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let $A$ be a subset of the reals, $f_n,f$ are positive lebesgue measurable functions that $f_n$ converge to f pointwise and $\int_{A}f_n dm$ converge to $\int_{A}f dm$ with $\int_{A}f dm<\infty$. prove $\int_{A} \min(f_n,f) dm$ converge to $\int_{A}f dm$

Can someone give me a hint?

Pedro
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Rann
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1 Answers1

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Observe $|\min(f_n,f) - f| \le |f_n - f|$ so that $\min(f_n,f) \to f$ pointwise. Since $|\min (f_n,f)| = \min (f_n,f) \le f$ the dominated convergence theorem implies $$\lim_{n \to \infty} \int_A \min (f_n,f) \, dm = \int_A f \, dm$$

Umberto P.
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