# Are Borel probability measures on second countable, locally compact, Hausdorff spaces automatically regular?

This is a quote from the wikipedia entry on regular measures:

Any Borel probability measure on a locally compact Hausdorff space with a countable base for its topology ... is regular.

Is this actually true? If so could someone point me to a reference where I could find a proof of the statement?

Edit: I should add that the definition of regularity used in wikipedia is that $\mu$ is regular if

$$\mu(A)=\sup\{\mu(F):F\subset A\text{ and F is compact}\}=\inf\{\mu(B):A\subset B\text{ and B is open}\}$$

where the infimum/supremum is taken over the space's Borel sigma algebra.

• – Giuseppe Negro Jan 8 '14 at 18:34
• @jkn Did you try the references on the Wikipedia page? – user940 Jan 8 '14 at 18:39
• @Bryon Schmuland no I didn't--I don't have ready access to them (this is why I asked for someone to provide one which contains the statement for certain). Sorry for any inconveniences. – jkn Jan 8 '14 at 18:45