A Simple Proof of Vitali's Theorem for Signed Measures

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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.)