${}^{\ast }$Department of Mathematics, Tafila Technical University, Jordan, e-mail: abu-sirhan2@ttu.edu.jo, https://orcid.org/0000-0003-1360-5554

1 Introduction

In this article, $(X,\Vert \cdot \Vert )$ will be referred to as a Banach space over the field $R$ of real numbers, $G$ as a closed subspace of $X$, and $(\mathrm{\Omega },\mathrm{\Sigma },\mu )$ as a $\sigma$-finite nontrivial measure space, i.e. $\mathrm{\Omega }$ is a countable union of measurable sets each with finite measure and there exists at least $A\in \mathrm{\Sigma }$ with $\mathrm{\infty }>\mu (A)>0$. For $p\ge 1$, we suppose $L^p(\mu ,X)$ is the space of all Bochner $p$-integrable functions from $\mathrm{\Omega }$ to $X$, $L(\mu ,X)$ is the linear space of all $\mu$-equivalence classes of strongly measurable functions from $\mathrm{\Omega }$ to $X$, see [ 10 ] . If a subset $M$ of $L(\mu ,X)$ is closed with regard to the pointwise limits of sequences, we say that it is closed. For $E\subseteq \mathrm{\Omega }$ and a function $f:\mathrm{\Omega }\to X$, ${\chi }_E$ is the characteristic function of the set $E$ and ${\chi }_E\otimes f$ is the function ${\chi }_E\otimes f(s)={\chi }_E(s)f(s)$. A function $f:\mathrm{\Omega }\to X$ is said to be simple if

where $x_i\in X$ and $E_i=f^{-1}(x_i)$ is measurable for $i=1,2,\dots ,n$. A function $f:\mathrm{\Omega }\to X$ is said to be strongly measurable if there exists a sequence of simple functions $\{f_n\}$ with

Let $M$ be a closed subspace of $L(\mu ,X)$, we write $L^p(M)$ for the Banach space of all functions in $M$ such that ${\int }_{\mathrm{\Omega }}\Vert f(s){\Vert }^p$ is finite. An element $g_0\in G$ is said to be a best coapproximation of $x\in X$ if

The set of all elements of best coapproximation of $x$ in $G$ will be denoted by $R_G(x)$. If $R_G(x)$ is nonempty for any $x\in X$, $G$ is said to be a coproximinal subspace of $X$. This new kind of approximation has been introduced by C. Franchetti and M. Furri (1972) [ 4 ] to characterize real Hilbert spaces among real reflexive Banach spaces.

P.L. Papini and I. Singer (1979) [ 8 ] then went into greater depth on the best coapproximation. It has lately been studied in $L^p(\mu ,X)$; for example, see [ 1 , 2 , 6 , 7 , 5 ] . With finite measure spaces, though, [ 5 , 6 , 7 ] have dealt. This paper’s goal is to demonstrate the connection between the pointwise coproximinality of $L^p(\mu ,G)$ in $L^p(\mu ,X)$ and the pointwise coproximinality of $L(\mu ,G)$ in $L(\mu ,X)$. Pointwise coproximinality is a counterpart of pointwise proximinality, it should be noted, see [ 9 , 2 ] .

2 Pointwise Coproximinality

Assume that $M$ is pointwise coproximinal and there exist $A\in \mathrm{\Sigma }$, $f\in M$, such that ${\chi }_A\otimes f\notin M$. By assumption there exists ${\varphi }_A$ in $M$ such that

and for all $\varphi \in M.$ Taking $\varphi =0$ one obtains

For $\varphi =f$ one obtains

By 1, ${\varphi }_A(s)=0a.e.s\in A^c$ and by 2 ${\varphi }_A(s)=f(s)a.e.s\in A$, that is, ${\chi }_A\otimes f={\varphi }_A\in M$. The proof is completed by the contradiction.

The proof follows from lemma 1 and the fact that the set of all simple functions in $L(\mu ,X)$ is dense in $L(\mu ,X)$.

