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Say $f \colon X \to \mathbb{R}$ is a lower semi-continuous on a compact space $X$. Let $\mathcal{P}(X)$ denote the space of Borel probability measures on $X$, and let $f^* \colon \mathcal{P}(X) \to \mathbb{R}$ be given by $f^*(\mu) = \mu(f) = \int_X f(x)\,\mathrm{d}\mu(x)$. Is $f^*$ lower semi-continuous in the weak* topology?

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    $\begingroup$ Is $X$ a metric space? If so, then this should follow pretty directly from the characterization that a function is lower semi-continuous iff it is an increasing pointwise limit of continuous functions. Let $f_n \uparrow f$ pointwise; then by monotone convergence, $f_n^* \uparrow f^*$ pointwise. $\endgroup$ – Nate Eldredge Oct 31 '16 at 0:59

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