On Simultaneous Farthest Points in 𝐿^∞(𝐼,𝑋)

Abstract

Let 𝑋 be a Banach space and let 𝐺 be a closed bounded subset of 𝑋. For (𝑥₁,𝑥₂,…,𝑥_𝑚)∈𝑋^𝑚, we set  𝜌(𝑥₁,𝑥₂,…,𝑥_𝑚,𝐺)=sup{max_{1≤𝑖≤𝑚}‖𝑥_𝑖−𝑦‖∶𝑦∈𝐺}. The set 𝐺 is called simultaneously remotal if, for any (𝑥₁,𝑥₂,…,𝑥_𝑚)∈𝑋^𝑚, there exists 𝑔∈𝐺 such that  𝜌(𝑥₁,𝑥₂,…,𝑥_𝑚,𝐺)=max_{1≤𝑖≤𝑚}‖𝑥_𝑖−𝑔‖. In this paper, we show that if 𝐺 is separable simultaneously remotal in 𝑋, then the set of ∞-Bochner integrable functions, 𝐿^∞(𝐼,𝐺), is simultaneously remotal in 𝐿^∞(𝐼,𝑋). Some other results are presented.

1. Introduction

Let 𝑋 be a Banach space and 𝐺 a bounded subset of 𝑋. For 𝑥∈𝑋, set 𝜌(𝑥,𝐺)=sup{‖𝑥−𝑦‖∶𝑦∈𝐺}. A point 𝑔₀∈𝐺 is called a farthest point of 𝐺 if there exists 𝑥∈𝑋 such that ‖𝑥−𝑔₀‖=𝜌(𝑥,𝐺). For 𝑥∈𝑋, the farthest point map 𝐹_𝐺(𝑥)={𝑔∈𝐺∶‖𝑥−𝑔‖=𝜌(𝑥,𝐺)}, that is, the set of points of 𝐺 farthest from 𝑥. Note that this set may be empty. Let 𝑅(𝐺,𝑋)={𝑥∈𝑋∶𝐹_𝐺(𝑥)≠𝜙}. We call a closed bounded set 𝐺 remotal if 𝑅(𝐺,𝑋)=𝑋 and densely remotal if 𝑅(𝐺,𝑋) is a norm dense in 𝑋. The concept of remotal sets in Banach spaces goes back to the sixties. However, almost all the results on remotal sets are concerned with the topological properties of such sets, see [1–4]. Remotal sets in vector valued continuous functions was considered in [5]. Related results on Bochner integrable function spaces, 𝐿^𝑝(𝐼,𝑋), 1≤𝑝≤∞, are given in [6–8].

The problem of approximating a set of points 𝑥₁,𝑥₂,…,𝑥_𝑚 simultaneously by a point 𝑔 (farthest point) in a subset 𝐺 of 𝑋 can be done in several ways, see [9]. Here, we will use the following definition.

Definition 1.1. Let 𝐺 be a closed bounded subset of 𝑋. A point 𝑔∈𝐺 is called a simultaneous farthest point of (𝑥₁,𝑥₂,…,𝑥_𝑚)∈𝑋^𝑚 if 𝜌𝑥₁,𝑥₂,…,𝑥_𝑚,𝐺=sup_ℎ∈𝐺max_{1≤𝑖≤𝑚}𝑥_𝑖−ℎ=max_{1≤𝑖≤𝑚}𝑥_𝑖−𝑔. (1.1) We call a closed bounded set 𝐺 of a Banach space 𝑋 simultaneously remotal if each 𝑚-tuple (𝑥₁,𝑥₂,…,𝑥_𝑚)∈𝑋^𝑚 admits a farthest point in 𝐺 and simultaneously densely remotal if the set of points 𝑅(𝐺,𝑋^𝑚)={(𝑥₁,𝑥₂,…,𝑥_𝑚)∈𝑋^𝑚∶𝐹_𝐺(𝑥₁,𝑥₂,…,𝑥_𝑚)≠𝜙}, where 𝐹_𝐺𝑥₁,𝑥₂,…,𝑥_𝑚=𝑔∈𝐺∶𝜌𝑥₁,𝑥₂,…,𝑥_𝑚,𝐺=max_{1≤𝑖≤𝑚}𝑥_𝑖−𝑔 (1.2) is norm dense in 𝑋^𝑚.

Clearly, if 𝑚=1, then simultaneously remotal is precisely remotal.

In this paper we consider the problem of simultaneous farthest point for bounded sets of the form 𝐿^∞(𝐼,𝐺) in the Banach space 𝐿^∞(𝐼,𝑋), where 𝑋 is a Banach space.

