Let $X$ be a locally compact and Hausdorff space, and let $\mu$ be a positive measure on the Borel sets of $X$ (here $\mu$ is not necessarily regular).
Then the linear map $L : C_c(X) \to \Bbb C$ defined by $L : f \mapsto \int_X f \;d\mu$ can be represented, by Riesz theorem, as $$L(f) = \int_X f \;d\nu$$ where $\nu$ is a regular Borel measure (here $C_c(X)$ denotes the set of continuous functions on $X$ that are compactly supported).
My question is:
Is my previous reasoning correct? That is : have I proven that for every measure $\mu$ we can find a regular Borel measure $\nu$ with respect to which integration is the same as the integration w.r.t $\mu$?
This would mean (in some sense) that we can always only handle with regular measures. (Or am I misinterpreting something?)
In Folland, A Guide to Advanced Real Analysis, page 59-60 : "If $\mu$ is a Borel measure on $X$ such that $\mu(K) < \infty$ for every compact $K$ of $ X$, then […] It is a fundamental fact, and a rich source of measures, that every positive linear functional on $C_c(X)$ is of this form, and that the measure can be taken to be regular."
Thank you for your comments!