Consider a Polish metric space $(\mathcal{X}, d)$ and let $\mathcal{X_0}$ be a $G_\delta$ subset. Let $\mathcal{P}$ and $\mathcal{P}_0$ be the classes of probability measures on $(\mathcal{X}, \mathcal{B}_{\mathcal{X}})$ and $(\mathcal{X}_{0},\mathcal{B}_{\mathcal{X_0}})$,respectively, where $\mathcal{B}_{\mathcal{X}}$ and $\mathcal{B}_{\mathcal{X}}$ are the Borel $\sigma$-algebras induced by $d$ over the spaces $\mathcal{X}$ and $\mathcal{X}_0$, respectively.
It should be trivially true that $\forall P_0 \in \mathcal{P}_0, \exists!\, P\in \mathcal{P}$ such that $P_0$ is the restriction of $P$ to $\mathcal{X}_0$ and $\bar{P}(\mathcal{X}_0^c)=0$. Endow $\mathcal{P}$ and $\mathcal{P}_0$ with the topology of weak convergence of probability measures, let $\mathcal{B}_\mathcal{P}$ and $\mathcal{B}_\mathcal{P_0}$.
QUESTION Is the map $\phi:\mathcal{P}_0\mapsto \mathcal{P}:P_0 \mapsto P$ a Borel isomorphism, i.e. a measurable bijective function between two measurable standard Borel spaces?
My considerations:
it is known that $(\mathcal{P}, \mathcal{B}_\mathcal{P})$ is standard Borel: in fact, since $(\mathcal{X},d)$ is assumed Polish, the space $\mathcal{P}$ endowed with the topology of weak convergence is separable and completely metrizable by some metric $\rho$ (e.g., Ghosal and van der Vaart, 2017, Theorem A.3).
If: a) $\phi$ is bicontinuous w.r.t. the topology of weak convergence on $\mathcal{P}_0$ and the metric toplogy induced by $\rho$ in $\mathcal{P}$ b) its image is closed in $\mathcal{P}$; the result should follow. Indeed $\phi(\mathcal{P_0})$ would be a closed subset of the Polish space $(\mathcal{P},\rho)$, thus $(\mathcal{P}_0, \rho)$ would be a Polish subspace and $\mathcal{P}_0$ would be Borel isomorphic to a Polish space (thus standard Borel).
It seems somehow intuitive to me to conclude that: $$ \phi(\mathcal{P_0})=\{P \in \mathcal{P}: \, \text{supp}(P)=\bar{\mathcal{X}_0}, \, P(\bar{\mathcal{X}_0} \setminus \mathcal{X}_0)=0 \} $$ where $\bar{B}$ is the $d$-closure of a set $B \subset \mathcal{X}$ and $\text{supp}(P)$ is the support of the measure $P$. However it is unclear to me whether this set is closed under $\rho$ (or, more generally, w.r.t. the topology of weak convergence, as the latter does not account for sets whose boundary receives positive mass by the limiting probability measure of a converging sequence).