\(^\ast \)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, \((X,\| \cdot \| )\) will be referred to as a Banach space over the field \(R\) of real numbers, \(G\) as a closed subspace of \(X\), and \((\Omega ,\Sigma ,\mu )\) as a \(\sigma \)-finite nontrivial measure space, i.e. \(\Omega \) is a countable union of measurable sets each with finite measure and there exists at least \(A\in \Sigma \) with \( \infty {\gt}\mu (A){\gt}0\). For \(p\geq 1\), we suppose \(L^{p}(\mu ,X)\) is the space of all Bochner \(p\)-integrable functions from \(\Omega \) to \(X\), \(L(\mu ,X)\) is the linear space of all \(\mu \)-equivalence classes of strongly measurable functions from \(\Omega \) 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 \(E \subseteq \Omega \) and a function \(f:\Omega \rightarrow X\), \(\chi _{E}\) 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 \(f:\Omega \rightarrow X\) is said to be simple if

where \(x_{i}\in X\) and \(E_{i}=f^{-1}(x_{i})\) is measurable for \(i=1,2,\ldots ,n\). A function \(f:\Omega \rightarrow X\) 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 \(\int _{\Omega } \| f(s)\| ^{p}\) 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 \(A\in \Sigma \), \(f \in M\), such that \(\chi _{A} \otimes f \notin M\). By assumption there exists \(\phi _{A}\) in \(M\) such that

and for all \(\phi \in M.\) Taking \(\phi =0\) one obtains

For \(\phi =f \) one obtains

By 1, \(\phi _{A}(s)=0 \hspace{0.4cm} a.e. \hspace{0.2cm} s \in A^{c}\) and by 2 \(\phi _{A}(s)=f(s) \hspace{0.4cm} a.e. \hspace{0.2cm} s \in A\), that is, \( \chi _{A} \otimes f=\phi _{A} \in M\). 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 \(\Omega \). if \( A_{n} \in \Sigma \) for any \( n \in \mathbb {N} \), \( A_{i}\cap A_{j} = \phi \) for \( i \neq j \), and \( \mu (\Omega -\cup _{n \in \mathbb {N}} A_{n} )=0 \), then the set \(\mathcal{A}\) is said to be measurable partition of \(\Omega \) [ 3 ] .

Since \((\Omega ,\Sigma ,\mu ) \) is \(\sigma \)-finite, we may assume that \( \Omega =\bigcup _{n \in \mathbb {N} } A_{n}\), \(A_{n}\) is measurable, \( A_{n} \subseteq A_{n+1}\), and \( \mu (A_{n}) {\lt} \infty \) 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 \(f=\sum _{n=1}^{\infty } \chi _{D_{n}} \otimes f\).

\((1)\Rightarrow (2)\). 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 \(\lbrace D_{n} : n \in \Bbb {N} \rbrace \) measurable partition of \(\Omega \) such that

and \( \chi _{D_{n}} \otimes f \in L^{p} (\mu , X) \) for any \(n\) in \(\Bbb {N}\) and \(1\leq p \leq \infty \). By assumption, there exists \( \phi _{n} \in L^{p} (\mu , G) \) such that

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

and hence \(\phi _{n}=\chi _{D_{n}} \otimes \phi _{n}\). If we let \(\Bar {\phi }=\sum _{ n\in \Bbb {N}} \phi _{n}\), then \(\Bar {\phi } \in L(\mu ,G)\). We claim that

for all \(\phi \in L(\mu ,G)\). If the claim is incorrect, then \(\phi _{0} \in L(\mu ,G) \) exists such that

and let \( A_{0}=\lbrace s: \| \Bar {\phi }(s) - \phi _{0}(s) \| {\gt} \| f(s) - \phi _{0}(s) \| \rbrace \). By lemma 3, there exists a measurable partition \(\lbrace C_{n} : n \in \Bbb {N} \rbrace \) of \(\Omega \) such that

and \( \chi _{C_{n}} \otimes \phi _{0} \in L^{p} (\mu , G) \) for any \(n\) in \(\Bbb {N}\) and \(1\leq p \leq \infty \). Then \(\lbrace C_{n} \cap D_{m} : n,m \in \Bbb {N} \rbrace \) is a measurable partition of \(\Omega \). Hence there exist \( n_{0},m_{0} \in \Bbb {N} \) such that \(\mu (A_{0} \cap C_{m_{0}} \cap D_{n_{0}}){\gt}0. \) 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 \(\mu (B_{0}){\gt}0\), 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.