Exercise 1, § 4

Let G be a locally compact group, $\beta$ a left Haar measure on G, A a $\beta$-measurable subset of G, and $\nu$ a nonzero positive measure on G. Assume that sA is locally $\nu$-negligible for every $s \in G$. Show that A is locally $\beta$-negligible. (Reduce to the case that A is relatively compact and $\nu$ is bounded. Prove that $\nu$ and $\varphi_A \cdot \beta$ are convolvable and that $\nu * \beta \varphi_A = 0$, whence $0 = \| \nu * \varphi_A \cdot \beta \| = \| \nu \| \cdot \| \varphi_A \cdot \beta \|$.) Show, on admitting the continuum hypothesis, that this result may fail to hold if A is not assumed to be $\beta$-measurable. (Take $G = \mathbf{R}^2$, take for $\nu$ the Haar measure on the subgroup $\mathbf{R} \times \{0\}$, and apply Exer. 7 c) of Ch. V, §8.)