Exercise 6, § 7

Let T be a locally compact space, X a locally compact subspace of T, and $j$ the canonical injection of X into T. Let $\lambda$ be a positive measure on X such that, for every compact set $K \subset T$, $K \cap X$ is essentially $\lambda$-integrable. Show that the measure $\nu = j(\lambda)$ is the smallest positive measure $\rho$ on T such that $\rho_X = \lambda$, and that its support is the closure in T of the support of $\lambda$.