# Assumptions for definition of Radon measure

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.

First of all, I think $\mu$ should only be defined on a subset of $2^X$, namely the Borel sets.
Second, I think you want a definition so that the Banach space of Borel measures is the dual of $C_0(X)$, the continuous functions that converge to $0$ at $\infty$ when $X$ is embedded in its one point compactification.
• I don't understand the first line of your answer. If $\mu$ is defined only on the Borel sets, its Borel regularity is vacuous. Also, it is quite reasonable (and common) to define measures on $2^X$, without insisting on additivity (i.e., they are what other people call outer measures). – Post No Bulls Dec 8 '13 at 6:02