Abstract <p> We study boundedly ae-compact sets admitting, for any <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3242_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon>0\)</EquationSource> </InlineEquation>, an <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3242_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\tau\)</EquationSource> </InlineEquation>-continuous <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3242_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation>-selection, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3242_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau\)</EquationSource> </InlineEquation> is the topology of convergence in measure. Any such set in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3242_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_p\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3242_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(1\leqslant p<\infty\)</EquationSource> </InlineEquation>, is shown to be a sun. Given a nonempty set, it is shown that the existence of an <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3242_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\tau\)</EquationSource> </InlineEquation>-continuous <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3242_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation>-selection for each <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3242_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon>0\)</EquationSource> </InlineEquation> is equivalent to existence of a norm-norm continuous <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3242_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation>-selection for each <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3242_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon>0\)</EquationSource> </InlineEquation>. </p>

Solarity of Boundedly ae-Compact Sets

期刊论文
发表日期：

2025年6月1日

查看全文

I.G. Tsar’kov

摘要

Abstract

We study boundedly ae-compact sets admitting, for any \(\varepsilon>0\) , an \(n\tau\) -continuous \(\varepsilon\) -selection, where \(\tau\) is the topology of convergence in measure. Any such set in \(L_p\) , \(1\leqslant p<\infty\) , is shown to be a sun. Given a nonempty set, it is shown that the existence of an \(n\tau\) -continuous \(\varepsilon\) -selection for each \(\varepsilon>0\) is equivalent to existence of a norm-norm continuous \(\varepsilon\) -selection for each \(\varepsilon>0\) .