Let $\Omega$ be a (for my purposes: locally compact) Hausdorff space. Let $\mu$ be a Borel measure on $\Omega$ and let $A\subset\Omega$ be a Borel set.
- $\mu$ is called locally finite if $\mu(K)<\infty$ for all compact $K\subset\Omega$.
- $\mu$ is called inner regular on $A$ if $$\mu(A)=\sup\{\mu(K):K\subset A \ \text{compact}\}.$$
- $\mu$ is called outer regular on $A$ if $$\mu(A)=\inf\{\mu(U):U\supset A \ \text{open}\}.$$
According to this article, the following two definitions are equivalent:
Definition 1. A borel measure $\mu$ is a Radon measure if $\mu$ is locally finite and inner regular on all Borel sets.
Definition 2. A Borel measure $\mu$ is a Radon measure if $\mu$ is locally finite, inner regular on all open sets and outer regular on all Borel sets.
Why are these definitions equivalent? Here is another article discussing these two definitions.
