# Is integration against a Borel function with respect to a regular measure regular?

This question comes from trying to understand the proof of the Riesz representation theorem for complex measures in Rudin's RCA. (Theorem 6.19)

Let $X$ be a locally compact Hausdorff space, let $\mu$ be a regular Borel measure (positive or complex) on $X$, and let $f$ be an integrable Borel function on $X$. Is it necessarily the case that the measure $\lambda$ defined by $d\lambda=f~d\mu$ is regular? I was able to show that this is true when $\mu$ is positive and finite, in which case it suffices to show $\lambda$ is inner regular. When I drop the finiteness assumption, though, I get stuck showing outer regularity even in the simple case when $f$ is a characteristic function $\chi_{A}$ of a Borel set.

Inner regularity of $\lambda$ suffices, since $\lambda$ is a bounded measure given that $f$ is integrable. But inner regularity of $\lambda$ is easy -- consider characteristic functions, then positive functions, then integrable functions.