Let $X$ be a complete metric space. Are all $\tau$-additive Borel measures on $X$ tight?

In Bogachev's "Measure Theory", vol. 2, in the proof of Theorem 8.9.4 (end of page 213) it says:

Note that if $X$ is a complete metric space, then $\mathcal{M_\tau}(X)=\mathcal{M}_t(X)$, and $\mathcal{M}_t(X) \subset \mathcal{M_\tau}(X)$ for any metric space.

There:

- $\mathcal{M}_t(X)$ is the set of
*tight Baire measures*, which are the same as tight Borel measures since we are in a metric space; - $\mathcal{M}_\tau(X)$ is the set of $\tau$-additive Borel measures.

Now the second part of the statement is basically implied (at least for locally finite measures) by Proposition 7.2.2 (page 74), which says that every Radon measure is $\tau$-additive. However, there is no proof for the first part of the statement.

Does anyone have a reference of the proof, or directly the proof?