Let $K$ be a compact metric space, endowed with its Borel $\sigma$-algebra.
Denote $P(K)$ the set of Borel probability measures on $K$.
It is known that, endowed with the weak $\ast$, $P(K)$ is compact metrizable.
Moreover if $f : K\to \mathbb{R}$ is semi-continuous, then $\mu \mapsto \int_K f d\mu$ is semi-continuous.
I have the following questions:
1) If $(\mu_k)_{k\in\mathbb{N}}$ weakly $\ast$ converges to $\mu$, do we have that $\mu_k(B(x,r))\to \mu(B(x,r))$, where $B(x,r)$ is the closed ball of center $x\in K$ and $r>0$?
2) If not. Is there a topology in which $P(K)$ is compact, if $\mu_k\to \mu$ then $\mu_k (B(x,r)) \to \mu(B(x,r))$ for all $x\in K$ and $r>0$, and $\mu\mapsto\mu(B(x,r))$ is continuous (or at least semi-continuous)?
Thanks.