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Fatou's Lemma: Let $\{f_n\} \rightarrow f$ pointwise a.e. on $E$, then $\int_E f \leq \liminf \int_E f_n$.

Generalization: Prove that if $\{ f_n \}$ is a sequence of nonnegative measurable functions on $E$, then $$\int_E \liminf f_n \leq \liminf \int_E f_n$$

This is from Royden 4e (pg 85).

The general form does not require pointwise convergence. Is this so simple as to observe that the $\inf$ of simple functions will be less than the integral of the same? Otherwise I need some guidance. When can I pass the limit through the integral.

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1 Answer 1

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If you are allowed to use monotone convergence theorem then

Observe that if $g_{n} := \inf_{k\geq n}f_{k}$ then $\int g_{n} \leq \int f_{n}$ and $g_{n}$ increases to $\lim \inf f_{n}$.

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  • $\begingroup$ I can use MCT. So, we've constructed a sequence (of functions) that is guaranteed to be less than the $f$'s because of the nature of infimums. This is the trick? Thank you $\endgroup$
    – Jared
    Commented Jun 4, 2013 at 3:50
  • $\begingroup$ Always! Glad to be able to help. $\endgroup$ Commented Jun 4, 2013 at 3:54
  • $\begingroup$ @VishalGupta I know this is an old post, but would you mind explaining what you mean by $\inf_{k\geq n}$? Thanks! $\endgroup$ Commented Oct 24, 2017 at 18:07
  • $\begingroup$ @MathStudent1324 I mean infimum over the set of functions $f_{k}$ for $k \geq n$ $\endgroup$ Commented Oct 24, 2017 at 18:40

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