I have been thinking about the following concept:
Let $X$ be a topological space, $(X, \mathcal{B}_X, \mu)$ be a measure space, $\mathcal{P}(X)$ the set of probability measures on $(X, \mathcal{B}_X)$, $(\mu_{n})_{n \in \mathbb{N}} \subset \mathcal{P}(X) $ and $E \in \mathcal{A}$.
Is there a term to describe the following property:
for each $U \in \mathcal{N}_{E}$ (neighborhood of $E$), $\mu_n(U^c) \rightarrow 0$
It looks like weak convergence type of property. Naturally I would like to refer to it as "$(\mu_n)_{n \in \mathbb{N}}$ concentrating on $E$", but I think that is already taken.