Throughout this paper, 𝑋 is a Banach space, 𝐺 is a closed bounded subset of 𝑋 and 𝐿^∞(𝐼,𝑋), the space of all 𝑋-valued essentially bounded functions on the unit interval 𝐼. For 𝑓∈𝐿^∞(𝐼,𝑋), we set ‖𝑓‖_∞=esssup{‖𝑓(𝑠)‖∶𝑠∈𝐼}. For 𝐺⊂𝑋, we set 𝐿^∞(𝐼,𝐺)={𝑓∈𝐿^∞(𝐼,𝑋)∶𝑓(𝑠)∈𝐺, almost all 𝑠∈𝐼}.

2. Distance Formula

The farthest distance formula is important in the study of farthest point. In this section, we compute the ∞-farthest distance from an element 𝑓∈𝐿^∞(𝐼,𝑋) to a bounded set 𝐿^∞(𝐼,𝐺). We begin with the following proposition.

Proposition 2.1. Let 𝑓₁,𝑓₂,…,𝑓_𝑚, then max_{1≤𝑖≤𝑚}‖𝑓_𝑖‖_∞ = esssupmax_{1≤𝑖≤𝑚}(‖𝑓_𝑖(𝑡)‖).

Proof. For 1≤𝑖≤𝑚, 𝑓_𝑖∞=esssup𝑓_𝑖(𝑡)≤esssupmax_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)≤max_{1≤𝑖≤𝑚}𝑓_𝑖∞. (2.1) Hence, max_{1≤𝑖≤𝑚}‖𝑓_𝑖‖_∞=esssupmax_{1≤𝑖≤𝑚}(‖𝑓_𝑖(𝑡)‖)

Theorem 2.2. Let 𝑋 be a Banach space and let 𝐺 be a closed bounded subset of 𝑋. If a function Φ∶𝐼→ℝ defined by Φ(𝑡)=𝜌(𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡),𝐺), where 𝑓₁,𝑓₂,…,𝑓_𝑚∈𝐿^∞(𝐼,𝑋), then Φ∈𝐿^∞(𝐼) and 𝜌𝑓₁,𝑓₂,…,𝑓_𝑚,𝐿^∞(𝐼,𝐺)=‖Φ‖_∞=sup_{𝑔∈𝐿∞()𝐼,𝐺}max_{1≤𝑖≤𝑚}𝑓_𝑖−𝑔_∞. (2.2)

