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Royden states that to prove Fatou's lemma it is ncessary and sufficent to show that if $h$ is any bounded measurable function of finite support for which $0\leq h\leq f$ on $E$, then $$ \int _E h\leq \lim \inf \int_E f_n$$

Why did Royden choose to construct such a function instead of working directly with $f$?


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2 Answers 2

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By definition, $\int f$ is the supremum of $\int h$ for $0 \leq h \leq f$ bounded measurable of finite support. So at the very end, you just need to take the sup over such $h$ to get $\int f \leq \liminf \int f_n$. He uses such $h$ is order to apply the bounded dominated convergence theorem, which he proves earlier. This allows him to prove Fatou in terms of dominated convergence instead of giving a completely independent proof (and potentially repeating a bunch of work).

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  • $\begingroup$ Thank you! Very insightful. $\endgroup$
    – chuck
    Mar 26, 2021 at 1:22
  • $\begingroup$ I just want to follow up on this a bit further. You are taking the supremum of the set of functions $h$ that are bounded by $f$ right? $\endgroup$
    – chuck
    Mar 26, 2021 at 2:20
  • $\begingroup$ Instead of taking the supremum over a uncountable set of functions, can we prove it by contradiction in the end by assuming $\int f>\lim \inf f_n$ and use $\int_E h\leq\lim \inf \int_E f_n$ to draw the conclusion. Because I am not sure how the supremum over a uncountable set of functions works. $\endgroup$
    – chuck
    Mar 26, 2021 at 2:45
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    $\begingroup$ While it would be possible to do it the way you suggest, I would highly encourage you to instead familiarize yourself with how to take the supremum of a set. This is, after all, the definition of the integral. There is no need to search for a contradiction. $\endgroup$
    – nullUser
    Mar 26, 2021 at 3:19
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You can also let $g_n=inf_{i \geq n} \{f_n\}$ and note that $\int g_n \leq \int f_n$ for each n. Moreover $g_n$ increases monotonically to lim inf $f_n$ . Hence by monotone convergence theorem $\int g_n = \int lim inf f_n$. Since $\int g_n \leq \int f_n$ we have $\int lim inf f_n \leq lim inf \int f_n$. The final step is to note that pointwise convergence implies lim inf $f_n$ = f

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