A Simple Proof of Vitali's Theorem for Signed Measures

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, $\sigma$-additive and translation invariant set function $\mathcal{L}$ from the power set of the real line to the extended real numbers with $\mathcal{L}([0,1])=1$. (Note that $\mathcal{L}$ is not assumed to be non-negative.)