Proof. Let 𝑓₁,𝑓₂,…,𝑓_𝑚∈𝐿^∞(𝐼,𝑋). Being strongly measurable, there exist 𝑚 sequences of simple functions (𝑓_𝑖𝑛), 1≤𝑖≤𝑚 such that ‖𝑓_𝑖𝑛(𝑡)−𝑓_𝑖(𝑡)‖→0 as 𝑛→∞ for almost all 𝑡∈𝐼. We may write 𝑓_𝑗𝑛=∑^𝑚(𝑛)_𝑖=1𝜒_{𝐴(𝑖,𝑛)}𝑥^(𝑗)_𝑖𝑛. Since 𝜌(𝑥₁,𝑥₂,…,𝑥_𝑚,𝐺) is a continuous function of (𝑥₁,𝑥₂,…,𝑥_𝑚)∈𝑋^𝑚, the inequality 𝜌𝑓_1𝑛(𝑡),𝑓_2𝑛(𝑡),…,𝑓_𝑚𝑛(𝑡),𝐺−𝜌𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡),𝐺≤max_{1≤𝑖≤𝑚}𝑓_𝑖𝑛(𝑡)−𝑓_𝑖(𝑡)⟶0 (2.3) implies that 𝜌𝑓_1𝑛(𝑡),𝑓_2𝑛(𝑡),…,𝑓_𝑚𝑛(𝑡),𝐺−𝜌𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡),𝐺⟶0. (2.4) Set Φ_𝑛(𝑡)=𝜌(𝑓_1𝑛(𝑡),𝑓_2𝑛(𝑡),…,𝑓_𝑚𝑛(𝑡),𝐺). Then, Φ_𝑛(𝑡)=sup_𝑔∈𝐺max_{1≤𝑘≤𝑚}𝑓_𝑘𝑛(𝑡)−𝑔=sup_𝑔∈𝐺max_{1≤𝑘≤𝑚}^𝑚(𝑛)_𝑖=1𝜒_𝐴𝑖𝑛(𝑡)𝑥^(𝑘)_𝑖𝑛=−𝑔_{𝑚(𝑛)𝑖=1}𝜒_𝐴𝑖𝑛(𝑡)sup_𝑔∈𝐺max_{1≤𝑘≤𝑚}𝑥^(𝑘)_𝑖𝑛−𝑔. (2.5) So Φ_𝑛 is a simple function for each 𝑛 and Lim_𝑛→∞‖Φ_𝑛(𝑡)−Φ(𝑡)‖=0 for almost all 𝑡∈𝐼. Hence Φ is measurable. Furthermore, for each 𝑤∈𝐿^∞(𝐼,𝐺), max_{1≤𝑖≤𝑚}𝑓_𝑖−𝑤_∞=max_{1≤𝑖≤𝑚}esssup𝑓_𝑖(𝑡)(𝑡=−𝑤)esssupmax_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)−𝑤(𝑡),Proposition≤2.1esssupsup_𝑔∈𝐺max_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)=𝜌𝑓−𝑔₁(𝑡),𝑓₂(𝑡),𝑓₃(𝑡),…,𝑓_𝑚(𝑡),𝐺_∞=‖Φ‖_∞. (2.6) Thus, ‖Φ‖_∞≥𝜌𝑓₁,𝑓₂,…,𝑓_𝑚,𝐿^∞(𝐼,𝐺). (2.7) To prove the reverse inequality. Let 𝜖>0 be given, since countably valued functions are dense in 𝐿^∞(𝐼,𝑋), there exist countably valued functions 𝛾₁,𝛾₂,…,𝛾_𝑚 in 𝐿^∞(𝐼,𝑋) such that ‖𝑓_𝑖−𝛾_𝑖‖<𝜖, 1≤𝑖≤𝑚. We may write 𝛾_𝑗=∑^∞_𝑖=1𝜒_𝐴𝑖𝑥_𝑖^(𝑗), as in [10]. We may assume ∑^∞_𝑖=1𝜒_𝐴𝑖=1, 𝜇(𝐴_𝑖)>0 for all 𝑖. For each 𝑖, Choose ℎ_𝑖∈𝐺 such that max_{1≤𝑘≤𝑚}𝑥_𝑖^(𝑘)−ℎ_𝑖>𝜌𝑥_𝑖⁽¹⁾,𝑥_𝑖⁽²⁾,…,𝑥_𝑖^(𝑚),𝐺−𝜖. (2.8) Now, set 𝑔∈𝐿^∞(𝐼,𝐺) as 𝑔(𝑠)=∑^∞_𝑖=1𝜒_𝐴𝑖(𝑠)ℎ_𝑖. The inequality 𝛾_𝑗−𝑔_∞≤𝛾_𝑗−𝑓_𝑗∞+𝑓_𝑗−𝑔_∞ (2.9) implies max_{1≤𝑗≤𝑚}𝛾_𝑗−𝑔_∞≤𝜖+max_{1≤𝑖≤𝑚}𝑓_𝑗−𝑔_∞. (2.10) Further, max_{1≤𝑗≤𝑚}𝑓_𝑗−𝑔_∞≥max_{1≤𝑗≤𝑚}𝛾_𝑗−𝑔_∞−𝜖=max_{1≤𝑗≤𝑚}esssup𝛾_𝑗(𝑡)(𝑡=−𝑔)−𝜖esssupmax_{1≤𝑗≤𝑚}^∞_𝑖=1𝜒_𝐴𝑖(𝑡)𝑥_𝑖^(𝑗)−ℎ_𝑖−𝜖,Proposition=2.1esssupmax^∞_{1≤𝑗≤𝑚𝑖=1}𝜒_𝐴𝑖(𝑡)𝑥_𝑖^(𝑗)−ℎ_𝑖=−𝜖esssup^∞_𝑖=1𝜒_𝐴𝑖(𝑡)max_{1≤𝑗≤𝑚}𝑥_𝑖^(𝑗)−ℎ_𝑖≥−𝜖esssup^∞_𝑖=1𝜒_𝐴𝑖(𝑡)𝜌𝑥_𝑖⁽¹⁾,𝑥_𝑖⁽²⁾,…,𝑥_𝑖^(𝑚)=,𝐺−2𝜖esssup𝜌𝛾₁(𝑡),𝛾₂(𝑡),…,𝛾_𝑚(𝑡),𝐺−2𝜖. (2.11) For 1≤𝑗≤𝑚 and 𝑎∈𝐺, the inequality 𝑓_𝑗(𝑡)−𝑎≤𝑓_𝑗(𝑡)−𝛾_𝑗(𝑡)+𝛾_𝑗(𝑡)−𝑎 (2.12) implies 𝜌𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡),𝐺≤max_{1≤𝑗≤𝑚}𝑓_𝑗(𝑡)−𝛾_𝑗(𝑡)+𝜌𝛾₁(𝑡),𝛾₂(𝑡),…,𝛾_𝑚(𝑡),𝐺. (2.13) Therefore, max_{1≤𝑗≤𝑚}𝑓_𝑗−𝑔_∞≥ess𝜌𝑓sup¹(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡),𝐺−max_{1≤𝑗≤𝑚}𝛾_𝑗(𝑡)−𝑓_𝑗(𝑡≥‖Φ‖)−2𝜖_∞−esssupmax_{1≤𝑗≤𝑚}𝛾_𝑗(𝑡)−𝑓_𝑗(𝑡≥‖Φ‖)−2𝜖_∞−3𝜖. (2.14) Hence, 𝜌(𝑓₁,𝑓₂,…,𝑓_𝑚,𝐿^∞(𝐼,𝐺))+3𝜖≥‖Φ‖_∞. Since 𝜖 was arbitrary, we have ‖Φ‖_∞=𝜌(𝑓₁,𝑓₂,…,𝑓_𝑚,𝐿^∞(𝐼,𝐺)).

