Let $A$ and $B$ be compact subsets of a metric space. Denote by $\mathcal{P}(A)$ the set of (Borel) probability measures in $A$, and similarly define $\mathcal{P}(A \cup B)$. They are compact metric. Define $C$ as the set of all elements of $\mathcal{P}(A \cup B)$ which assign probability 1 to $A$.
Question: Is there an embedding between $C$ and $\mathcal{P}(A)$?
As far as I can see, when $A$ and $B$ are finite (and assuming without loss they are also disjoint), the answer is yes because one can identify $\mathcal{P}(A)$ with a closed subset of $\mathcal{P}(A \cup B)$. But I am a little lost on how to (try to) show it in general.