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 $\nu_1=\frac {\nu} {\nu (\mathbb R^{d})}$.