Measurable homomorphisms of locally compact groups

Abstract:

Let $G$ and $H$ be locally compact groups and $\varphi$ a homomorphism from $G$ into $H$. Suppose that ${\varphi ^{ - 1}}\left ( U \right )$ is measurable for every open set $U \subset H$. It is known under some conditions, for example, if $H$ is $\sigma$-compact, that $\varphi$ is continuous. Here it is shown that this result is true without any countability restrictions on $G$ and $H$. The proof depends on the observation that the regular representation of $H$ is a homomorphism.