# Royden “Real Analysis”, 3rd edition, chapter 4, exercise 9

Let $$\langle f_n\rangle$$ be a sequence of nonnegative measurable functions on $$(-\infty,\infty)$$ such thta $$f_n\rightarrow f$$ a.e., and suppose that $$\int f_n\rightarrow\int f<\infty$$. Then for each measurable set $$E$$ we have $$\int_E f_n \rightarrow \int_E f$$.

I know that by Fatou Lemma we have $$\int_E f\leq\liminf_{n\rightarrow \infty} \int_E f_n$$, but I have no idea about how to use the finiteness of $$\int f$$ to conclude the problem.

## 1 Answer

You have $$|f - f_n| \le |f| + |f_n| = f + f_n$$ so that $$f + f_n - |f - f_n| \ge 0$$. Apply Fatou to this sequence. Since $$f + f_n - |f - f_n| \to 2f$$ you have $$\int 2f \le \liminf \int (f + f_n - |f - f_n|).$$

You have to be a bit careful with the $$\liminf$$ because it is generally not additive, but by hypothesis you have $$\int f_n \to \int f$$ so that $$\liminf \int (f + f_n - |f - f_n|) = 2 \int f - \limsup \int |f - f_n|.$$ Thus $$\limsup \int |f - f_n| \le 0.$$

Consequently for any set $$E$$ you get $$\left| \int_E f_n - \int_E f \right| \le \int_E |f - f_n| \le \int |f - f_n| \to 0.$$