一致凸Banach空间测度的正性原理

Pub Date : 2022-06-27 DOI:10.1090/spmj/1722

E. Riss

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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</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</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</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</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</mml:mi>\n <mml:mo>≤</mml:mo>\n <mml:mi>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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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</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</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</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</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</mml:mi>\\n <mml:mo>≤</mml:mo>\\n <mml:mi>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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-06-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/spmj/1722\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1722","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

摘要

Banach空间X X被认为满足小球的正性原理，如果对于X X上的每个有限Borel测度μ，对于半径小于1的所有球B，不等式μ。证明了没有一致凸的无穷维可分Banach空间X X服从小球的正性原理。

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Positivity principle for measures on uniformly convex Banach spaces

A Banach space X X is said to satisfy the positivity principle for small balls if for every finite Borel measures μ \mu and ν \nu on X X , 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 X X obeys the positivity principle for small balls.

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