It is well known that Riesz Representation Theorem states that every positive linear functional $\Psi$ on $C_c(X),$ where $X$ is a locally compact Hausdorff space, can be realize as integration $$\Psi(f)=\int_X f(x)d\mu(x)$$ for a unique regular Borel measure $\mu$ on $X.$
Question: In the Wiki article linked above, there is this sentence: The theorem is named for Frigyes Riesz (1909) who introduced it for continuous functions on the unit interval.
I would like to know that how Riesz proved the statement himself. It would be good if someone can provide me a paper where he provided a proof.