\(^{\ast }\)Department of Mathematics, Tafila Technical University, Tafila, Jordan, e-mail: abu-sirhan@ttu.edu.jo.

1 Introduction

Let \(Z\) be a Banach space and \(Y\) be a closed subspace of \(\ Z\). For a subset \(B\) of \(Z,\) define

An element \(y^{\ast }\in Y\) is said to be a best simultaneous approximant to the subset \(B\) if

The theory of best simultaneous approximation has been studied by many authors. Most of these works have dealt with the space of continuous functions with values in a Banach space e.g. [18, 10, 4]. Some recent results for best simultaneous approximation in the Banach space of \(P\)-Bochner integrable (essentially bounded) functions have been obtained in [5, 6, 9, 16, 19]. We consider here a problem of simultaneous approximation completing the work done in \(\left[ 6,19\right] \). In this paper \(\left( \Omega ,\Sigma ,\mu \right) \) stands for a complete \(\sigma \)-finite measure space and \(L^{\infty }\left( \mu ,X\right) \) the Banach space of all essentially bounded functions on \(\left( \Omega ,\Sigma ,\mu \right) \) with values in a Banach space \(X,\) endowed with the usual norm

In [19], it is shown that if \(G\) is a reflexive subspace of a Banach space \(X \), then \(L^{\infty }\left( \mu ,G\right) \) is simultaneously proximinal in \(L^{\infty }\left( \mu ,X\right) .\) In [6], it is shown that if \(G\) is a separable \(w^{\ast }\)-closed subspace of a dual space \(X\), then \(L^{\infty }\left( \mu ,G\right) \) is simultaneously proximinal in \(L^{\infty }\left( \mu ,X\right) .\) The aim of this paper is to show that if \(G\) is a closed separable subspace of a Banach space \(X\), then \(L^{\infty }\left( \mu ,G\right) \) is simultaneously proximinal in \(L^{\infty }\left( \mu ,X\right) \) if and only if \(G\) is simultaneously proximinal in \(X.\)

2 Preliminary Results

The following Theorem is a generalization of the distance formula given in [6]. Here \(\chi _{A}\) is denoted to the characteristic function of \(A.\)

for \(i=1,2,...,m,\) and for almost all \(t\)’s. We may write, [9],

\(\ \) where \(A\left( n,j\right) \) are disjoint and \(\overset {k\left( n\right) }{\underset {j=1}{\cup }}A\left( n,j\right) =\Omega .\) Define \(d_{n}\left( \cdot \right) :\Omega \rightarrow \mathbb {R} \) by

Then

and

for almost all \(s\). Thus \(d\left( \left\{ f_{i}\left( \cdot \right) :1\leq i\leq m\right\} ,\ G\right) \) is measurable.

Now, for any \(h\in L^{\infty }\left( \mu ,G\right) ,\)

Hence

To prove the reverse inequality, let \(\epsilon {\gt}0\) be given and \(w_{i},\ i=1,2,...,m,\) be countably valued functions in \(L^{\infty }\left( \mu ,X\right) \) such that

We may write \(w_{i}=\sum _{k=1}^{\infty }\chi _{A_{k}}\left( \cdot \right) \ x_{\left( i,k\right) },\) where \(A_{k}\) are disjoint, \(\overset {\infty }{\underset {k=1}{\cup }}A_{k}=\Omega ,\) and \(\mu \left( A_{k}\right) {\gt}0\), for all \(k.\) Let \(h_{k}\in G\) be such that

for all \(k.\) Let \(g=\sum _{k=1}^{\infty }\chi _{A_{k}}\left( \cdot \right) h_{k}.\) It is clear that \(g\in L^{\infty }\left( \mu ,G\right) .\) Further

for almost all \(s\), and for all \(z\in G.\) Then

for almost all \(s,\) therefore \(g\in L^{\infty }\left( \mu ,G\right) .\) By Theorem 2.1,

Therefore \(g\) is a best simultaneous approximation for \(f_{1},f_{2},..,f_{m}\) in
\(L^{\infty }\left( \mu ,G\right) .\)

The condition in Corollary 2.2 is sufficient; \(g\left( s\right) \) is a best simultaneous approximation of \(f_{1}\left( s\right) ,f_{2}\left( s\right) ,...,f_{m}\left( s\right) \) for almost all \(s\) in \(G,\) implies \(g\) is a best simultaneous approximation of \(f_{1},f_{2},...,f_{m}\) in \(L^{\infty }\left( \mu ,G\right) .\) For the converse, we need the following easy lemma.

for all \(s\in \Omega .\)

Thus \(\underset {1\leq i\leq m}{\sup }\left\Vert f_{i}-g\right\Vert _{\infty }\geq 1.\) Since \(g\in L^{\infty }\left( \mu ,G\right) \) was arbitrary, then

To prove the converse inequality. Let \(\epsilon {\gt}0\) be given. Let \(f_{1}^{^{\prime }},f_{2}^{^{\prime }},...,f_{m}^{^{\prime }}\in L^{\infty }\left( \mu ,X\right) \) be countably valued functions such that \(f_{i}^{^{\prime }}\left( \Omega \right) \subset \) \(f_{i}\left( \Omega \right)\),
\(i=1,2,\ldots ,m,\) and

