From wikipedia, Radon measure is defined as
Radon measure is a measure on the $\sigma$-algebra of Borel sets of a Hausdorff topological space $X$ that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets.
I am reading a proof on the $\sigma$-compact, locally compact metric space, that a local finite Borel measure is a Radon measure. It seems to me the proof shows it is inner regular on all Borel sets, which is stronger than the definition of Radon measure. Am I correct? What is the correct definition of inner regularity? I think the correct statement is
If $X$ is a locally compact, $\sigma$-compact metric space and $\mu$ is a locally finite Borel measure on $X$, then it is both inner and outer regular (or simply regular).
