I have come across two different versions of the Riesz-Markov theorem, one identifies the dual with Radon measures and the other with Baire measures.
From Wikipedia:
Let $X$ be a locally compact Hausdorff space. For any positive linear functional $\varphi$ on the space of continuous compactly supported functions, there is a unique Radon measure $\mu$ on $X$ such that $$\forall f \in C_c(X): \quad \varphi(f) = \int_X f(x) d\mu(x).$$
From Reed & Simon's book on functional analysis:
Let $X$ be a locally compact space. A positive linear functional on the algebra of continuous functions of compact support is of the form $\ell(f) = \int f d\mu$ for some Baire measure $\mu$. A positive linear functional on the algebra of continuous functions vanishing at infinity comes from a measure $\mu$ with total finite mass, that is, $\sup_{A \in \mathscr{B}} \mu (A) < \infty$.
Are these two statements equivalent? My guess is that the two can be shown to be equivalent by using the unique extension of each Baire measure to a regular Borel measure (which I assume will be Radon?).