Exercise 14, § 4

a) Let G be a locally compact group, $f$ a lower semi-continuous positive function on G, $\mu$ a positive measure on G. Show that the function

is lower semi-continuous on G (cf. Ch. IV, §1, No. 1, Th. 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 Lebesgue measure on $\mathbf{R}$; let $f(x, y) = (1 - |xy - 2|)^+$; show that the function

is everywhere finite on G, but is not continuous and is not integrable for Lebesgue measure on G.