Corollary 2.25 . [019W]

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Corollary 2.25.

Let $(f_{j})_{j}$ a decreasing net of usc functions on $X$ converging pointwise to a (usc) function $f$ , and $(\mu_{j})_{j}$ a net of positive Radon measures on $X$ converging weakly to a positive Radon measure $\mu$ . Then

\limsup_{j}\int f_{j}\mu_{j}\leq\int f\mu.