Exercise 21, § 5

Let $X$ be a locally compact space, $\mu$ a positive measure on $X$, and $(U_\alpha)_{\alpha \in A}$ an open covering of $X$. For every $\alpha \in A$, let $f_\alpha$ be a mapping of $U_\alpha$ into a set $G$. Assume that for every pair of indices $\alpha, \beta$, the set of points $x \in U_\alpha \cap U_\beta$ such that $f_\alpha(x) \neq f_\beta(x)$ is locally $\mu$-negligible. Show that there exists a mapping $f$ of $X$ into $G$ such that, for every $\alpha \in A$, the set of $x \in U_\alpha$ such that $f(x) \neq f_\alpha(x)$ is locally μ-negligible. (Consider first the case that X is compact, by covering X by a finite number of sets U_α; pass to the general case with the help of Prop. 14 of No. 9.)