This question concerns theorem 2.14 (Riesz representation theorem) of Rudin's book, in particular, his claim that if $\mu_1,\mu_2$ are measures satisfying the hypothesis of the theorem then, $\mu_1=\mu_2$.
However, the question will be made in order to be self contained.
Let $(X,\mathfrak{M},\mu)$ be a measure space, where $X$ is a locally compact Hausdorff space. Consider the following conditions (regularity of measures):
(c) For every $E\in\mathfrak{M}$ $$\mu(E)=\inf\{\mu(V):\ E\subset V,\ V\ \mbox{open}\}.$$
(d) For every open set $E$ and for every $E\in\mathfrak{M}$, with $\mu(E)$, we have that $$\mu(E)=\sup\{\mu(K):\ K\subset E,\ K\ \mbox{compact}\}.$$
Assume that $\mu_1,\mu_2$ are measures satisfying (c) and (d). Based on conditions (c), (d) above, why it is then sufficient to prove that $\mu_1(K)=\mu_2(K)$ for all $K$ compact, in order to prove that $\mu_1=\mu_2$?
thanks.
By one hand, if $E$ is open or if $E\in\mathfrak{M}$ with $\mu_1(E),\mu_2(E)<\infty$, condition (d) implies that if $$\mu_1(K)=\mu_2(K),\ \forall \ K\subset E,\ K\ \mbox{compact},$$
then $\mu_1(E)=\mu_2(E)$.
On the other hand, if $E\in\mathfrak{M}$ and $\mu_1(E)=\infty$ or $\mu_2(E)=\infty$ then, once $\mu_1,\mu_2$ satifies (c), you need to know how does $\mu_1,\mu_2$ behaves in open sets, but again, once for open sets, you can use condition (d), the same argument as above implies that you only need to know the behavior of the measures for compact sets.