Impossibility of decoding a translation invariant measure from a single set of positive Lebesgue measure

Abstract

Let $\mu$ be a translation invariant measure on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ and let $\lambda$ denote the Lebesgue measure on $\mathbb{R}^d$. If there exists an open set $U$ such that $0<\mu(U)=\lambda(U)<\infty$, it is a simple exercise to show that $\mu=\lambda|_{\mathcal{B}(\mathbb{R}^d)}$. Is the same conclusion true if $U$ is merely a Borel set? The main purpose of this short note is to construct a measure that provides a negative answer to this question. Incidentally, this construction provides a new example of a translation invariant measure with a rich domain and range that is not Hausdorff, a problem previously studied by Hirst.