$L^1$-spaces of vector measures with vector density

TL;DR

This paper explores the structure of $L^1$-spaces associated with vector measures that have vector densities, establishing connections with various integrability concepts like Pettis and Bochner integrals.

Contribution

It introduces a framework linking $L^1$-spaces of vector measures with different integrability types, clarifying their relationships and properties.

Findings

Identifies connections between $L^1_w( u_F)$, $L^1( u_F)$, and $L^1(| u_F|)$ spaces.

Provides conditions under which vector measures have densities with respect to positive measures.

Clarifies the relationship between vector measure integrability and classical function spaces.

Abstract

Let $F$ be a function with values in a Banach space. When $F$ is locally (Pettis or Bochner) integrable with respect to a locally determined positive measure, a vector measure $\nu_F$ with density $F$ defined on a $\delta$-ring is obtained. We present the existing connection between the spaces $L^1_w(\nu_F)$, $L^1(\nu_F)$ and $L^1(|\nu_F|)$ and the spaces of Dunford, Pettis or Bochner integrable functions.

Equations

$$\Vert F{\Vert }_{\mathbb{D}}=\underset{x^{\ast }\in B_{X^{\ast }}}{\sup }{\int }_{\Omega }|⟨F,x^{\ast }⟩|d\mu ,\forall F\in \mathbb{D}(\mu ,X).$$

$$\Vert F{\Vert }_{\mathbb{D}}=\underset{x^{\ast }\in B_{X^{\ast }}}{\sup }{\int }_{\Omega }|⟨F,x^{\ast }⟩|d\mu =\underset{x^{\ast }\in B_{X^{\ast }}}{\sup }\Vert T{x}^{\ast }{\Vert }_{L^1(\mu )}\le \Vert T\Vert .■$$

$$|\lambda |(A):=\sup \left\{\sum _{j=1}^n|\lambda (A_j)|:\{A_j\}\in {\pi }_A\right\}.$$

$$|\nu |(A):=\sup \left\{\sum _{j=1}^n\Vert \nu (A_j){\Vert }_X:\{A_j\}\in {\pi }_A\right\}.$$

$$\Vert \nu \Vert (A):=\sup \{|⟨\nu ,x^{\ast }⟩|(A):x^{\ast }\in B_{X^{\ast }}\},$$

$$⟨\nu ,x^{\ast }⟩(B)=⟨\nu (B),x^{\ast }⟩,\forall B\in \mathcal{R}.$$

$$\Vert f{\Vert }_{\nu }:=\sup \left\{{\int }_{\Omega }|f|d|⟨\nu ,x^{\ast }⟩|:x^{\ast }\in B_{X^{\ast }}\right\},$$

$$⟨x_A,x^{\ast }⟩={\int }_{A}fd⟨\nu ,x^{\ast }⟩,\forall x^{\ast }\in X^{\ast }.$$

$${\tilde{\nu }}_F(A)=\mathbb{P}-{\int }_{A}Fd\mu ,\forall A\in \Sigma .$$

$$\Vert {\tilde{\nu }}_F\Vert (A)=\underset{x^{\ast }\in B_{X^{\ast }}}{\sup }{\int }_A|⟨F,x^{\ast }⟩|d\mu ,\forall A\in \Sigma .$$

$$\begin{array}[]{lll}\displaystyle{\left\langle\mathbb{P}-\int_{\bigcup_{n}A_{n}}Fd\mu,x^{*}\right\rangle}&=&\displaystyle{\int_{\bigcup_{n}A_{n}}\langle F,x^{*}\rangle d\mu\vspace{3 pt}}\\ &=&\displaystyle{\sum_{n=1}^{\infty}\int_{A_{n}}\langle F,x^{*}\rangle d\mu=\sum_{n=1}^{\infty}\left\langle\mathbb{P}-\int_{A_{n}}Fd\mu,x^{*}\right\rangle}.\end{array}$$

$$|⟨{\tilde{\nu }}_F,x^{\ast }⟩|(A)={\int }_A|⟨F,x^{\ast }⟩|d\mu .$$

$$|{\tilde{\nu }}_F|(A)={\int }_A\Vert F{\Vert }_{X}d\mu ,\forall A\in \Sigma .$$

$$\mathbb{B}(\mu ,X)\subset \mathbb{B}(\mu ,X{)}^{loc}\subset \mathbb{P}(\mu ,X{)}^{loc}.$$

$$F(t):=f(t)x.$$

$${\int }_B\Vert F{\Vert }_{X}d\mu ={\int }_B|f|\Vert x{\Vert }_{X}d\mu =\left({\int }_B|f|d\mu \right)\Vert x{\Vert }_X.$$

$${\nu }_F(B):=\mathbb{P}-{\int }_{B}Fd\mu ,\forall B\in {\Sigma }^f,$$

$$\Vert {\nu }_F\Vert (A)=\underset{x^{\ast }\in B_{X^{\ast }}}{\sup }{\int }_A|⟨F,x^{\ast }⟩|d\mu ,\forall A\in \Sigma .$$

$$\begin{array}[]{lll}\nu_{F}(B)&=&\displaystyle{\mathbb{P}-\int_{B}Fd\mu=\mathbb{P}-\int_{B}\chi_{B}Fd\mu\vspace{5 pt}}\\ &=&\displaystyle{\sum_{n=1}^{\infty}\mathbb{P}-\int_{B_{n}}\chi_{B}Fd\mu=\sum_{n=1}^{\infty}\mathbb{P}-\int_{B_{n}}Fd\mu=\nu_{F}(B_{n})}.\end{array}$$

$$|{\mu }_{⟨F,x^{\ast }⟩}|(A)={\int }_A|⟨F,x^{\ast }⟩|d\mu ,\forall A\in \Sigma .$$

