# 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 Commented Mar 20, 2015 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? Commented Mar 20, 2015 at 17:52
• Hint: consider the functions $g_n = f_n - f_1$. They form an increasing sequence of positive functions. Apply Beppo Levi. Commented Mar 20, 2015 at 17:55
• @Brandon : No, Fatou's lemma is not in the background, this lemma will only be covered on chapter 7. Thanks. Commented Mar 20, 2015 at 18:39
• @iwriteonbananas : No again, Beppo Levi is not even listed on the book's index. Thanks again. Commented Mar 20, 2015 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 $$0\leq \int g-g_n \leq \int g- s_n \to 0$$