Exercise 8, § 7

Let X, Y be two locally compact spaces, $\pi : X \to Y$ a continuous mapping, and $\nu$ a positive measure on Y that is concentrated on $\pi(X)$. Assume that for every $y \in \pi(X)$, there exist a neighborhood V of y in Y and a positive measure $\lambda_V$ on $\pi^{-1}(V)$ such that the image of $\lambda_V$ under the mapping $\pi_V : \pi^{-1}(V) \to V$ that coincides with $\pi$ is equal to the induced measure $\nu_V$. Show that under these conditions, there exists a positive measure $\mu$ on X such that $\pi(\mu) = \nu$. (Consider the set $\mathscr{G}$ of pairs $(G, \lambda)$, where G is open in Y and $\lambda$ is a positive measure on X such that $\pi(\lambda) = \nu_G$. Order $\mathscr{G}$ by the relation «$G_1 \subset G_2$ and $\lambda_1 \leq \lambda_2$ » between $(G_1, \lambda_1)$ and $(G_2, \lambda_2)$. First show that the ordered set $\mathscr{G}$ is inductive, then complete the argument with the help of Zorn’s lemma and Exer. 7.)