Corollary 2.3. Let 𝑔 be a strongly measurable function from 𝐼 to a closed bounded subset 𝐺 of a Banach space 𝑋, and 𝑓₁,𝑓₂,…,𝑓_𝑚∈𝐿^∞(𝐼,𝑋). If 𝑔(𝑡) is a simultaneous farthest point of 𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡) in 𝐺, then 𝑔 is a simultaneous farthest point of 𝑓₁,𝑓₂,…,𝑓_𝑚 in 𝐿^∞(𝐼,𝐺).

Proof. By assumption, max_{1≤𝑖≤𝑚}(‖𝑓_𝑖(𝑡)−𝑔(𝑡)‖)=𝜌(𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡),𝐺) for almost 𝑡∈𝐼. Since 𝐺 is bounded, it follows that 𝑔∈𝐿^∞(𝐼,𝐺) and esssupmax_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)−𝑔(𝑡)=esssup𝜌𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡),𝐺. (2.15) Theorem 2.2 and Proposition 2.1 implies that max_{1≤𝑖≤𝑚}𝑓_𝑖−𝑔_∞=esssupmax_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)−𝑔(𝑡)=𝜌𝑓₁,𝑓₂,…,𝑓_𝑚,𝐿^∞(𝐼,𝐺) (2.16) and 𝑔 is a simultaneous farthest point of 𝑓₁,𝑓₂,…,𝑓_𝑚 in 𝐿^∞(𝐼,𝐺).

3. Remotal Sets in 𝐿^∞(𝐼,𝑋)

In this section, we raise the question: if 𝐺 is a simultaneously remotal set in 𝑋, is 𝐿^∞(𝐼,𝐺) simultaneously remotal in 𝐿^∞(𝐼,𝑋)? We get a positive answer to this question in the case that 𝐺 is a separable simultaneously remotal subset of 𝑋 or in the case that Span 𝐺 is finite dimensional subspace of 𝑋. We begin with the following theorem.

Theorem 3.1. If 𝐺 is simultaneously densely remotal in 𝑋, then 𝐿^∞(𝐼,𝐺) is simultaneously densely remotal in 𝐿^∞(𝐼,𝑋).

