A Simple Proof of Vitali's Theorem for Signed Measures

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.)