Upon reviewing some basic real analysis I have encountered two different definitions for Radon measure. Let the underlying space $X$ be locally compact and Hausdorff. Folland's Real Analysis gives the definition
A Radon measure is a Borel measure that is finite on all compact sets, outer regular on Borel sets, and inner regular on open sets.
Folland goes on to prove that a Radon measure is inner regular on $\sigma$-finite sets, and remarks that full inner regularity is too much to ask for, especially in the context of the Riesz representation theorem for positive linear functionals on $C_c(X)$. Folland's approach seems to match the approach taken by Rudin, if I recall.
However, I've heard from others, as well as Wikipedia, that a Radon measure is defined as a Borel measure that is locally finite (which means finite on compact sets for LCH spaces) and inner regular, and no mention of outer regularity.
Neither definition seems to connect well with Bourbaki's approach of defining Radon measures as positive linear functionals on $C_c(X)$, because, at least according to Wikipedia's article on the Riesz representation theorem, a positive linear functional on $C_c(X)$ uniquely corresponds to a regular Borel measure, which is stronger than Radon in either of the two definitions given above.
Sadly I do not have any more advanced analysis treatises to compare against, so I was hoping somebody could clear up this discrepancy.