Exercise 14, § 3

a) Let $G$ be a locally compact group, $f$ a positive lower semi-continuous function in $G$, $\mu$ a positive measure on $G$. Prove that the function $x \to \int^* f(s^{-1}x)d\mu(s) = g(x)$ is lower semi-continuous in $G$ (cf. Chap. IV, § 1, No. 1, Theorem 1); for $\mu$ and $f$ to be convolvable, it is necessary and sufficient that $g$ be integrable for a left Haar measure on $G$.

b) On the group $G = \mathbf{R} \times \mathbf{R}$, let $\mu$ be the measure $\varepsilon_0 \otimes \lambda$, where $\lambda$ is the Lebesgue measure on $\mathbf{R}$; let $f(x, y) = (1 - |xy - 2|)^+$; prove that the function $g(x, y) = \int f(x - s, y - t)d\mu(s, t)$ is finite everywhere in $G$ but is neither continuous nor integrable for Lebesgue measure on $G$.