My reference is L. Simon's Lectures on Geometric Measure Theory. He defines a measure on a set $X$ as a countably subadditive function $\mu:2^X\to[0,\infty]$ with $\mu(\emptyset)=0.$
When $X$ is a locally compact and separable topological space, he defines a measure $\mu$ on $X$ to be Radon if it is Borel regular (i.e. every set is contained in a Borel set with the same measure, and all Borel sets are measurable) and finite on compact sets.
Why make the assumption that $X$ is locally compact and separable? They seem extraneous to me.