Proof. Let 𝑓₁,𝑓₂,…,𝑓_𝑚∈𝐿^∞(𝐼,𝑋). Then, there exist 𝛾₁,𝛾₂,…,𝛾_𝑚 simple functions such that ‖𝑓_𝑖−𝛾_𝑖‖_∞<𝜖/2, 1≤𝑖≤𝑚. Now, we can write without lost of generality, 𝛾_𝑖=∑^𝑛_𝑘=1𝜒_𝐴𝑘𝑥_𝑘^(𝑖). Since 𝐺 is simultaneously densely remotal, then there exist an 𝑚-tuple (𝑦_𝑘⁽¹⁾,𝑦_𝑘⁽²⁾,…,𝑦_𝑘^(𝑚)) and 𝑔_𝑘∈𝐺 such that max_{1≤𝑖≤𝑚}‖𝑥_𝑘^(𝑖)−𝑦_𝑘^(𝑖)‖<𝜖/2 and max_{1≤𝑖≤𝑚}𝑦_𝑘^(𝑖)−𝑔_𝑘=𝜌𝑦_𝑘⁽¹⁾,𝑦_𝑘⁽²⁾,…,𝑦_𝑘^(𝑚),𝐺=sup_𝑔∈𝐺max_{1≤𝑖≤𝑚}𝑦_𝑘^(𝑖)−𝑔. (3.1) Set 𝜙=∑^𝑛_𝑘=1𝜒_𝐴𝑘𝑔_𝑘 and ℎ_𝑖=∑^𝑛_𝑘=1𝜒_𝐴𝑘𝑦_𝑘^(𝑖). Then, max_{1≤𝑖≤𝑚}ℎ_𝑖−𝜙_∞=max_{1≤𝑖≤𝑚}esssupℎ_𝑖(𝑡)(𝑡=−𝜙)esssupmax_{1≤𝑖≤𝑚}^𝑛_𝑘=1𝜒_𝐴𝑘(𝑡)𝑦_𝑘^(𝑖)−𝑔_𝑘,Proposition=2.1esssupmax^𝑛_{1≤𝑖≤𝑚𝑘=1}𝜒_𝐴𝑘(𝑡)𝑦_𝑘^(𝑖)−𝑔_𝑘≥esssupmax^𝑛_{1≤𝑖≤𝑚𝑘=1}𝜒_𝐴𝑘(𝑡)𝑦_𝑘^(𝑖)−𝑔 (3.2) for every 𝑔∈𝐺. In particular, for any 𝑤∈𝐿^∞(𝐼,𝐺), using Proposition 2.1, max_{1≤𝑖≤𝑚}ℎ_𝑖−𝜙_∞≥esssupmax_{1≤𝑖≤𝑚}ℎ_𝑖(𝑡)−𝑤(𝑡)≥max_{1≤𝑖≤𝑚}esssupℎ_𝑖(𝑡)−𝑤(𝑡)=max_{1≤𝑖≤𝑚}ℎ_𝑖−𝑤_∞, (3.3) Hence, 𝜙 is a farthest point from the 𝑚-tuple (ℎ₁,ℎ₂,…,ℎ_𝑚). But max_{1≤𝑖≤𝑚}ℎ_𝑖−𝛾_𝑖∞=max_{1≤𝑖≤𝑚}esssup^𝑛_𝑘=1𝜒_𝐴𝑘𝑦_𝑘^(𝑖)−𝑥_𝑘^(𝑖)=max_{1≤𝑖≤𝑚}esssup^𝑛_𝑘=1𝜒_𝐴𝑘𝑦_𝑘^(𝑖)−𝑥_𝑘^(𝑖)=esssupmax^𝑛_{1≤𝑖≤𝑚𝑘=1}𝜒_𝐴𝑘𝑦_𝑘^(𝑖)−𝑥_𝑘^(𝑖),Proposition=2.1esssup^𝑛_𝑘=1𝜒_𝐴𝑘max_{1≤𝑖≤𝑚}𝑦_𝑘^(𝑖)−𝑥_𝑘^(𝑖)≤𝜖2. (3.4) Hence, max_{1≤𝑖≤𝑚}𝑓_𝑖−ℎ_𝑖∞≤max_{1≤𝑖≤𝑚}𝑓_𝑖−𝛾_𝑖+max_{1≤𝑖≤𝑚}𝛾_𝑖−ℎ_𝑖≤𝜖2+𝜖2=𝜖. (3.5) This complete the proof of the theorem.

For a remotal set 𝐺⊆𝑋, the map 𝑇∶𝑋^𝑚→2^𝐺 defined by 𝑇𝑥₁,𝑥₂,…,𝑥_𝑚=𝐹𝑥₁,𝑥₂,…,𝑥_𝑚=𝑥,𝐺𝑔∈𝐺∶𝜌₁,𝑥₂,…,𝑥_𝑚,𝐺=max_{1≤𝑖≤𝑚}𝑥_𝑖−𝑔 (3.6) is a multivalued map in general. Hence for any 𝑓∈𝐿^∞(𝐼,𝑋), the map 𝑇∘𝑓 is a multivalued map from 𝐼 into 𝐺.

