$-integrable (on $T$) if the net $(\sigma(P))_{P\in(\mathcal{P},\leq)}$ is convergent in $X$, where the family $\mathcal{P}$ of all finite partitions of $T$ is ordered by the relation ”$\leq$” given in Definition 2.1.ii). The limit of $(\sigma(P))_{P\in(\mathcal{P},\leq)}$ is called the Gould integral of $f$ with respect to $m$, denoted by $(G)\displaystyle{\int_{T}}fdm.$

Remark 4.4**.**

$f$ is Gould $m$-integrable if and only if there exists $a\in X$ such that for every $\varepsilon>0$, there exists a finite partition $P_{\varepsilon}$ of $T$, so that for every other finite partition $P=\{A_{i}\}_{i=1,\ldots,n}$ of $T$, with $P\geq P_{\varepsilon}$ and every $t_{i}\in A_{i},i=1,\ldots,n,$ we have $\|\sigma(P)-a\|<\varepsilon.$

Proposition 4.5**.**

Let $m:\mathcal{A}\rightarrow[0,+\infty)$ be a complete $\sigma$-additive measure of finite variation, and $f:T\rightarrow\mathbb{R}$ a bounded function. Then $f$ is Gould $m$-integrable if and only if $f$ is Riemann-Lebesgue $m$-integrable and the two integrals coincide.

Proof.

It is an easy consequence of [15, Theorem 5.1] and of [33, Theorem 8]. ∎

Example 4.6**.**

We show now that, without assuming additivity of $m$, the previous equivalence does not hold. Suppose that $T=\mathbb{N}$, with $\mathcal{A}=\mathcal{P}(\mathbb{N})$, and define the scalar measure $m$ as follows:

[TABLE]

Then, the constant function $f(n)\equiv 1$ is clearly simple-Birkhoff integrable, with null integral, and also RL integrable with the same integral: it suffices to take as partition $P_{\varepsilon}$ the finest possible one, consisting of all singletons.

However, this mapping is not Gould-integrable. In fact, if $P_{\varepsilon}$ is any finite partition of $\mathbb{N}$, some of its sets are infinite, so the quantity $\sigma(P_{\varepsilon})$ coincides with the number of the infinite sets of $P_{\varepsilon}$. And the quantity $\sigma(P)$ is unbounded when $P$ runs over the family of all finer partitions of $P_{\varepsilon}$.

Theorem 4.7**.**

*Suppose $m:\mathcal{A}\rightarrow[0,+\infty)$ is monotone, $\sigma$-subadditive and of finite variation and let $f:T\rightarrow\mathbb{R}$ be a bounded function. Then $f$ is Gould $m$-integrable if and only if $f$ is Riemann-Lebesgue $m$-integrable.

In this case, $(G)\displaystyle{\int_{T}}fdm={\scriptstyle(|RL|)}\displaystyle{\int_{T}}fdm.$*

Proof.

Since $m$ is a submeasure of finite variation, according to [21, Theorem 2.15], $f$ is Gould $m$-integrable if and only if $f$ is Gould $\overline{m}$-integrable and $(G)\displaystyle{\int_{T}}fdm=(G)\displaystyle{\int_{T}}fd\overline{m}.$ On the other hand, from Remark 2.5 and Proposition 4.5, Gould $\overline{m}$-integrability is equivalent to Riemann-Lebesgue and the two integrals coincide. Now the conclusion follows by [15, Theorem 5.5]. ∎

In the general case only partial results can be obtained, for example:

Theorem 4.8**.**

Suppose that $m:\mathcal{A}\rightarrow[0,+\infty)$ satisfies property ($\sigma$), is monotone, null-additive set function and $A\in\mathcal{A}$ is an atom of $m$. If a bounded function $f:T\rightarrow\mathbb{R}$ is Riemann-Lebesgue $m$-integrable on $A$, then $f$ is Gould $m$-integrable on $A$ and $(G)\displaystyle{\int_{A}}fdm={\scriptstyle(|RL|)}\displaystyle{\int_{A}}fdm.$

Proof.

It follows in an analogous way as the previous theorem, using in this case [15, Theorem 5.2]. ∎

Acknowledgements

The first and the last authors have been partially supported by University of Perugia – Department of Mathematics and Computer Sciences– Ricerca di Base 2018 ”Metodi di Teoria dell’Approssimazione, Analisi Reale, Analisi Nonlineare e loro applicazioni” and the last author by GNAMPA - INDAM (Italy) ”Metodi di Analisi Reale per l’Approssimazione attraverso operatori discreti e applicazioni” (2019).