I am following big Rudin and I have arrived at the representation theorem. Before doing the full long proof I would like to know what results are based on this theorem that for completeness I state below:
Let $X$ be a locally compact Hausdorff space. For any positive linear functional $\psi$ on $C_c(X)$, there is a unique regular Borel measure $μ$ on $X $such that:
$$ \psi(f) = \int_X f(x) \, d \mu(x) \quad $$
for all f in Cc(X).
(Rudin proves it in a more general setting ).
I apologize if I should already understand why this is important and what results are based on this but in my defence I am fairly new to measure theory.