Before proceeding we remind the reader for some facts regarding mult-valued maps. For a Banach space 𝑋 and a measurable space 𝐼 a function 𝑓∶𝐼→𝑋 is said to be strongly measurable if it is the pointwise limit of a sequence of simple functions almost everywhere. On the other hand 𝑓 is said to measurable in the classical sense if 𝑓⁻¹(𝐾) is measurable in 𝐼 for any closed set 𝐾 in 𝑋, see [10]. A multivalued function 𝑇∶𝐼→𝑋 is said to measurable in the classical sense if 𝑇⁻¹(𝐾) is measurable in 𝐼 for any closed set 𝐾 in 𝑋, here 𝑇⁻¹(𝐾)={𝑡∈𝐼∶𝑇(𝑡)∩𝐾≠𝜙}. A measurable in the classical sense 𝑔 may be extracted from a measurable multivalued function 𝑇∶𝐼→𝑋 where 𝑋 is a separable Banach space provided that 𝑇(𝑡) is a closed subset of 𝑋 for each 𝑡∈𝐼 and such that 𝑔(𝑡)∈𝑇(𝑡) for each 𝑡∈𝐼, see [11, page 289].

Theorem 3.2. Let 𝐺 be a closed bounded simultaneously remotal in 𝑋 such that span(𝐺) is finite dimensional subspace of 𝑋, then 𝐿^∞(𝐼,𝐺) is simultaneously remotal in 𝐿^∞(𝐼,𝑋).

Proof. First, we will prove that 𝑇 is a closed valued map. If 𝑇(𝑥₁,𝑥₂,…,𝑥_𝑚) is finite set, then it is closed. If 𝑇(𝑥₁,𝑥₂,…,𝑥_𝑛) is not finite, let 𝑦∈𝑇(𝑥₁,𝑥₂,…,𝑥_𝑚), then there exists 𝑦_𝑛∈𝑇(𝑥₁,𝑥₂,…,𝑥_𝑚) such that 𝑦_𝑛→𝑦. This implies max_{1≤𝑖≤𝑚}𝑥_𝑖−𝑔≤max_{1≤𝑖≤𝑚}𝑥_𝑖−𝑦_𝑛, (3.7) for every 𝑔∈𝐺. Taking the limit as 𝑛→∞, we get max‖𝑥_𝑖−𝑔‖≤max‖𝑥_𝑖−𝑦‖ for every 𝑔∈𝐺, and this implies that 𝑦∈𝑇(𝑥₁,𝑥₂,…,𝑥_𝑚) and 𝑇 is a closed multivalued map.
To prove that 𝑇 is measurable in the classical sense let 𝐵 be any closed subset of 𝐺. If (𝑥₁,𝑥₂,…,𝑥_𝑚)∈𝑇⁻¹(𝐵), then there exists (𝑥₁^(𝑛),𝑥₂^(𝑛),…,𝑥_𝑚^(𝑛))∈𝑇⁻¹(𝐵) that converges to (𝑥₁,𝑥₂,…,𝑥_𝑚). Since (𝑥₁^(𝑛),𝑥₂^(𝑛),…,𝑥_𝑚^(𝑛))∈𝑇⁻¹(𝐵), then 𝑇(𝑥₁^(𝑛),𝑥₂^(𝑛),…,𝑥_𝑚^(𝑛))∩𝐵≠𝜙. Choose 𝑦_𝑛∈𝑇(𝑥₁^(𝑛),𝑥₂^(𝑛),…,𝑥_𝑚^(𝑛))∩𝐵. Then, 𝑦_𝑛 has a convergent subsequence 𝑦_𝑛𝑘→𝑦 being a sequence in a bounded closed subset of a finite dimensional space. But max_{1≤𝑖≤𝑚}𝑥^(𝑛𝑘)_𝑖−𝑦_𝑛𝑘≥max_{1≤𝑖≤𝑚}𝑥^(𝑛𝑘)_𝑖−𝑔 (3.8) for every 𝑔∈𝐺. Taking the limit as 𝑛_𝑘→∞, we get max_{1≤𝑖≤𝑚}𝑥_𝑖−𝑦≥max_{1≤𝑖≤𝑚}𝑥_𝑖−𝑔 (3.9) for every 𝑔∈𝐺. Hence 𝑦∈𝑇(𝑥₁,𝑥₂,…,𝑥_𝑚)∩𝐵≠∅. Therefore (𝑥₁,𝑥₂,…,𝑥_𝑚)∈𝑇⁻¹(𝐵) and 𝑇⁻¹(𝐵) is closed. Hence, 𝑇 is measurable and if 𝛾 is the vector valued map 𝛾∶𝐼→𝑋^𝑚, 𝛾(𝑡)=(𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡)), then 𝑇∘𝛾 is a measurable closed multivalued map. By Theorem 6.6.4 in [11, page 289], 𝑇∘𝛾 has a measurable selection say 𝑔. Further, max_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)−𝑔(𝑡)≥max_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)−ℎ(𝑡), (3.10) for every ℎ∈𝐿^∞(𝐼,𝐺) and so esssupmax_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)−𝑔(𝑡)=max_{1≤𝑖≤𝑚}esssup𝑓_𝑖(𝑡)−𝑔(𝑡)≥max_{1≤𝑖≤𝑚}esssup𝑓_𝑖(𝑡)−ℎ(𝑡). (3.11) Hence, max_{1≤𝑖≤𝑚}𝑓_𝑖−𝑔_∞≥max_{1≤𝑖≤𝑚}𝑓_𝑖−ℎ_∞, (3.12) for every ℎ∈𝐿^∞(𝐼,𝐺) and 𝐿^∞(𝐼,𝐺) is remotal in 𝐿^∞(𝐼,𝑋).