$$⟨{\nu }_F,x^{\ast }⟩(B)=⟨\mathbb{P}-{\int }_{B}Fd\mu ,x^{\ast }⟩={\int }_B⟨F,x^{\ast }⟩d\mu ={\mu }_{⟨F,x^{\ast }⟩}(B).$$

$$|⟨{\nu }_F,x^{\ast }⟩|(A)={\int }_A|⟨F,x^{\ast }⟩|d\mu .$$

$$|{\nu }_F|(B)=|{\nu }_{{\chi }_{B}F}|(B)={\int }_B\Vert F{\Vert }_{X}d\mu .$$

$$|{\nu }_F|(A)=\underset{B\in {\Sigma }_A^f}{\sup }|{\nu }_F|(B)=\underset{B\in {\Sigma }_A^f}{\sup }{\int }_B\Vert F{\Vert }_{X}d\mu ={\int }_A\Vert F{\Vert }_{X}d\mu ,\forall A\in \Sigma .■$$

$${\nu }_F(B)=\mathbb{B}-{\int }_{B}Fd\mu =\left({\int }_{B}fd\mu \right)x,\forall B\in {\Sigma }^f.$$

$$\begin{array}[]{lll}\|\nu_{F}\|(A)&=&\displaystyle{\sup_{x^{\ast}\in B_{X^{\ast}}}\int_{A}|\langle F,x^{\ast}\rangle|d\mu=\sup_{x^{\ast}\in B_{X^{\ast}}}\int_{A}|f||\langle x,x^{\ast}\rangle|d\mu\vspace{8 pt}}\\ &=&\displaystyle{\left(\int_{A}|f|d\mu\right)\sup_{x^{\ast}\in B_{X^{\ast}}}|\langle x,x^{\ast}\rangle|=\left(\int_{A}|f|d\mu\right)\|x\|_{X}\vspace{8 pt}}\\ &=&\displaystyle{\int_{A}\|F\|_{X}d\mu=|\nu_{F}|(A)}.\end{array}$$

$$\begin{array}[]{lll}\displaystyle{\int_{\Omega}|s|d|\langle\nu_{F},x^{\ast}\rangle|}&=&\displaystyle{\sum_{j=1}^{n}}|a_{j}|\hskip 1.0pt|\langle\nu_{F},x^{\ast}\rangle|(A_{j})=\displaystyle{\sum_{j=1}^{n}}|a_{j}|\int_{A_{j}}|\langle F,x^{\ast}\rangle|d\mu\vspace{5 pt}\\ &=&\displaystyle{\int_{\Omega}\sum_{j=1}^{n}}|a_{j}|\chi_{A_{j}}|\langle F,x^{\ast}\rangle|d\mu=\int_{\Omega}|\langle sF,x^{\ast}\rangle|d\mu.\end{array}$$

$${\int }_{\Omega }sd⟨{\nu }_F,x^{\ast }⟩={\int }_{\Omega }⟨sF,x^{\ast }⟩d\mu .$$

$$\begin{array}[]{lll}|\langle gF,x^{\ast}\rangle|d\mu&=&\displaystyle{\lim_{n\rightarrow\infty}\int_{\Omega}|\langle s_{n}F,x^{\ast}\rangle|d\mu\vspace{3 pt}}\\ &=&\displaystyle{\lim_{n\rightarrow\infty}\int_{\Omega}s_{n}d|\langle\nu_{F},x^{\ast}\rangle|=\int_{\Omega}|g|d|\langle\nu_{F},x^{\ast}\rangle|},\end{array}$$

$${\int }_{\Omega }⟨gF,x^{\ast }⟩d\mu =\underset{n\to \infty }{\lim }{\int }_{\Omega }⟨s_{n}F,x^{\ast }⟩d\mu =\underset{n\to \infty }{\lim }{\int }_{\Omega }{s}_{n}d⟨{\nu }_F,x^{\ast }⟩={\int }_{\Omega }gd⟨{\nu }_F,x^{\ast }⟩.$$

Full text

††journal: Indagationes Mathematicae

1 Introduction

Let us consider a Banach space $X$ and a $\sigma$-algebra $\Sigma$ on $\Omega$. Since the integrability of $X$-valued functions with respect to a positive finite measure defined in $\Sigma$ was introduced by B. J. Pettis and S. Bochner in the thirties of the last century this theory has been well studied by several authors; see [6], [8], [12]. After that the theory of integration of scalar functions with respect to $X$-valued measures defined on $\Sigma$ was created by R. G. Bartle, N. Dunford and J. Schwartz in [2]. It turns out that the Pettis and the Bochner integrals define vector measures, these ones were studied among other by J. Diestel and J. J. Uhl in [6]. On the other hand N. Dinculeanu and J. J. Uhl introduced the concept of $\mathcal{R}$-locally (Pettis and Bochner) integrable function with respect to a locally $\sigma$-finite positive measure $\lambda$ defined on a $\delta$-ring $\mathcal{R}$; see [7]. The extension of the integration theory with respect to vector measures defined on $\delta$-rings was developed by D. R. Lewis [9], P. R. Masani and H. Niemi [10] and O. Delgado [4]. As one might expect the $\mathcal{R}$-locally (Pettis and Bochner) integrals define vector measures on $\mathcal{R}$.