We may write \(f_{i}^{^{\prime }}=\sum _{k=1}^{\infty }\chi _{A_{k}}\left( \cdot \right) \ x_{\left( i,k\right) },\) with the subsets \(A_{k}\) disjoint and measurable, and \(x_{\left( i,k\right) }\in \) \(f_{i}\left( \Omega \right) .\) For each \(k\) take \(h_{k}\in G\) such that

It is clear that \(g\) defined by

belongs to \(L^{\infty }\left( \mu ,G\right) \) and

Therefore,

Sufficiency of the condition is an immediate consequence of Corollary 2.2. We will show the necessity. Assume that \(L^{\infty }\left( \mu ,G\right) \) is simultaneously proximinal in \(L^{\infty }\left( \mu ,X\right) \) and take \(f_{1},f_{2},...,f_{m}\) in \(L^{\infty }\left( \mu ,X\right) \). Consider the non-negative measurable function

Take \(\Omega _{0}=\left\{ s\in \Omega :h\left( s\right) =0\right\} ,\) and for each \(n=1,2,...,\ \)take
\(\Omega _{n}=\left\{ s\in \Omega :n-1{\lt}h\left( s\right) \leq n\right\} .\) Of course, we may forget those \(\Omega _{n}\) which are \(\mu -\)null sets, so, without loss of generality, we will assume that \(\mu \left( \Omega _{n}\right) {\gt}0\) for all \(n.\) Now for each \(n=1,2,..., \) we define \(f_{in}:\Omega \rightarrow X\) by

It is clear that \(f_{in}\) \(\in L^{\infty }\left( \mu ,X\right) ,\) \(i=1,2,...,m, \) and also that

for all \(s\in \Omega _{n}.\) So, it follows from proceeding lemma that

On the other hand, using simultaneous proximinality of \(L^{\infty }\left( \mu ,G\right) ,\) we deduce that there exists \(g_{n}\in L^{\infty }\left( \mu ,G\right) \) such that

Therefore, we have

for almost all \(s\in \Omega _{n}.\) Then,

for almost all \(s\in \Omega _{n}.\) Thus

for almost all \(s\in \Omega _{n}.\) We do notice that, if \(s\in \Omega _{0},\) then \(f_{i}\left( s\right) =f_{j}\left( s\right) ,\) for 1\(\leq i\leq j\leq m. \) Hence, it is clear that \(g\) defined by

for all \(s\in \Omega \) enjoys the required property.

3 Main Result

The following lemmas will be used to prove our main result.

Let \(\Phi \) be a set-valued mapping, taking each point of a measurable space \(\Omega \) into a subset of a metric space \(X.\) We say that \(\Phi \) is weakly measurable if \(\Phi ^{-1}(O)\) is measurable in \(\Omega \) whenever \(O\) is open in \(X.\) Here we have put, for any \(A\) \(\subset \) \(X,\)

The following theorem is due to Kuratowski [17], it is known as Measurable Selection Theorem.

\(\left( 2\right) \Rightarrow \left( 1\right) :\) Let \(x_{1},x_{2},...,x_{m}\) be any finite number of elements in \(X.\) Define \(f_{i}:\Omega \rightarrow X,\) \(i=1,2,...,m,\) by

Using simultaneous proximinality of \(L^{\infty }\left( \mu ,G\right) \) and Theorem 2.4, we get \(g\in L^{\infty }\left( \mu ,G\right) \) such that \(g\left( s\right) \) is a best simultaneous approximation of
\(f_{1}\left( s\right) ,f_{2}\left( s\right) ,...,f_{n}\left( s\right) \) for almost all \(s. \) Choose \(s_{0}\in \Omega \) so that \(\ g\left( s_{0}\right) \) is a best simultaneous approximation of \(x_{1},x_{2},...,x_{m}\) in \(G.\)

\(\left( 1\right) \Rightarrow \left( 2\right) :\) Let \(f_{1},f_{2},...,f_{m}\) be any finite number of elements in \(L^{\infty }\left( \mu ,X\right) .\) For each \(s\in \Omega \) define

For each \(s\in \Omega ,\) \(\Phi \left( s\right) \) is closed, bounded, and nonvoid subset of \(G.\) We shall show that \(\Phi \) is weakly measurable. Let \(O\) be an open set in \(X,\) the set

can be also be described as

Since \(\left( \Omega ,\Sigma ,\mu \right) \) is complete, \(f_{i}\) is measurable in the classical sense for \(i=1,2,...,m,\) by Lemma 3.1. Since subtraction in \(X,\) \(\sup ,\) and the norm in \(X\) are continuous, then the map

is measurable for any set \(A.\) It follows that \(\Phi ^{-1}(O)\) is measurable. By Theorem 3.3, \(\Phi \) has a measurable selection; i.e., there exists a function  \(f:\Omega \rightarrow G\) such that \(f(s)\in \phi (s)\) for each \(s\epsilon \Omega \) \(\ \)and \(f\) is measurable in the classical sense. By Lemma 3.2, \(f\) is strongly measurable. Hence \(f\) is a best simultaneous approximation for \(f_{1},f_{2},...,f_{m}\) in \(L^{\infty }\left( \mu ,G\right) \) by Theorem 2.4.