A simple proof of Vitali's theorem for signed measures

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Abstract

There are several theorems named after the Italian mathematician Vitali. In this note we provide a simple proof of an extension of Vitali's Theorem on the existence of non-measurable sets. Specifically, we show, without using any decomposition theorems, that there does not exist a non-trivial, atom-less, σ-additive and translation invariant set function L from the power set of the real line to the extended real numbers with L([0,1]) = 1. (Note that L is not assumed to be non-negative.)
Original languageEnglish
Pages (from-to)654-660
JournalThe American Mathematical Monthly
Volume120
Issue number7
DOIs
Publication statusPublished - 2013

Keywords

  • Vitali's Theorem
  • Non-measureable sets
  • signed measures

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