When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the [Borel Hierarchy](http://en.wikipedia.org/wiki/Borel_hierarchy) and some transfinite induction. But, typically, I've lost the details.


So: is this true? Are related questions true? What are some good sources for this sort of questions? As motivation, a student pointed me to <http://en.wikipedia.org/wiki/Lp_space#Dense_subspaces> where it's claimed (without reference) that (up to a slight change of definition) the result is true for finite Borel measures on any metric space.


(I'm normally only interested in Locally Compact Hausdorff spaces, for which, e.g. Rudin's "Real and Complex Analysis" answers such questions to my satisfaction. But here I'm asking more about metric spaces).


To clarify, some definitions (thanks Bill!):


* I guess by "Borel" I mean: the sigma-algebra generated by the open sets.
* A measure $\mu$ is "outer regular" if $\mu(B) = \inf\{\mu(U) : B\subseteq U \text{ is open}\}$ for any Borel B.
* A measure $\mu$ is "inner regular" if $\mu(B) = \sup\{\mu(K) : B\supseteq K \text{ is compact}\}$ for any Borel B.
* A measure $\mu$ is "Radon" if it's inner regular and locally finite (that is, all points have a neighbourhood of finite measure).


So I don't think I'm quite interested in Radon measures (well, I am, but that doesn't completely answer my question): in particular, the original link to Wikipedia (about L^p spaces) seems to claim that any finite Borel measure on a metric space is automatically outer regular, and inner regular in the weaker sense with K being only closed.



Regular borel measures on metric spaces
The book *Probability measures on metric spaces* by K. R. Parthasarathy is my standard reference; it contains a large subset of the material in *Convergence of probability measures* by Billingsley, but is much cheaper! Parthasarathy shows that every finite Borel measure on a metric space is regular (p.27), and every finite Borel measure on a complete separable metric space, or on any Borel subset thereof, is tight (p.29). Tightness tends to fail when separability is removed, although I don't know any examples offhand.


(Definitions used in Parthasarathy's book: $\mu$ is regular if for every measurable set $A$, $\mu(A)$ equals the supremum of the measures of closed subsets of $A$ and the infimum of open supersets of $A$. We call $\mu$ tight if $\mu(A)$ is always equal to the supremum of the measures of *compact* subsets of $A$. Some other texts use "regular" to mean "regular and tight", so there is some room for confusion here.)


