Naively Haar null sets in Polish groups

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• 作者：Má ; rton Elekes^a ; ^b ; ¹ ; ^{elekes.marton@renyi.mta.hu" class="auth_mail" title="E-mail the corresponding author} ; Zoltá ; n Vidnyá ; nszky^a ; ¹ ; ^{vidnyanszky.zoltan@renyi.mta.hu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Polish groups ; Haar null ; Christensen ; Shy ; Universally measurable ; Problem FC
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：193-200
• 全文大小：297 K

文摘

Let (G,⋅)$(G,\cdot )$ be a Polish group. We say that a set fc605a34e6f" title="Click to view the MathML source">X⊂G$X\subset G$ is Haar null   if there exists a universally measurable set fcb51e70" title="Click to view the MathML source">U⊃X$U\supset X$ and a Borel probability measure μ   such that for every bc54a1e88ef9fa6c" title="Click to view the MathML source">g,h∈G$g,h\in G$ we have μ(gUh)=0$\mu (gUh)=0$. We call a set X naively Haar null if there exists a Borel probability measure μ   such that for every bc54a1e88ef9fa6c" title="Click to view the MathML source">g,h∈G$g,h\in G$ we have fc638439" title="Click to view the MathML source">μ(gXh)=0$\mu (gXh)=0$. Generalizing a result of Elekes and Steprāns, which answers the first part of Problem FC from Fremlin's list, we prove that in every abelian Polish group there exists a naively Haar null set that is not Haar null.