In this paper we consider a locally determined positive measure $\mu$ on $\Sigma$. Then the collection $\Sigma^{f}$ consisting of those subsets in $\Sigma$ which have $\mu$-finite measure is a $\delta$-ring and if $\mu$ is restricted to $\Sigma^{f}$, a locally $\sigma$-finite measure is obtained; so makes sense to consider vector functions which are $\Sigma^{f}$-locally (Pettis or Bochner) integrable, these functions will be simply called locally (Pettis or Bochner) integrable functions. In section 2 we recall the basic concepts relative to the Dunford, the Pettis and the Bochner integrals with respect to a positive measure and also the main results about vector measures defined on $\delta$-rings. In section 3 we study briefly the vector measure $\nu_{F}$ defined on $\Sigma^{f}$ by the integral of a locally Pettis integrable function $F:\Omega\rightarrow X$ over each $B\in\Sigma^{f}$ and a description of its corresponding semivariation is given. If additionally the function $F$ is locally Bochner integrable we provide a characterization of the variation of the measure $\nu_{F}$. Finally in section 4 we present the existing connection between the weakly $\nu_{F}$-integrable, $\nu_{F}$-integrable or $|\nu_{F}|$-integrable functions and the Dunford, Pettis or Bochner integrable functions as well as their corresponding integrals. In this way Theorems $8$ and $13$ established by G.F. Stefansson in [12] are generalized in two directions, namely the positive measure is no longer necessarily finite but locally determined and the vector function $F$ is now locally integrable. Besides we obtain some conditions to determine when a locally Pettis integrable function is in fact Pettis integrable.

2 Preliminary results

2.1 Bochner and Pettis integrals

Throughout the paper $\Omega$ will be a non empty set and $X$ stands for a Banach space over $\mathbb{K}$ ($\mathbb{R}$ or $\mathbb{C}$). We denote by $X^{*}$ and $B_{X}$ its dual space and its unit ball respectively. Let us consider a $\sigma$-algebra $\Sigma$ on $\Omega$ and a positive measure $\mu:\Sigma\rightarrow[0,\infty]$. The set $\Sigma^{f}$ consists of the subsets $B\in\Sigma$ such that $\mu(B)<\infty$ and $\mathcal{N}_{0}(\mu)$ is the collection of $\mu$-null sets. Recall that the measure $\mu$ is said to be semi-finite if for each set $A\in\Sigma$ such that $\mu(A)>0$, there exists a subset $B\in\Sigma^{f}$ satisfying that $B\subset A$ and $0<\mu(B)$. The measure $\mu$ is locally determined if it is semi-finite and $\Sigma=\{A\subset\Omega\ |\ A\cap B\in\Sigma,\ \forall\ B\in\Sigma^{f}\}$.

We denote by $St(\mu,X)$ the vector space of $X$-valued simple functions whose support has finite measure. An $X$-valued function $F:\Omega\rightarrow X$ is said to be strongly $\mu$-measurable if there exists a sequence $\{S_{n}\}\subset St(\mu,X)$, which converges pointwise to $F$ $\mu$-a.e. and to be weakly $\mu$-measurable if $\langle F,x^{*}\rangle:\Omega\rightarrow\mathbb{R}$ is strongly $\mu$-measurable for any $x^{*}\in X^{*}$. Clearly each strongly $\mu$-measurable function is a weakly $\mu$-measurable function. We say that two functions $F,G:\Omega\rightarrow X$ are weakly equal $\mu$-a.e. if $\langle F,x^{*}\rangle=\langle G,x^{*}\rangle$ $\mu$-a.e. for all $x^{*}\in X^{*}$. We will denote by $L^{0}(\mu,X)$ the vector space that consists of the equivalence classes that are obtained by identifying strongly $\mu$-measurable functions if they are equal $\mu$- a.e. and $L^{0}_{w}(\mu,X)$ the vector spaces formed by the equivalence classes that are obtained when we identified weakly $\mu$-measurable functions if they are weakly equal $\mu$-a.e. We write $\mathbb{B}(\mu,X)$ to indicate the Banach space of the Bochner integrable functions, namely the functions $F\in L^{0}(\mu,X)$ such that $\|F\|_{X}\in L^{1}(\mu)$, with the norm defined by $\|F\|_{1}=\int_{\Omega}\|F\|_{X}d\mu$. On the other hand a function $F\in L^{0}_{w}(\mu,X)$ is Dunford integrable when $\langle F,x^{*}\rangle\in L^{1}(\mu)$, $\forall\ x^{*}\in X^{*}$, if additionally for each $A\in\Sigma$ there exists a vector $x_{A}\in X$ such that $\int_{A}\langle F,x^{*}\rangle d\mu=\langle x_{A},x^{*}\rangle$; $\forall\ x^{*}\in X^{*}$, the function $F$ is Pettis integrable and the vector $x_{A}$ is called the Pettis integral of $F$ over $A$ and it is denoted by $\mathbb{P}-\int_{A}Fd\mu$. We write $\mathbb{D}(\mu,X)$ and $\mathbb{P}(\mu,X)$ for the vector spaces consisting of the Dunford and Pettis integrable functions respectively.

Lemma 2.1

The space $\mathbb{D}(\mu,X)$ is a normed space with the norm given by

[TABLE]

Proof.

Observe that to obtain the conclusion it only remains to establish that $\|F\|_{\mathbb{D}}<\infty$, $\forall\ F\in\mathbb{D}(\mu,X)$. So, let us fix $F\in\mathbb{D}(\mu,X)$ and define $T:X^{\ast}\rightarrow L^{1}(\mu)$ by $T(x^{\ast})=\langle F,x^{\ast}\rangle,\ \forall\ x^{\ast}\in X^{\ast}$. Clearly $T$ is a well defined linear operator. Now take $\{x^{\ast}_{n}\}\subset X^{\ast}$ and $x^{\ast}\in X^{\ast}$ such that $x^{\ast}_{n}\rightarrow x$. Let us assume that there exists $g\in L^{1}(\mu)$ satisfying that $Tx^{\ast}_{n}\rightarrow g$ in $L^{1}(\mu)$. Proceeding as in [5, p.46] we get a subsequence $\{x^{\ast}_{n_{k}}\}\subset\{x^{\ast}_{n}\}$ such that $\langle F,x^{\ast}_{n_{k}}\rangle=Tx^{\ast}_{n_{k}}\rightarrow g$ $\mu$-a.e. On the other hand $\langle F(t),x^{\ast}_{n}\rangle\rightarrow\langle F(t),x^{\ast}\rangle$, $\forall\ t\in\Omega$. Thus, $T(x^{\ast})=\langle F(t),x^{\ast}\rangle=g$, $\mu$-a.e. By the closed graph theorem we get that $T$ is bounded. Therefore

[TABLE]

Since $\mathbb{P}(\mu,X)\subset\mathbb{D}(\mu,X)$ it turns out that $\mathbb{P}(\mu,X)$ is also a normed space with the same norm $\|\cdot\|_{\mathbb{D}}$ which will be denoted by $\|\cdot\|_{\mathbb{P}}$ in this case. It is well know that $\mathbb{B}(\mu,X)\subset\mathbb{P}(\mu,X)$ with $\|F\|_{\mathbb{P}}\leq\|F\|_{1}$ and $\mathbb{B}-\int_{A}Fd\mu=\mathbb{P}-\int_{A}Fd\mu$, $\forall\ F\in\mathbb{B}(\mu,X)$.

