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I am working on a problem$^{(1)}$ similar to this 2013 posting:

Suppose that $f_n$ is a sequence of integrable, non-negative functions, so that $\forall x$, $f_n(x)$ decreases to $f(x)$. Show the following is true: $$\int f_n d \mu \rightarrow \int f d \mu.$$

Aside from the apparent differences between the two questions, this problem comes from early chapter of Measure & Integration class, so that Monotone Convergence Theorem is not in the background. Here are what I have been attempting to do $-$ rightly or wrongly:

Since $f_n(x)$ decreases to $f(x)$, this implies that $$\lim_{n \to \infty} f_n(x) = f(x). \tag{1}$$

Taking integral on both sides, $$\begin{align} \int \lim_{n \to \infty} f_n(x) d \mu = \int f(x) d \mu \tag{2}\\ \lim_{n \to \infty} \int f_n(x) d \mu = \int f(x) d \mu \tag{3}\\ \end{align}$$

implying that

$$\int f_n d \mu \rightarrow \int f d \mu. \tag{4}$$

But I am not sure if the move from (2) to (3) is valid. Please let me know what I should do instead. Thank you for your time and help.


(1) Richard F. Bass' Real Analysis, 2nd. edition, chapter 6: The Lebesgue Integration, Exercise 6.6, page 50.

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  • $\begingroup$ The interchange of integration and limit in step (3) is generally not permitted $\endgroup$ – iwriteonbananas Mar 20 '15 at 17:51
  • $\begingroup$ The move from (2) to (3) is essentially the meat of the problem. In general, we cannot arbitrarily move limits in and out of integrals. Have you discussed Fatou's lemma? $\endgroup$ – user31415926535 Mar 20 '15 at 17:52
  • $\begingroup$ Hint: consider the functions $g_n = f_n - f_1$. They form an increasing sequence of positive functions. Apply Beppo Levi. $\endgroup$ – iwriteonbananas Mar 20 '15 at 17:55
  • $\begingroup$ @Brandon : No, Fatou's lemma is not in the background, this lemma will only be covered on chapter 7. Thanks. $\endgroup$ – Amanda.M Mar 20 '15 at 18:39
  • $\begingroup$ @iwriteonbananas : No again, Beppo Levi is not even listed on the book's index. Thanks again. $\endgroup$ – Amanda.M Mar 20 '15 at 18:43
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Disregard my previous reply.

  • Set $g_n = f_1 - f_n \geq 0$, then $g_n \nearrow g$ where $g = f_1 - f$.
  • (The hard part) Construct $s_n$ simple functions so that $s_n \leq g_n$ and $s_n \nearrow g$.
  • Then $$0\leq \int g-g_n \leq \int g- s_n \to 0$$
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