{"title":"Positivity principle for measures on uniformly convex Banach spaces","authors":"E. Riss","doi":"10.1090/spmj/1722","DOIUrl":null,"url":null,"abstract":"<p>A Banach space <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is said to satisfy the <italic>positivity principle</italic> for small balls if for every finite Borel measures <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu\">\n <mml:semantics>\n <mml:mi>μ<!-- μ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\mu</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"nu\">\n <mml:semantics>\n <mml:mi>ν<!-- ν --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\nu</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, the inequalities <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu left-parenthesis upper B right-parenthesis less-than-or-equal-to nu left-parenthesis upper B right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>μ<!-- μ --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>B</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>≤<!-- ≤ --></mml:mo>\n <mml:mi>ν<!-- ν --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>B</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mu (B) \\leq \\nu (B)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for all balls B of radius less than 1 imply that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu less-than-or-equal-to nu\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>μ<!-- μ --></mml:mi>\n <mml:mo>≤<!-- ≤ --></mml:mo>\n <mml:mi>ν<!-- ν --></mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mu \\leq \\nu</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. It is shown that no uniformly convex infinite-dimensional separable Banach space <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> obeys the positivity principle for small balls.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1722","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A Banach space XX is said to satisfy the positivity principle for small balls if for every finite Borel measures μ\mu and ν\nu on XX, the inequalities μ(B)≤ν(B)\mu (B) \leq \nu (B) for all balls B of radius less than 1 imply that μ≤ν\mu \leq \nu. It is shown that no uniformly convex infinite-dimensional separable Banach space XX obeys the positivity principle for small balls.
期刊介绍：
This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.
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