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

2024/06/10 by Aleksandar Bulj, Bulj, Aleksandar
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.2406.06065

openalex publication_date 2024/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let μ be a translation invariant measure on (ℝ^d,B(ℝ^d)) and let λ denote the Lebesgue measure on ℝ^d. If there exists an open set U such that 0<μ(U)=λ(U)<∞, it is a simple exercise to show that μ=λ|_B(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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