On closedness of law-invariant convex sets in rearrangement invariant spaces

被引：0
作者：
Made Tantrawan
Denny H. Leung
机构：
[1] National University of Singapore,Department of Mathematics
[2] Universitas Gadjah Mada,Department of Mathematics Faculty of Mathematics and Natural Sciences
来源：
Archiv der Mathematik | 2020年 / 114卷
关键词：
Convex sets; Law-invariant; Rearrangement invariant spaces; Fatou property; 46A55; 46E30; 46A20;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}$$\end{document}. In particular, we show that order closedness, σ(X,Xn∼)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ({\mathcal {X}},{\mathcal {X}}_n^\sim )$$\end{document}-closedness, and σ(X,L∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ({\mathcal {X}},L^\infty )$$\end{document}-closedness of a law-invariant convex set in X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}$$\end{document} are equivalent, where Xn∼\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}_n^\sim $$\end{document} is the order continuous dual of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {X}}$$\end{document}. We also provide some application to proper quasiconvex law-invariant functionals with the Fatou property.
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页码：175 / 183
页数：8
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