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

Question

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.

@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 08 '14 at 18:45

score 3 · Accepted Answer · answered Jan 08 '14 at 18:30

3

Proposition 7.2.3. in Measure Theory by Donald L. Cohn.

answered Jan 08 '14 at 18:30

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