The equality $\mathcal M_\tau(X)=\mathcal M_t(X)$ for a complete metric space $X$ follows from three facts:

1. For any finitely additive measure $\mu$ on $X$ its support $supp(\mu)$ (i.e., the set of points $x\in X$ whose any neighborhood $O_x$ has positive measure $\mu(O_x)$) has countable cellularity and hence is separable.

2. For a $\tau$-additive measure $\mu$ the complement $X\setminus supp(\mu)$ has measure zero, so the measure $\mu$ ``lives'' on its support.

3. Each countably additive Borel measure on a Polish (= separable completely metrizable space) is Radon. This follows from the existence of metrizable compactifications of Polish spaces.

The same argument shows that the equality $\mathcal M_\tau(X)=\mathcal M_t(X)$ holds for any metrizable space $X$ whose any closed separable subspace is universally measurable (i.e., measurable with respect to any $\sigma$-additive Borel measure on some metrizable compactification of $X$).

The equality $\mathcal M_\tau(X)=\mathcal M_t(X)$ for a complete metric space $X$ follows from three facts:

1. For any finitely additive measure $\mu$ on $X$ its support $supp(\mu)$ (i.e., the set of points $x\in X$ whose any neighborhood $O_x$ has positive measure $\mu(O_x)$) has countable cellularity and hence is separable.

2. For a $\tau$-additive measure $\mu$ the complement $X\setminus supp(\mu)$ has measure zero, so the measure $\mu$ ``lives'' on its support (in the sense that $\mu(supp(\mu))=1$).

3. Each countably additive Borel measure on a Polish (= separable completely metrizable space) is Radon. This follows from the existence of metrizable compactifications of Polish spaces.

The same argument shows that the equality $\mathcal M_\tau(X)=\mathcal M_t(X)$ holds for any metrizable space $X$ whose any closed separable subspace is universally measurable (i.e., measurable with respect to any $\sigma$-additive Borel measure on some metrizable compactification of $X$).

The equality $\mathcal M_\tau(X)=\mathcal M_t(X)$ for a complete metric space $X$ follows from two facts:

1. for any $\tau$-additive measure $\mu$ on $X$ its support $supp(\mu)$ (i.e., the set of points $x\in X$ whose any neighborhood $O_x$ has positive measure $\mu(O_x)$) has countable cellularity and hence is separable.

2. Each $\sigma$-additive Borel measure on a Polish (= separable completely metrizable space) is Radon. This follows from the existence of metrizable compactifications of Polish spaces.

The equality $\mathcal M_\tau(X)=\mathcal M_t(X)$ for a complete metric space $X$ follows from three facts:

1. For any finitely additive measure $\mu$ on $X$ its support $supp(\mu)$ (i.e., the set of points $x\in X$ whose any neighborhood $O_x$ has positive measure $\mu(O_x)$) has countable cellularity and hence is separable.

2. For a $\tau$-additive measure $\mu$ the complement $X\setminus supp(\mu)$ has measure zero, so the measure $\mu$ ``lives'' on its support.

3. Each countably additive Borel measure on a Polish (= separable completely metrizable space) is Radon. This follows from the existence of metrizable compactifications of Polish spaces.

The same argument shows that the equality $\mathcal M_\tau(X)=\mathcal M_t(X)$ holds for any metrizable space $X$ whose any closed separable subspace is universally measurable (i.e., measurable with respect to any $\sigma$-additive Borel measure on some metrizable compactification of $X$).