Probability measures on a dense subset

Question

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?

Answer 1

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 $\lim_{m \to \infty} \sum_{i=1}^n p_i \delta_{x_{im}} = P$.