2.2 Integration with respect to measures defined on $\delta$-rings

A family $\mathcal{R}$ of subsets of $\Omega$ is a $\delta$-ring if $\mathcal{R}$ is a ring which is closed under countable intersections. We denote by $\mathcal{R}^{loc}$ the $\sigma$-algebra of all sets $A\subset\Omega$ such that $A\cap B\in\mathcal{R}$, $\forall\ B\in\mathcal{R}$. Given $A\in\mathcal{R}^{loc}$ we indicate by $\mathcal{R}_{A}$ the $\delta$-ring $\{B\subset A:B\in\mathcal{R}\}$ and by $\pi_{A}$ the collection of finite families of pairwise disjoint sets in $\mathcal{R}_{A}$. Note that if $\Omega\in\mathcal{R}$, then $\mathcal{R}$ is a $\sigma$-algebra, and in this case we have that $\mathcal{R}^{loc}=\mathcal{R}$. Moreover, for each $B\in\mathcal{R}$ it turns out that $\mathcal{R}_{B}$ is a $\sigma$-algebra.

A scalar measure is a function $\lambda:\mathcal{R}\rightarrow\mathbb{K}$ satisfying that if $\{B_{n}\}\subset\mathcal{R}$, is a family of pairwise disjoint sets such that $\bigcup_{n=1}^{\infty}B_{n}\in\mathcal{R}$, then $\sum_{n=1}^{\infty}\lambda(B_{n})=\lambda\left(\bigcup_{n=1}^{\infty}B_{n}\right)$. The variation of $\lambda$ is the countably additive measure $|\lambda|:\mathcal{R}^{loc}\rightarrow[0,\infty]$ defined by

[TABLE]

A function $f\in L^{0}(\mathcal{R}^{loc})$ is $\lambda$-integrable if $f\in L^{1}(|\lambda|)$. We denote by $L^{1}(\lambda)$ the vector space consisting of the equivalence classes of $\lambda$-integrable functions when we identify two functions if they are equal $|\lambda|$-a.e.

A set function $\nu:\mathcal{R}\rightarrow X$ is a vector measure if for any collection $\{B_{n}\}\subset\mathcal{R}$ of pairwise disjoint sets satisfying that $\bigcup_{n=1}^{\infty}B_{n}\in\mathcal{R}$ we have $\sum_{n=1}^{\infty}\nu(B_{n})=\nu(\bigcup_{n=1}^{\infty}B_{n})$. A vector measure $\nu$ is called strongly additive if $\nu(B_{n})\rightarrow 0$ whenever $\{B_{n}\}$ is a disjoint sequence in $\mathcal{R}$. The variation of $\nu$ is the positive measure $|\nu|$ defined in $\mathcal{R}^{loc}$ by

[TABLE]

The semivariation of $\nu$ is the function $\|\nu\|:\mathcal{R}^{loc}\rightarrow[0,\infty]$ given by

[TABLE]

where $|\langle\nu,x^{*}\rangle|$ is the variation of the scalar measure $\langle\nu,x^{*}\rangle:\mathcal{R}\rightarrow\mathbb{K}$, defined by

[TABLE]

The semivariation of $\nu$ is finite in $\mathcal{R}$ and for any $A\in\mathcal{R}^{loc}$ satisfies $\|\nu\|(A)\leq|\nu|(A)$. A set $A\in\mathcal{R}^{loc}$ is said to be $\nu$-null if $\|\nu\|(A)=0$. We will denote by $\mathcal{N}_{0}(\nu)$ the collection of $\nu$-null sets. It turns out that $\mathcal{N}_{0}(\nu)=\mathcal{N}_{0}(|\nu|)$. Moreover $A\in\mathcal{N}_{0}(\nu)$ if and only if $\nu(B)=0$, $\forall\ B\in\mathcal{R}_{A}$. We say that two functions $f,g\in L^{0}(\mathcal{R}^{loc})$ are equal $\nu$-a.e. if they are equal outside of a set in $\mathcal{N}_{0}(\nu)$. We define $L^{0}(\nu)$ as the space of equivalence classes of functions in $L^{0}(\mathcal{R}^{loc})$, where two functions are identified when they are equal $\nu$-a.e.

A function $f\in L^{0}(\mathcal{R}^{loc})$ is weakly $\nu$-integrable, if $f\in L^{1}(\langle\nu,x^{*}\rangle)$, for each $x^{*}\in X^{*}$. We will denote by $L^{1}_{w}(\nu)$ the subspace of $L^{0}(\nu)$ of all weakly $\nu$-integrable functions. With the norm given by

[TABLE]

$L^{1}_{w}(\nu)$ is a Banach space.

