Infinite products of infinite measures

Abstract

Let $(X_i,{\mathscr B}_i,m_i)$ $(i\in{\mathbb N})$ be a sequence of regular Borel measure spaces. There is a Borel measure $\mu$ on $\prod_{i\in{\mathbb N}}{X_i}$ such that if $K_i\subseteq X_i$ is compact for all $i\in{\mathbb N}$ and $\prod_{i\in{\mathbb N}}{m_i(K_i)}$ converges, then $\mu({\prod_{i\in{\mathbb N}}{K_i}})=\prod_{i\in{\mathbb N}}{m_i(K_i)}$.