A Simple Proof of Vitali's Theorem for Signed Measures

TL;DR

This paper presents a straightforward proof of an extension of Vitali's Theorem, demonstrating the non-existence of certain non-measurable, translation-invariant signed measures on the real line without using decomposition theorems.

Contribution

It provides a simplified proof of Vitali's Theorem for signed measures, avoiding complex decomposition theorems and extending the classical result.

Findings

No non-trivial, atom-less, σ-additive, translation-invariant signed measure exists with measure 1 on [0,1]

The proof does not rely on decomposition theorems

Extends Vitali's Theorem to 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.)