Extending isometrically invariant measures on ℝn—a solution to Ciesielski’s query

Abstract

We prove that if \(m\colon \mathcal{M} \to [0,+\infty]\) is an isometrically invariant \(\sigma\)-finite countably additive measure on \(\mathbb{R} ^{n}\), then there exists a countably additive isometrically invariant extension \(m'\colon\mathcal{M}' \to [0,+\infty]\) of \(m\) such that the canonical embedding \(e\colon{\mathcal{M}/ m} \to {\mathcal{M}'/ m'}\) of measure algebras defined by \(e([A]_{m})=[A]_{m'}\) is not surjective. This answers a question of Ciesielski \cite{C2}.