Exercise 11, § 3

Let $K$ be a metrizable compact space, $\mu$ a positive measure on $K$, $f$ a $\mu$-measurable mapping of $K$ into a locally compact space $T$, and $\nu$ a positive measure on $T$; I and $U$ have the same meaning as in Exer. 10.

a) Suppose that for every $\mu$-negligible set $N \subset K$, $f(N)$ is $\nu$-negligible. Then $U$ is $\nu$-negligible and, for every $\mu$-measurable subset $A \subset K$, $f(A)$ is $\nu$-measurable.

b) In order that $I$ be $\nu$-negligible and the image under $f$ of every $\mu$-negligible set be $\nu$-negligible, it is necessary and sufficient that $f$ satisfy the following condition: for every $\varepsilon > 0$, there exists a $\delta > 0$ such that, for every subset $A \subset K$ that is $\mu$-measurable and is such that $\mu(A) \leq \delta$, $f(A)$ is $\nu$-measurable and $\nu(f(A)) \leq \varepsilon$. (To see that the condition is necessary, argue by contradiction, by considering a decreasing sequence $(A_n)$ of $\mu$-measurable sets whose intersection is $\mu$-negligible, but such that $\nu(f(A_n)) \geq \alpha > 0$; then take the intersection of the $f(A_n)$ and $\mathbf{C}I$.)