Exercise 18, § 3

Let $T_1$ and $T_2$ be two topological spaces and $f$ a bijective continuous mapping of $T_1$ onto $T_2$. Assume that, for every bounded countably additive function $m_1$ on the Borel tribe $\mathscr{B}(T_1)$, there exists a sequence of compact subsets $K_n$ of $T_1$ such that $m_1(T_1 - \bigcup_n K_n) = 0$. Let $m_2$ be a bounded countably additive function on $\mathscr{B}(T_2)$. In order that there exist a bounded countably additive function $m_1$ on $\mathscr{B}(T_1)$ such that $m_2(A) = m_1(f^{-1}(A))$ for all $A \in \mathscr{B}(T_2)$, it is necessary and sufficient that the following condition be fulfilled: there exists a sequence of compact subsets $K_n$ of $T_1$ such that $m_2(T_2 - \bigcup_n f(K_n)) = 0$.