Let $\mathcal{A}=\{A_n:n\in \mathbb{N}\}$ be a family of countable subset of $\mathrm{\Omega }$. if $A_n\in \mathrm{\Sigma }$ for any $n\in \mathbb{N}$, $A_i\cap A_j=\varphi$ for $i\ne j$, and $\mu (\mathrm{\Omega }-{\cup }_{n\in \mathbb{N}}{A}_n)=0$, then the set $\mathcal{A}$ is said to be measurable partition of $\mathrm{\Omega }$ [ 3 ] .

Since $(\mathrm{\Omega },\mathrm{\Sigma },\mu )$ is $\sigma$-finite, we may assume that $\mathrm{\Omega }=\bigcup _{n\in \mathbb{N}}{A}_n$, $A_n$ is measurable, $A_n\subseteq A_{n+1}$, and $\mu (A_n)<\mathrm{\infty }$ for any $n\in \mathbb{N}$. For each $n\in \mathbb{N}$, let

$C_n=A_n\cap B_n$, and $D_n=C_n-C_{n-1}.$ Then $\{D_n,n\in \mathbb{N}\}$ is a measurable partition, ${\chi }_{D_n}\otimes f\in L^p(M)$ by lemma 1, and $f=\sum _{n=1}^{\mathrm{\infty }}{\chi }_{D_n}\otimes f$.

$(1)\Rightarrow (2)$. Let$f\in L^p(\mu ,X)$. Then $f\in L(\mu ,X)$ and it has a best pointwise coapproximation $f_0$ from $L(\mu ,G)$. By lemma 2, $f_0$ is a best pointwise coapproximation to $f$ from $L^p(\mu ,G)$.

$(2)\Rightarrow (1)$. Let $f\in L(\mu ,X)$. By lemma 3, there exists $\{D_n:n\in \mathbb{N}\}$ measurable partition of $\mathrm{\Omega }$ such that

and ${\chi }_{D_n}\otimes f\in L^p(\mu ,X)$ for any $n$ in $\mathbb{N}$ and $1\le p\le \mathrm{\infty }$. By assumption, there exists ${\varphi }_n\in L^p(\mu ,G)$ such that

for all $\varphi \in L(\mu ,G)$. Taking $\varphi =0$ yields,

and hence ${\varphi }_n={\chi }_{D_n}\otimes {\varphi }_n$. If we let $\text{Bar}\varphi =\sum _{n\in \mathbb{N}}{\varphi }_n$, then $\text{Bar}\varphi \in L(\mu ,G)$. We claim that

for all $\varphi \in L(\mu ,G)$. If the claim is incorrect, then ${\varphi }_0\in L(\mu ,G)$ exists such that

and let $A_0=\{s:\Vert \text{Bar}\varphi (s)-{\varphi }_0(s)\Vert >\Vert f(s)-{\varphi }_0(s)\Vert \}$. By lemma 3, there exists a measurable partition $\{C_n:n\in \mathbb{N}\}$ of $\mathrm{\Omega }$ such that

and ${\chi }_{C_n}\otimes {\varphi }_0\in L^p(\mu ,G)$ for any $n$ in $\mathbb{N}$ and $1\le p\le \mathrm{\infty }$. Then $\{C_n\cap D_m:n,m\in \mathbb{N}\}$ is a measurable partition of $\mathrm{\Omega }$. Hence there exist $n_0,m_0\in \mathbb{N}$ such that $\mu (A_0\cap C_{m_0}\cap D_{n_0})>0.$ If $s\in A_0\cap C_{m_0}\cap D_{n_0}$, then

Let

Then $A_0\cap C_{m_0}\cap D_{n_0}\subseteq B_0$ and $\mu (B_0)>0$, which contradicts 1.

It is obvious that pointwise coproximinality of $L(\mu ,G)$ in $L(\mu ,x)$ follows from coproximinality of $L(\mu ,G)$ in $L(\mu ,x)$. Then $L^p(\mu ,G)$ is pointwise coproximinal in $L^p(\mu ,X)$ by theorem 1. Therefore $L^p(\mu ,G)$ is coproximinal in $L^p(\mu ,X)$ and the result follows from theorem 2.