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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304425&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304425&_rdoc=1&_issn=0022247X&md5=a842db9297a7b48e5c607cf47382fa62" title="Click to view the MathML source">(G,⋅)class="mathContainer hidden">class="mathCode">$(G,\cdot )$ be a Polish group. We say that a set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304425&_mathId=si2.gif&_user=111111111&_pii=S0022247X16304425&_rdoc=1&_issn=0022247X&md5=c0610ad2517ee9501c63afc605a34e6f" title="Click to view the MathML source">X⊂Gclass="mathContainer hidden">class="mathCode">$X\subset G$ is Haar null   if there exists a universally measurable set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304425&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304425&_rdoc=1&_issn=0022247X&md5=6a392dcceaf88cf4dceba3c5fcb51e70" title="Click to view the MathML source">U⊃Xclass="mathContainer hidden">class="mathCode">$U\supset X$ and a Borel probability measure μ   such that for every class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304425&_mathId=si4.gif&_user=111111111&_pii=S0022247X16304425&_rdoc=1&_issn=0022247X&md5=1772e7b85d718e4dbc54a1e88ef9fa6c" title="Click to view the MathML source">g,h∈Gclass="mathContainer hidden">class="mathCode">$g,h\in G$ we have class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304425&_mathId=si5.gif&_user=111111111&_pii=S0022247X16304425&_rdoc=1&_issn=0022247X&md5=5e265069786e28e046ca740aa855266d" title="Click to view the MathML source">μ(gUh)=0class="mathContainer hidden">class="mathCode">$\mu (gUh)=0$. We call a set X naively Haar null if there exists a Borel probability measure μ   such that for every class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304425&_mathId=si4.gif&_user=111111111&_pii=S0022247X16304425&_rdoc=1&_issn=0022247X&md5=1772e7b85d718e4dbc54a1e88ef9fa6c" title="Click to view the MathML source">g,h∈Gclass="mathContainer hidden">class="mathCode">$g,h\in G$ we have class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304425&_mathId=si6.gif&_user=111111111&_pii=S0022247X16304425&_rdoc=1&_issn=0022247X&md5=13a3220d6ba76d891f9aaf41fc638439" title="Click to view the MathML source">μ(gXh)=0class="mathContainer hidden">class="mathCode">$\mu (gXh)=0$. 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.