Induce measure between topological spaces. Let $X$ and $Y$ be topological spaces, suppose that the function $f:X\rightarrow  Y$ is a continuous surjection. Let $\mu$ be a regular measure on $X$, and $M_X$ the set of $\mu$-measurable sets in $X$. We can induce a measure on $Y$ define in the $\sigma$-algebra $M_Y=\{A\subset Y: f^{-1}(A)\in M_X\}$ as 
$$\nu(A)=\mu(f^{-1}[A])$$
What conditions do we have to impose in order that the measure $\nu$ be regular?
I supposed that $\mu$ were regular but I got nothing.
Any clue?

Let $A\in M_Y$. Then $$\nu(A)=\mu\big(f^{-1}(A)\big)=\sup\big\{\mu(K):K\subseteq f^{-1}(A),K \text{ compact}\big\},$$
since $\mu$ is regular. Now for $K\subseteq f^{-1}(A)$ with $K$ compact, $f(K)\subseteq A$ is compact too and $K\subseteq f^{-1}\big(f(K)\big)$. So $\nu\big(f(K)\big)=\mu\Big(f^{-1}\big(f(K)\big)\Big)\geq\mu(K)$ and $\nu$ is regular.