A function $f\in L^{1}_{w}(\nu)$ is $\nu$-integrable, if for each $A\in\mathcal{R}^{loc}$ there exists a vector $x_{A}\in X$, such that

[TABLE]

In this case the vector $x_{A}$ is denoted by $\int_{A}fd\nu$. With the norm $\|\cdot\|_{\nu}$ the subset of all $\nu$-integrable functions is a closed subspace of $L^{1}_{w}(\nu)$ and it will be denoted by $L^{1}(\nu)$. Therefore $L^{1}(\nu)$ is also a Banach space . We indicate by $S(\mathcal{R})$ the collection of simple functions in $L^{0}(\mathcal{R}^{loc})$ which have support in $\mathcal{R}$. It turns out that $S(\mathcal{R})$ is a dense subspace of $L^{1}(\nu)$. Finally the integral operator $I_{\nu}:L^{1}(\nu)\rightarrow X$ defined by $I_{\nu}(f)=\int_{\Omega}fd\nu$, is linear and bounded.

3 Vector measures with vector density

Recall that $(\Omega,\Sigma,\mu)$ is a positive measure space and $X$ a Banach space. Given a vector function $F\in\mathbb{P}(\mu,X)$ let us define the set function $\tilde{\nu}_{F}:\Sigma\rightarrow X$ by

[TABLE]

In the case when $\mu$ is finite it is well known that $\tilde{\nu}_{F}$ is a vector measure [6, Thm. II. 3.5]. The next result generalizes this fact, it can be established in the similar way, using the Orlicz-Pettis Theorem [6, Cor. I.4.4].

Theorem 3.1

The set function $\tilde{\nu}_{F}$ defined on (2) is a vector measure with semivariation

[TABLE]

Proof.

Let us fix $x^{*}\in X^{*}$ and take a pairwise disjoint countable collection $\{A_{n}\}\subset\Sigma$, then

[TABLE]

So, $\tilde{\nu}_{F}$ is weakly $\sigma$-aditive, by the Orlicz-Pettis Theorem [6, Cor. I.4.4] $\tilde{\nu}_{F}$ is a vector measure. On the other hand since $\langle\tilde{\nu}_{F},x^{*}\rangle(A)=\int_{A}\langle F,x^{*}\rangle$, $\forall$ $A\in\Sigma$, $\mu$ is a positive measure and $\langle F,x^{*}\rangle\in L^{1}(\mu)$ from [11, Thm. 6.13] we get that

[TABLE]

It is follows (3). $\blacksquare$

Since $\mathbb{B}(\mu,X)\subset\mathbb{P}(\mu,X)$ we have the following consequence. In order to get (4) we can proceed as in [6, Thm. II.2.4 iv)].

Corollary 3.2

Let $F\in\mathbb{B}(\mu,X)$. Then $\tilde{\nu}_{F}$ defined on (2) is a vector measure with bounded variation such that

[TABLE]

Hereafter let us consider a locally determined positive measure $\mu:\Sigma\rightarrow[0,\infty]$. It turns out that $\Sigma^{f}$ is a $\delta$-ring such that $(\Sigma^{f})^{loc}=\Sigma$ and the restriction of $\mu$ to $\Sigma^{f}$, denoted by $\lambda$, is a scalar measure such that $|\lambda|=\mu$ ([1, Lemma 4.3]).

Now we will study a kind of vector functions that include vector measures having Pettis or Bochner functions as density functions.

Definition 3.3

Let $F:\Omega\rightarrow X$ be a vector function.

1. a)

The function $F$ is locally Pettis integrable if $F$ is weakly $\mu$-measurable function and $\chi_{B}F\in\mathbb{P}(\mu,X)$, $\forall$ $B\in\Sigma^{f}$. The collection of equivalence classes obtained by identifying locally Pettis integrable functions if they are weakly equal $\mu$-a.e. will be denoted by $\mathbb{P}(\mu,X)^{loc}$. 2. b)

$F$ es locally Bochner integrable if $F$ is strongly $\mu$-measurable function and $\chi_{B}F\in\mathbb{B}(\mu,X)$, $\forall$ $B\in\Sigma^{f}$. The collection of equivalence classes obtained by identifying locally Bochner integrable functions if they are equal $\mu$-a.e. will be denoted by $\mathbb{B}(\mu,X)^{loc}$.

Remark 3.4

Observe that $\mathbb{P}(\mu,X)^{loc}$ and $\mathbb{B}(\mu,X)^{loc}$ are vector spaces. Additionally we have that

[TABLE]

The following example shows that the containment $\mathbb{B}(\mu,X)\subset\mathbb{B}(\mu,X)^{loc}$ can be proper.

Example 3.5

Let us fix $x\in X$ and assume that $f:\Omega\rightarrow\mathbb{R}$ is such that $\chi_{B}f\in L^{1}(\mu)$, $\forall\ B\in\Sigma^{f}$ (c.f. [10, Def. 2.14 c)]). Now define $F:\Omega\rightarrow X$ by

[TABLE]

Let us see that $\chi_{B}F\in\mathbb{B}(\mu,X)$, $\forall$ $B\in\Sigma$. Since $f\chi_{B}\in L^{0}(\Sigma)$, for all $B\in\Sigma^{f}$, it turns out that $f\in L^{0}(\Sigma)$. Take $\{s_{n}\}\subset S(\Sigma)$ such that $s_{n}\rightarrow f$ and $|s_{n}|\leq|f|$, $\forall$ $n\in\mathbb{N}$. Fix $B\in\Sigma^{f}$. For each $n\in\mathbb{N}$, define $S_{n}:\Omega\rightarrow X$ by $S_{n}(t)=\chi_{B}s_{n}(t)x$. Since $\chi_{B}f\in L^{1}(\mu)$, we have that $\chi_{B}s_{n}\in L^{1}(\mu)$. And so $S_{n}\in St(\mu,X)$ and $S_{n}(t)\rightarrow\chi_{B}f(t)x=\chi_{B}F(t)$, $\forall$ $t\in\Omega$ indicating that $\chi_{B}F$ is strongly $\mu$-measurable.

Further

[TABLE]

Since $\chi_{B}f\in L^{1}(\mu)$, it follows that $\displaystyle\int_{B}\|F\|_{X}d\mu<\infty$, and so $\chi_{B}F\in\mathbb{B}(\mu,X)$. Thus $F\in\mathbb{B}(\mu,X)^{loc}$. Finally observe that $F\in\mathbb{B}(\mu,X)$ if and only if $f\in L^{1}(\mu)$.

