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.
Best Answer
One standard example is the reals numbers times the reals with the discrete topology: $X = \mathbb{R} \times \mathbb{R}_d$.
This is a locally compact metrizable space. The compact subsets intersect only finitely many horizontal lines and each of those non-empty intersections must be compact. A Borel set $E\subset X$ intersects each horizontal slice $E_y$ in a Borel set.
Consider the following Borel measure where $\lambda$ is Lebesgue measure on $\mathbb{R}$: $$ \mu(E) = \sum_{y} \lambda(E_y). $$ This is easily checked to define an inner regular Borel measure and its null sets are precisely those Borel sets that intersect each horizontal line in a null set. In particular, the diagonal $\Delta = \{(x,x) : x \in \mathbb{R}\}$ is a null set. However, every open set containing $\Delta$ must intersect each horizontal line in a set of positive measure, so it must have infinite measure and hence $\mu$ is not outer regular.
Now define $\nu$ by the same formula as $\mu$ if $E$ intersects only countably many horizontal lines, and set $\nu(E) = \infty$ if $E$ intersects uncountably many horizontal lines. Now this measure $\nu$ is inner regular on open sets and outer regular on Borel sets.
Finally, you can check that $\mu$ and $\nu$ assign the same integral to compactly supported continuous functions in $X$.