Exercise 9, § 1

Let T be a topological space. If $\lambda$ and $\mu$ are two positive measures on T such that $\lambda^*(U) = \mu^*(U)$ for every open set U in T, then $\lambda = \mu$ (first prove that $\lambda^*(f) = \mu^*(f)$ for every lower semi-continuous function $f \geq 0$ on T, then that $\lambda^* = \mu^*$).