As a consequence of Theorem 3.1 we obtain the following result.

Proposition 3.6

Let $F\in\mathbb{P}(\mu,X)^{loc}$. Then the set function $\nu_{F}:\Sigma^{f}\rightarrow X$ defined by

[TABLE]

is a vector measure such that

[TABLE]

Proof.

Let us show that $\nu_{F}$ is a vector measure. Let $\{B_{n}\}\subset\Sigma^{f}$ be a disjoint collection such than $B:=\bigcup_{n=1}^{\infty}B_{n}\in\Sigma^{f}$. Since $\chi_{B}F\in\mathbb{P}(\mu,X)$, by Theorem 3.1 we have

[TABLE]

Thus $\nu_{F}$ is a vector measure. Now fix $x^{*}\in X^{*}$. Since $F\in\mathbb{P}(\mu,X)^{loc}$ we have $\langle F,x^{*}\rangle\in L^{1}_{loc}(\lambda)$. From [10, Thm. 2.31] we have that the variation of the scalar measure $\mu_{\langle F,x^{*}\rangle}:\Sigma^{f}\rightarrow\mathbb{K}$ defined by $\mu_{\langle F,x^{*}\rangle}(B)=\int_{B}\langle F,x^{*}\rangle d\mu$, $\forall\ B\in\Sigma^{f}$ is given by

[TABLE]

On the other hand notice that for each $B\in\Sigma^{f}$, we have

[TABLE]

Therefore

[TABLE]

From this we get (7). $\blacksquare$

Observe that from (7) we have that if $A\in\mathcal{N}_{0}(\mu)$, then $\|\nu_{F}\|(B)=0$, $\forall$ $B\in\Sigma^{f}_{A}$. So $\mathcal{N}_{0}(\mu)\subset\mathcal{N}_{0}(\nu_{F})$. Also observe that $\|\nu_{F}\|(B)=\|\chi_{B}F\|_{\mathbb{P}}$, $\forall$ $B\in\Sigma^{f}$. Moreover from the Dunford integrability definition we get our next result.

Corollary 3.7

Let $F\in\mathbb{P}(\mu,X)^{loc}$. Then $F$ is Dunford integrable if and only if $\nu_{F}$ has bounded semivariation. In particular if $F\in\mathbb{P}(\mu,X)$, then $\|\nu_{F}\|(A)=\|\chi_{A}F\|_{\mathbb{P}}$.

In the case that $F$ is locally Bochner integrable we have a useful characterization of the variation of $\nu_{F}$, as follows.

Proposition 3.8

If $F\in\mathbb{B}(\mu,X)^{loc}$, then $\displaystyle{|\nu_{F}|(A)=\int_{A}\|F\|_{X}d\mu}$, $\forall$ $A\in\Sigma$.

Proof.

Take $B\in\Sigma^{f}$. Notice that $\nu_{F}(A\cap B)=\nu_{\chi_{B}F}(A)$, $\forall$ $A\in\Sigma$. Since $\chi_{B}F\in\mathbb{B}(\mu,X)$ we have that

[TABLE]

Hence

[TABLE]

Remark 3.9

Let us note that if $F:\Omega\rightarrow X$ is a strong $\mu$-measurable function, by the previous result we obtain that:

$F\in\mathbb{B}(\mu,X)$ if and only if $F\in\mathbb{B}(\mu,X)^{loc}$ and $\nu_{F}$ has bounded variation.

Example 3.10

Let us return to the example 3.5. It was shown there that $F$ defined in (5) is locally Bochner integrable. In particular $F$ is locally Pettis integrable. Let us obtain now the vector measure $\nu_{F}$, its variation and its semivariation. See that for each $B\in\Sigma^{f}$

[TABLE]

Now take $A\in\Sigma$, from (7) and (10)

[TABLE]

Therefore $\|\nu_{F}\|=|\nu_{F}|$ in this case.

4 The space of $\nu_{F}$-integrable functions

When $F$ is a locally Pettis or Bochner integrable function we have constructed the vector measure $\nu_{F}$ defined on the $\delta$-ring $\Sigma^{f}$. In the present section we will study the spaces $L^{0}(\nu_{F})$, $L^{1}_{w}(\nu_{F})$, $L^{1}(\nu_{F})$ and $L^{1}(|\nu_{F})|)$ associated to this vector measure through the operator $M_{F}$ which to each function $g$ assigns the function $gF$. The following lemmas allow us to conclude that $M_{F}:L^{0}(\nu_{F})\rightarrow L^{0}_{w}(\mu,X)$ or $L^{0}(\mu,X)$ is well defined. Clearly $M_{F}$ is a linear operator.

Lemma 4.1

Let $F:\Omega\rightarrow X$ be a function and $g\in L^{0}(\Sigma)$.

1. i)

If $F$ is strongly $\mu$-measurable, then $gF$ is strongly $\mu$-measurable. 2. ii)

If $F$ is weakly $\mu$-measurable, then $gF$ is weakly $\mu$-measurable.

Proof.

i) Observe that if $\varphi\in S(\Sigma)$ and $S\in St(\mu,X)$, then $\varphi S\in St(\mu,X)$. Let us assume that $F$ is strongly $\mu$-measurable. Take $\{\varphi_{n}\}\subset S(\Sigma)$ and $\{S_{n}\}\subset St(\mu,\Sigma)$ such that $\varphi_{n}\rightarrow g$ and $S_{n}\rightarrow F$, $\mu$-a.e. Thus $\varphi_{n}S_{n}\in St(\mu,X)$, $\forall\ n\in\mathbb{N}$ and $\varphi_{n}S_{n}\rightarrow gF$, $\mu$-a.e. It follows that $gF$ is strongly $\mu$-measurable.

ii) By definition if $F$ is weakly $\mu$-measurable, we have that for each $x^{*}\in X^{*}$, the function $\langle F,x^{*}\rangle$ is strongly $\mu$-measurable. Using i) we obtain that $\langle gF,x^{*}\rangle=g\langle F,x^{*}\rangle\in L^{0}(\mu,X)$ $\forall\ x^{*}\in X^{*}$. $\blacksquare$

Lemma 4.2

Let $F\in\mathbb{P}(\mu,X)^{loc}$, $\{g_{n}\}\subset L^{0}(\Sigma)$ and $g,h\in L^{0}(\Sigma)$.