Finally, following the steps in the prove of Theorem 3.6 in [6], we proof the following main result in the paper.

Theorem 3.3. Let 𝐺 be a separable simultaneously remotal subset of 𝑋. Then 𝐿^∞(𝐼,𝐺) is simultaneously remotal in 𝐿^∞(𝐼,𝑋).

Proof. Let (𝑓₁,𝑓₂,…,𝑓_𝑚)∈(𝐿^∞(𝐼,𝑋))^𝑚. Using Corollary 2.3, it is sufficient to show that there exists a measurable function 𝑔 defined on 𝐼 such that 𝑔(𝑡) is a simultaneous farthest point of (𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡)) in 𝐺. Since 𝑓₁,𝑓₂,…,𝑓_𝑚 are strongly measurable. We may assume that 𝑓₁(𝐼),𝑓₂(𝐼),…,𝑓_𝑚(𝐼) are separable sets in 𝑋. So there exist {𝐼_𝑛𝑖}^∞_𝑛=1a countable partition of 𝐼 such that diam(𝑓_𝑖(𝐼_𝑛𝑖))<1/2, for each 𝑖, 1≤𝑖≤𝑚, for all 𝑛; where diam(𝑆)=sup𝑥−𝑦∶𝑥,𝑦∈𝑆. (3.13) Consider the partition 𝐼_{𝑘1𝑘2⋯𝑘𝑚}=⋂^𝑚_𝑖=1𝐼_𝑘𝑖, where (𝐼_𝑘1,𝐼_𝑘2,…,𝐼_𝑘𝑚)⊆∏^𝑚_𝑖=1{𝐼_𝑛𝑖}, for 1≤𝑛<∞. Then diam(𝑓_𝑖(𝐼_{𝑘1𝑘2⋯𝑘𝑚}))<1/2. For simplicity write 𝐼_{𝑘1𝑘2⋯𝑘𝑚} as {𝐼_𝑛}^∞_𝑛=1. For each 𝑡∈𝐼, let 𝑔₀(𝑡) be a simultaneous farthest point from (𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡)) in 𝐺. Define the map 𝑔₀ from 𝐼 into 𝐺 by 𝑔₀(𝑡) is a simultaneous farthest point from (𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡)). Apply Lemma 3.1 in [6] with 𝜖=1/2 and 𝐼=𝐼_𝑛=𝐴, we get countable partitions in each 𝐼_𝑛 and therefore countable partition in the whole of 𝐼 in measurable sets {𝐸_𝑛}^∞_𝑛=1 and a sequence of subsets {𝐴_𝑛}^∞_𝑛=1 such that 𝐴_𝑛⊆𝐸_𝑛,𝜇^∗𝐴_𝑛=𝜇𝐸_𝑛,diam𝑔₀𝐴_𝑛<12anddiam𝑓_𝑖𝐸_𝑛<12,1≤𝑖≤𝑚. (3.14) Repeat the same argument in each 𝐸_𝑛 with 𝜖=1/2², 𝐼=𝐸_𝑛, and 𝐴=𝐴_𝑛. For each 𝑛, we get a countable partition {𝐸_(𝑛,𝑘)∶1≤𝑘<∞} of 𝐸_𝑛 in measurable sets and a sequence {𝐴_(𝑛,𝑘)∶1≤𝑘<∞} of subsets of 𝐼 such that 𝐴_(𝑛,𝑘)⊆𝐸_(𝑛,𝑘)∩𝐴_𝑛,𝜇^∗𝐴_(𝑛,𝑘)=𝜇𝐸_(𝑛,𝑘),diam𝑔₀𝐴_(𝑛,𝑘)<12²anddiam𝑓_𝑖𝐸_(𝑛,𝑘)<12²,𝑖=1,2,…,𝑚. (3.15) Now, we will use