# Integral of limit of a function

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.

• The interchange of integration and limit in step (3) is generally not permitted – iwriteonbananas Mar 20 '15 at 17:51
• 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? – user31415926535 Mar 20 '15 at 17:52
• Hint: consider the functions $g_n = f_n - f_1$. They form an increasing sequence of positive functions. Apply Beppo Levi. – iwriteonbananas Mar 20 '15 at 17:55
• @Brandon : No, Fatou's lemma is not in the background, this lemma will only be covered on chapter 7. Thanks. – Amanda.M Mar 20 '15 at 18:39
• @iwriteonbananas : No again, Beppo Levi is not even listed on the book's index. Thanks again. – Amanda.M Mar 20 '15 at 18:43

• 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 $$0\leq \int g-g_n \leq \int g- s_n \to 0$$