1. i)

If $g=h$, $\nu_{F}$-a.e., then $gF=hF$, weakly $\mu$-a.e. 2. ii)

If $g_{n}\rightarrow g$, $\nu_{F}$-a.e., then $\langle g_{n}F,x^{*}\rangle\rightarrow\langle gF,x^{*}\rangle$, $\mu$-a.e., $\forall\ x^{*}\in X^{*}$.

Proof.

i) Choose $N\in\mathcal{N}_{0}(\nu_{F})$ such that $g(t)=h(t)$, $\forall\ t\in N^{c}$. Then $g\chi_{N^{c}}F=h\chi_{N^{c}}F$; moreover $\chi_{N}F=0$ weakly $\mu$ -a.e. implies that $g\chi_{N}F=h\chi_{N}F=0$, weakly $\mu$-a.e. Thus $gF=hF$, weakly $\mu$-a.e.

ii) Let $N\in\mathcal{N}_{0}(\nu_{F})$ such that $g_{n}(t)\rightarrow g(t)$, $\forall\ t\in N^{c}$. So $g_{n}\chi_{N^{c}}\rightarrow g\chi_{N^{c}}F$ and $g_{n}\chi_{N}F=g\chi_{N}F=0$, weakly $\mu$-a.e. Then for each $x^{*}\in X^{*}$, $\langle g_{n}\chi_{N^{c}}F,x^{*}\rangle\rightarrow\langle g\chi_{N^{c}}F,x^{*}\rangle$ and $\langle g_{n}\chi_{N}F,x^{*}\rangle=\langle g\chi_{N}F,x^{*}\rangle=0$, $\mu$-a.e. Therefore $\langle g_{n}F,x^{*}\rangle\rightarrow\langle gF,x^{*}\rangle$, $\mu$-a.e., $\forall\ x^{*}\in X^{*}$. $\blacksquare$

Proposition [12, Prop.8] established by G. F. Stefansson for the case that $F\in\mathbb{P}(\mu,X)$ and $\mu$ is a finite positive measure defined on a $\sigma$-algebra is generalized in the next theorem.

Theorem 4.3

For $F\in\mathbb{P}(\mu,X)^{loc}$ and $g\in L^{0}(\Sigma)$, we have that

1. i)

$g\in L^{1}_{w}(\nu_{F})$ if, and only if, $gF\in\mathbb{D}(\mu,X)$. Moreover the restriction to $L^{1}_{w}(\nu_{F})$ of the operator $M_{F}$ is a linear isometry from $L^{1}_{w}(\nu_{F})$ into $\mathbb{D}(\mu,X)$. 2. ii)

$g\in L^{1}(\nu_{F})$ if, and only if, $gF\in\mathbb{P}(\mu,X)$. Moreover $M_{F}:L^{1}(\nu_{F})\rightarrow\mathbb{P}(\mu,X)$, the restriction of the operador $M_{F}$, is a linear isometry such that $I_{\nu_{F}}=I_{\mathbb{P}}\circ M_{F}$.

Proof.

Fix $x^{\ast}\in X^{\ast}$ and consider $\displaystyle{s=\sum_{j=1}^{n}a_{j}\chi_{A_{j}}\in S(\Sigma)}$. By hypothesis $\chi_{B}F\in\mathbb{P}(\mu,X)$, $\forall\ B\in\Sigma^{f}$. It follows that $sF\in\mathbb{P}(\mu,X)$. From (9) we obtain

[TABLE]

Thus $s\in L^{1}_{w}(\nu_{F})$ if and only if $sF\in\mathbb{D}(\mu,X)$.

Proceeding in the same way, it follows from (8) that

[TABLE]

Now take $g\in L^{0}(\nu_{F})^{+}$ and $\{s_{n}\}\subset S(\Sigma)$ such that $0\leq s_{n}\uparrow g$, $\nu_{F}$-a.e. From lemma 4.2 we obtain $\langle s_{n}F,x^{\ast}\rangle\rightarrow\langle gF,x^{\ast}\rangle$, $\mu$-a.e. Then, $|\langle s_{n}F,x^{\ast}\rangle|\uparrow|\langle gF,x^{\ast}\rangle|$, $\mu$-a.e. By the Monotone Convergence Theorem and (11) it turns out that

[TABLE]

showing that $g\in L^{1}_{w}(\nu_{F})$ if and only if $gF\in\mathbb{D}(\mu,X)$.

By the Dominate Convergence Theorem and (8)

[TABLE]

We conclude from here that $g\in L^{1}(\nu_{F})$ if and only if $gF\in\mathbb{P}(\mu,X)$ and

[TABLE]

Since the involved sets are vector spaces and each $g\in L^{0}(\nu_{F})$ is a linear combination of non negative functions, we obtain the first part in i) and ii).