mathematical induction for each 𝑛; let Δ_𝑛 be the set of 𝑛-tuples of natural numbers and Δ=⋃{Δ_𝑛∶1≤𝑛<∞}. On this Δ consider the partial order defined by (𝑚₁,𝑚₂,…,𝑚_𝑖)<(𝑛₁,𝑛₂,…,𝑛_𝑗) if and only if 𝑖≤𝑗 and 𝑚_𝑘=𝑛_𝑘, 𝑘=1,2,…,𝑖. Then by induction for each 𝑛, we can find a countable partition {𝐸_𝛼∶𝛼∈Δ_𝑛} of 𝐼 of measurable sets and a collection {𝐴_𝛼∶𝛼∈Δ_𝑛} of subsets of 𝐼 such that:(1)𝐴_𝛼⊆𝐸_𝛼 and 𝜇^∗(𝐴_𝛼)=𝜇(𝐸_𝛼),(2)𝐸_𝛽⊆𝐸_𝛼 and 𝐴_𝛽⊆𝐴_𝛼 if 𝛼≤𝛽,(3)diam(𝑓_𝑖(𝐸_𝛼))<1/2^𝑛 for 𝑖=1,2,…,𝑚 and diam(𝑔₀(𝐴_𝛼))<1/2^𝑛 for 𝛼∈Δ_𝑛.
We may assume 𝐴_𝛼≠𝜙, for all 𝛼. For each 𝛼∈Δ, let 𝑡_𝛼∈𝐴_𝛼 and define 𝑔_𝑛 from 𝐼 into 𝐺 by 𝑔_𝑛(𝑡)=∑_{𝛼∈Δ𝑛}𝜒_𝐸𝛼(𝑡)𝑔₀(𝑡_𝛼). Then, for each 𝑡∈𝐼 and 𝑛≤𝑚, we have 𝑔_𝑛(𝑡)−𝑔_𝑚(𝑡)=_{𝛼∈Δ𝑛}𝜒_𝐸𝛼(𝑡)𝑔₀𝑡_𝛼−_{𝛽∈Δ𝑚}𝜒_𝐸𝛽(𝑡)𝑔₀𝑡_𝛽≤_{𝛽∈Δ𝑚}𝜒_𝐸𝛽(𝑡)𝑔₀𝑡_𝛼−𝑔₀𝑡_𝛽≤_{𝛽∈Δ𝑚}𝑔₀𝑡_𝛼−𝑔₀𝑡_𝛽𝜒_𝐸𝛽≤12^𝑛. (3.16) Hence, (𝑔_𝑛(𝑡)) is a Cauchy sequence in 𝐺 for all 𝑡∈𝐼. Therefore (𝑔_𝑛(𝑡)) is a convergent sequence. Let 𝑔∶𝐼→𝐺 be defined to be the pointwise limit of (𝑔_𝑛). Since 𝑔_𝑛 is strongly measurable for each 𝑛, we have 𝑔 is strongly measurable. Further for 𝑡∈𝐼, 1≤𝑛<∞, and 𝑡∈𝐸_𝛼 for some 𝛼∈Δ_𝑛, we have max_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)−𝑔_𝑛(𝑡)=max_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)−𝑔₀𝑡_𝛼≥max_{1≤𝑖≤𝑚}𝑓_𝑖𝑡_𝛼−𝑔₀𝑡_𝛼−𝑓_𝑖(𝑡)−𝑓_𝑖𝑡_𝛼≥max_{1≤𝑖≤𝑚}𝑓_𝑖𝑡_𝛼−𝑔₀𝑡_𝛼−12^𝑛≥𝜌𝑓₁𝑡_𝛼,𝑓₂𝑡_𝛼,…,𝑓_𝑚𝑡_𝛼−1,𝐺2^𝑛. (3.17) For 1≤𝑖≤𝑚, the inequality 𝑓_𝑖(𝑡)−𝑎≤𝑓_𝑖(𝑡)−𝑓_𝑖𝑡_𝛼+𝑓_𝑖𝑡_𝛼≤1−𝑎2^𝑛+𝑓_𝑖𝑡_𝛼−𝑎 (3.18) implies that 𝜌𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡)≤1,𝐺2^𝑛+𝜌𝑓₁𝑡_𝛼,𝑓₂𝑡_𝛼,…,𝑓_𝑚𝑡_𝛼,𝐺. (3.19) Therefore, max_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)−𝑔_𝑛(𝑡)≥𝜌𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡)−1,𝐺2^𝑛−1. (3.20) Taking limit as 𝑛→∞, we get 𝜌𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡),𝐺=max_{1≤𝑖≤𝑚}𝑓_𝑖(𝑡)−𝑔(𝑡). (3.21) This completes the proof of the theorem.