Exercise 7, § 7

Let X, Y be two locally compact spaces, $\pi : X \to Y$ a continuous mapping, T a locally compact subspace of Y, $S = \pi^{-1}(T)$, $\pi_T : S \to T$ the mapping that coincides with $\pi$ in S, and $\nu$ a positive measure on Y. Show that the following three conditions are equivalent:

a) If $\nu_T$ is the measure induced by $\nu$ on T, there exists a positive measure $\mu_T$ on S such that $\pi_T(\mu_T) = \nu_T$.

b) There exists a positive measure $\lambda$ on X such that $\pi(\lambda) = \nu_T$.

c) There exists a positive measure $\lambda_S$ on S such that $(\pi|S)(\lambda_S) = \nu_T$.