# Metrizability of the space $C^0([0, T], \mathcal G)$

Given $$T>0$$, let us denote by $$\mathcal G$$ the set of the Borel measures $$\nu$$ on $$\mathbb R^d$$ bounded by a constant $$G$$ (i.e. $$\int_{\mathbb R^d}d\nu\leq G$$) and endowed with the weak* convergence topology (see The wide or weak* topology what I mean for weak * convergence topology) and, if necessary, with finite $$p^{\text{th}}$$ moment.

I want to know if the space $$C^0([0, T], \mathcal G)$$ is metrizable.

I know that it depends on the metrizability of the space $$\mathcal G$$. Indeed, once $$\mathcal G$$ is metrizable, $$C^0([0, T], \mathcal G)$$ has a natural topology and a natural metric, which is the sup norm.

In the web page above, it is written that the weak* convergence topology is metrizable but I do not know if it can be applied to the particular space $$\mathcal G$$ too. Moreover it is not specified how could be the metric (Wasserstein?).

So my question is: is $$\mathcal G$$ metrizable? If not, what are the conditions for which $$\mathcal G$$ can be metrizable? And what is the metric?

Thank You

The Levy-Prohorov metric $$d$$ metrizes weak* convergence of Borel probability measures on $$\mathbb R^{d}$$ [ https://en.wikipedia.org/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric ]. We can metrize $$\mathcal G$$ by $$D(\mu, \nu)=|\mu_1 (\mathbb R^{d})-\nu_1 (\mathbb R^{d})+|d(\mu, \nu)|$$ where $$\mu_1=\frac {\mu} {\mu (\mathbb R^{d})}$$ and $$\nu_1=\frac {\nu} {\nu (\mathbb R^{d})}$$.
• @Redeldio The metric I have defined does give weak* convergence. You only need the fact that $\mu_n \to \mu$ iff $\mu_n (\mathbb R^{d}) \to \mu (\mathbb R^{d})$ and $\frac {\mu_n} {\mu_n (\mathbb R^{d})} \to \frac {\mu } { \mu (\mathbb R^{d})}$ Sep 16, 2020 at 9:54
• @Redeldio If $\mu, \mu_n$ are finite measures on $X=\mathbb R^{d}$ then $\mu_n \to mu$ in weak* topology iff $\mu_n(X) \to \mu(X)$ and the probability measures $\frac {\mu_n} {\mu_n(X)}$ converge to the probability measure $\frac {\mu} {\mu(x)}$ in weak* topology. This is immediate from definition of weak* convergence. Sep 16, 2020 at 23:22