Lemma 1

Let X, B be two locally compact spaces, π a mapping of X into B, ν a positive measure on B. Let b ↦ λb (b ∈ B) be a ν-adequate family of positive measures on X such that, for every b ∈ B, the measure λ_b is concentrated on $\overline{\pi}(b)$. Set $\mu = \int \lambda_b d\nu(b)$, and assume that the mapping π is μ-measurable.

a) If N ⊂ B is locally ν-negligible, then $\overline{\pi}(N)$ is locally μ-negligible. b) If f is a ν-measurable function on B (with values in a topological space), then $f \circ \pi$ is μ-measurable on X.

Let K be a compact subset of X. We are to prove that $\overline{\pi}(N) \cap K$ is μ-negligible and that the restriction of $f \circ \pi$ to K is μ-measurable. Now, K is the union of a μ-negligible set and a sequence of compact subsets $K_n$ such that $\pi|K_n$ is continuous. It suffices to show that $\overline{\pi}(N) \cap K_n$ is μ-negligible and that the restriction of $f \circ \pi$ to $K_n$ is μ-measurable. We may therefore assume henceforth that $\pi|K$ is continuous. Then $\pi(K) = K'$ is compact. Since $\overline{\pi}(N) \cap K = \overline{\pi}(N \cap K') \cap K$ and $N \cap K'$ is ν-negligible, we may assume henceforth that N is ν-negligible. Then N is contained in a ν-negligible set $N'$ that is a countable intersection of open sets (Ch. IV, §4, No. 6, Cor. 2 of Th. 4). Since $\pi$ is μ-measurable, $\overline{\pi}(N')$ is μ-measurable (Ch. IV, §5, No. 5, Prop. 7), therefore $\overline{\pi}(N') \cap K$ is μ-integrable, and (Ch. V, §3, No. 4, Th. 1)

*Cf. Ch. V, §6, No. 6, Cor. 1 of Prop. 10; the second edition of Ch. V was not available when Ch. VII was published.

Now, if $b \notin N'$, $\pi^{-1}(N') \cap K$ is $\lambda_b$-negligible since $\lambda_b$ is concentrated on $\pi^{-1}(b)$ by hypothesis. Therefore $\mu(\pi^{-1}(N') \cap K) = 0$. A fortiori, $\pi^{-1}(N) \cap K$ is $\mu$-negligible and a) has indeed been proved. On the other hand, there exists a partition of $K'$ formed by a $\nu$-negligible set $M$ and a sequence $(K'_n)$ of compact sets such that $f|_{K'_n}$ is continuous. Then the restriction of $f \circ \pi$ to each set $\pi^{-1}(K) \cap K'_n$ is continuous, and $\pi^{-1}(M) \cap K$ is $\mu$-negligible by a), therefore the restriction of $f \circ \pi$ to $K$ is indeed $\mu$-measurable.

Exercises