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arXiv · 2406.06065

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

Aleksandar Bulj
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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.

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Aleksandar Bulj. 2024-06-10. Impossibility of decoding a translation invariant measure from a single set of positive Lebesgue measure. https://arxiv.org/abs/2406.06065

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