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I'm trying to prove the Monotone Convergence Theorem for decreasing sequences, namely if

Let $(X,\mathcal{M},\mu)$ be a measure space and suppose $\{f_n\}$ are non-negative measurable functions decreasing pointwise to $f$. Suppose also that $f_1 \in \mathscr{L}(\mu)$. Then $$\int_X f~d\mu = \lim_{n\to\infty}\int_X f_n~d\mu.$$

Why does this statement not follow from LDCT with $f_n$ being dominated by $f_1$?

I'm also aware of the solutions with $g_n=f_1-f_n$, but the question asks to prove it using Fatou's lemma

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  • $\begingroup$ In Fatou's lemma, the inequality goes the other way. So the inequality $\lim\int f_n\,d\mu\le\int f\,d\mu$ doesn't follow from Fatou. $\endgroup$
    – grand_chat
    Oct 9, 2015 at 1:26
  • $\begingroup$ @grand_chat why does the statement not follow from LDCT with $g=f_1$? $\endgroup$ Oct 9, 2015 at 1:59
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    $\begingroup$ I think it does follow from LDCT. $\endgroup$
    – user99914
    Oct 9, 2015 at 3:01
  • $\begingroup$ @JohnMa Then why does every solution use the $g_n=f_1-f_n$ trick and MCT? $\endgroup$ Oct 9, 2015 at 3:03
  • $\begingroup$ I guess that's because LDCT is an overkill? (It seems that LDCT is the last theorem to prove, either Fatou or MCT comes first) $\endgroup$
    – user99914
    Oct 9, 2015 at 3:05

1 Answer 1

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As you say in the comments, the problem is posed at the end of the first chapter of Rudin's Real and Complex Analysis, so it is not clear whether Rudin intends the student to use the Monotone Convergence Theorem, Fatou's Lemma or the Dominated Convergence Theorem.

You are right that the proof is a trivial application of DCT. However, it is certainly an instructive exercise to prove it using just MCT or Fatou's Lemma.

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