Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$

Question

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.

Let $X$ be a topological space (for convenience, it might be Polish or compact) and let $f \colon \mathcal{P}(\mathcal{P}(X)) \to \mathcal{P}(X)$ be defined by $$ f(\mu)(E) = \int \nu(E) \mu(d\nu). $$ I can prove that each $f(\mu)$ is well-defined.

Is $f$ continuous?