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A Simple Proof of Vitali’s Theorem for Signed Measures

T. Samuel
The American Mathematical Monthly
Vol. 120, No. 7 (August–September 2013), pp. 654-659
DOI: 10.4169/amer.math.monthly.120.07.654
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Page Count: 6
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A Simple Proof of Vitali’s Theorem for Signed Measures
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Abstract Several theorems are 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 nonmeasurable sets. Specifically we show, without using any decomposition theorems, that there does not exist a nontrivial, 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 nonnegative.)

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