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.