Is Lebesgue measure the only \(\sigma\)-finite invariant Borel measure? (Q2497373)

This is an interesting and well-written paper on elementary geometric measure theory. The main results are summarized as follows, where \(I_n\) refers to the group of Euclidean isometries, \({\mathfrak B}_{n}\) to the Borel \(\sigma\)-algebra and \(\lambda_n\) to the \(n\)-dimensional Lebesgue measure, each with respect to \({\mathbb R}^n\). Theorem. Let \({\mathfrak A}\) be an \(I_n\)-invariant \(\sigma\)-algebra which contains \({\mathfrak B}_{n}\). Then there exists an \(I_n\)-invariant, \(\sigma\)-finite measure \(\mu\) on \({\mathfrak A}\) such that the following holds for each \(B \in {\mathfrak B}_{n}\): (1) \(\lambda_n(B) = 0 \Rightarrow \mu (B) =0\); (2) \(\lambda_n(B) > 0 \Rightarrow \mu (B) = \infty\); (3) there exists \(c \in [0,\infty]\) such that \(\mu(B) = c \lambda_n (B)\).