Hypothesis for establishing that the measure given by the Riesz representation theorem is a probability measure

Question

I am trying to study, through the Riesz representation theorem applied to a space of compactly supported continuous functions, what hypotheses must be met to establish that the measure given by the theorem (which is unique, regular and Borel) is a probability measure. I have tried to find literature on this but I cannot find it. Any help? Thanks.

Answer 1

The theorem gives you an expression of form $$\Psi(f) ~=~ \int_X f\,\mathrm{d}\mu.$$ The functional $\Psi$ is positive if and only if $\mu$ is a positive measure and in this case one has $$\|\Psi\| ~=~ \mu(E).$$ If I am not mistaken, there is more general statement (when $X$ is a compact Hausdorff space), that $$(\mathcal{M}_\mathrm{Rad},\|\cdot\|_\mathrm{TV})~\cong~ (C_0(X),\|\cdot\|_\infty)^*,$$ where the isomorphism is isometric, giving $\|\Psi\|=\|\mu\|_\mathrm{TV}=:|\mu|(X)$. To conclude, you want $\Psi$ to be a positive with $\|\Psi\|=1$.

Edit: when Googling about this theorem, you might want to also try "Riesz-Markov" and "Riesz-Markov-Kakutani" to avoid search result about the standard RRT.