Proving Convergence of Integrals for Sequences of Functions

[Artusartos]

Homework Statement

Show that under the hypothesis of Theorem 17 we have \int |f_n - f| \rightarrow 0.

Theorem 7:

Let <g_n> be a sequence of integrable functions which converges a.e. to an integrable function g. Let <f_n> be a dequence of measurable functions such that |f_n| \leq g_n and <f_n> converges to f a.e.. If \int g = lim \int g_n then \int f lim \int f_n.

Homework Equations

The Attempt at a Solution

By linearity, \int (f_n-f) = \int f_n - \int f. So if we take the limit of \int f_n - \int f, we get lim_{n \rightarrow \infty} \int f_n - lim_{n \rightarrow \infty} \int f. But we know from Theorem 17 that \int f = lim \int f_n. So if we replace lim_{n \rightarrow \infty} \int f_n by \int f. So \int f - \int f = 0. I feel like there is something wrong with my proof, because it's so simple...so can anybody tell me if I'm wrong?

Thanks in advance

 

[jbunniii]

You showed that $\int (f_n - f) \rightarrow 0$, but that was already known from theorem 17. You were asked to show that $\int |f_n - f| \rightarrow 0$, which is a stronger statement because
$$\left|\int f_n - f\right| \leq \int |f_n - f|$$
What you are trying to prove looks like a slight generalization of the dominated convergence theorem, so examining how that proof works should be useful, i.e., you know you will probably have to use Fatou's lemma.