Let $D\subseteq X$ be a dense subset of a separable metric space $X$. Let $P(D)$ and $P(X)$ respectively denote the probability measures on $D$ and on $X$ with their weak topologies. Then, if we view $P(D)$ as a subset of $P(X)$ via the "inclusion" $\iota:P(D)\rightarrow P(X)$ defined by:
$$
\iota(\mu)\mapsto \mu(\cdot \cap D)
$$
is this subset dense?



Probability measures on a dense subset
Yes. First the probability measures with finite support are dense in $P(X)$. Second, if $P = \sum\_{i=1}^n p\_i \delta\_{x\_i}$ with $x\_i \in X$ and $\sum\_{i=1}^n p\_i = 1$, let $(x\_{im})\_{m \in \mathbb{N}}$ be sequences in $D$ with $\lim\_{m \to \infty} x\_{im} = x\_i$. Then $\lim\_{m \to \infty} \sum\_{i=1}^n p\_i \delta\_{x\_{im}} = P$.


