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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?

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Yes. Moreover, if $X$ is a metric space, then your map is Lipschitz with respect to the corresponding transportation metrics.

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  • $\begingroup$ Thank you. It looks like mine was an easy question, and hence off-topic for this forum. Sorry for that. Since I could not (and still cannot) find an answer by myself, may I ask you some references (papers or textbooks) that might help me? $\endgroup$
    – user66910
    May 25, 2015 at 13:52
  • $\begingroup$ Use the fact that any continuous function $f$ on $X$ determines a continuous function $F(\mu)=\langle f,\mu\rangle$ on $\mathcal P(X)$. $\endgroup$
    – R W
    May 25, 2015 at 15:36