# The General Lebesgue Integral

1. Mar 1, 2013

### Artusartos

1. The problem statement, all variables and given/known data

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 $$\int f lim \int f_n$$.

2. Relevant equations

3. 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?

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
$$\left|\int f_n - f\right| \leq \int |f_n - f|$$