Finally take $g\in L^{1}_{w}(\nu_{F})$, since $|g|\geq 0$ we obtain equality (13) with a sequence $\{s_{n}\}\subset S(\Sigma)$ such that $0\leq s_{n}\uparrow|g|$, $\nu_{F}$-a.e. Taking the supremum over $x^{\ast}\in B_{X^{\ast}}$ it turns out that $\|g\|_{\nu_{F}}=\|gF\|_{\mathbb{D}}$. That is, $M_{F}$ restricted to $L^{1}_{w}(\nu_{F})$ is a linear isometry. Since $L^{1}(\nu_{F})$ and $\mathbb{P}(\mu,X)$ are subspaces of $L^{1}_{w}(\nu_{F})$ and $\mathbb{D}(\mu,X)$, respectively we conclude that $M_{F}$ restricted to $L^{1}(\nu_{F})$ is also an isometry. Moreover, from (14) it follows that $I_{\nu_{F}}=I_{\mathbb{P}}\circ M_{F}$. $\blacksquare$

Corollary 4.4

Let $F\in L_{w}^{0}(\mu,X)$. Then $F\in\mathbb{P}(\mu,X)$ if and only if $F\in\mathbb{P}(\mu,X)^{loc}$ and $\nu_{F}$ is strongly additive.

Proof.

Let assume that $F\in\mathbb{P}(\mu,X)$. Consider the vector measure $\tilde{\nu}_{F}:\mathcal{R}^{loc}\rightarrow X$ defined in (2). Since $\Sigma$ is a $\sigma$-algebra, it turns out that $\tilde{\nu}_{F}$ is strongly additive. Observe that $\nu_{F}$ is the restriction of $\tilde{\nu}_{F}$ to $\Sigma^{f}$, so it follows that it is strongly additive.

Now assume that $F\in\mathbb{P}(\mu,X)^{loc}$ and that $\nu_{F}$ is strongly additive. From [4, Cor. 3.2] we obtain that $\chi_{\Omega}\in L^{1}(\nu_{F})$. So, by the previous theorem $F=\chi_{\Omega}F\in\mathbb{P}(\mu,X)$. $\blacksquare$

Corollary 4.5

Let $F\in\mathbb{P}(\mu,X)^{loc}$. If $X$ does not contain any subspace isomorphic to $c_{0}$ and $\nu_{F}$ is bounded, then $F\in\mathbb{P}(\mu,X)$.

Proof.

Since $X$ does not contain any subspace isomorphic to $c_{0}$ and $\nu_{F}$ is bounded it turns out that $\nu_{F}$ is strongly additive [5, p. 36]. Then by the previous corollary $F\in\mathbb{P}(\mu,X)$. $\blacksquare$

The following result gives us the connection between the spaces $L^{1}(|\nu_{F}|)$ and $\mathbb{B}(\mu,X)$ through the operator $M_{F}$ in case that $F\in\mathbb{B}(\mu,X)^{loc}$. We will show that, as it occur when $F\in\mathbb{P}(\mu,X)^{loc}$, $M_{F}$ is a linear isometry in this case.

Proposition 4.6

Consider $F\in\mathbb{B}(\mu,X)^{loc}$ and $g\in L^{0}(|\nu_{F}|)$. Then $g\in L^{1}(|\nu_{F}|)$ if and only if $gF\in\mathbb{B}(\mu,X)$. Moreover $M_{F}:L^{1}(|\nu_{F}|)\rightarrow\mathbb{B}(\mu,X)$ is a linear isometry such that $I_{\nu_{F}}(g)=I_{\mathbb{B}}\circ M_{F}(g)$, $\forall\ g\in L^{1}(|\nu_{F}|)$.

Proof.

Clearly $M_{F}$ is a linear operator, we will see that its image is a subset of $\mathbb{B}(\mu,X)$. By lemma 4.2 we have that the restriction $M_{F}:L^{1}(|\nu_{F}|)\rightarrow\mathbb{B}(\mu,X)$ is well defined.

Since the norms in $L^{1}(|\nu_{F}|)$ and $\mathbb{B}(\mu,X)$ are different from those in $L^{1}(\nu_{F})$ and $\mathbb{P}(\mu,X)$ respectively, we need to establish that, under these norms, $M_{F}$ is also an isometry.

By hypothesis $F\in\mathbb{B}(\mu,X)^{loc}$, then from (10) it follows that $|\nu_{F}|(B)<\infty$, $\forall\ B\in\Sigma^{f}$. So, $S(\Sigma^{f})\subset L^{1}(|\nu_{F}|)$. Further for each $s=\sum_{j=1}^{n}a_{j}\chi_{A_{j}}\in S(\Sigma)$ we have that

[TABLE]

Therefore $s\in L^{1}(|\nu_{F}|)$ if and only if $sF\in\mathbb{B}(\mu,X)$. Now consider $g\in L^{0}(|\nu_{F}|)$ and take $\{s_{n}\}\subset S(\Sigma)$ such that $0\leq s_{n}\uparrow|g|$, $\nu_{F}$-a.e. Then $\|s_{n}F\|_{X}\uparrow\|gF\|_{X}$, $\mu$-a.e. By the Monotone Convergence Theorem

[TABLE]

Thus we have that $gF\in\mathbb{B}(\mu,X)$ if and only if $g\in L^{1}(|\nu_{F}|)$. Moreover $\|g\|_{|\nu_{F}|}=\|gF\|_{1}$.

The equality between the operators follows from Proposition 4.3. $\blacksquare$

Example 4.7

Consider again the function $F$ defined in (5). As we see in Example 3.5 $F\in\mathbb{B}(\mu,X)^{loc}$.

Take $g\in L^{1}(\nu_{F})$, from Proposition 4.3

[TABLE]

then $gf\in L^{1}(\mu)$. And so,

[TABLE]

By Lemma 4.1 $gF$ is strongly $\mu$-measurable. Thus we have that $gF\in\mathbb{B}(\mu,X)$ and by Proposition 4.6, $g\in L^{1}(|\nu_{F}|)$. We conclude that $L^{1}(|\nu_{F}|)=L^{1}(\nu_{F})$. And from [3, Prop. 5.4] it follows that $L^{1}(|\nu_{F}|)=L^{1}(\nu_{F})=L^{1}_{w}(\nu_{F})$ .

Acknowledgment

Many thanks to Professor F. Galaz-Fontes for the careful reading and valuable comments.