Let $\{f_n\}$ be a sequence of nonnegative measurable functions on $E$ that converges point wise on $E$ to $f$. Suppose $f_n \leq f$ for each $n$. Show that:
$$\lim_{n \rightarrow \infty} \int_E f_n = \int_E f.$$
So, I've already proved the monotone convergence theorem, the assumption of which is the same as what I'm trying to prove except for in the monotone convergence theorem the sequences of functions is increasing. I feel like there must be some clever trick to use the convergence theorem to prove this one… I've been thinking about it for awhile but I fear I'm stuck in tunnel vision. Insights appreciated!! Thanks!
Best Answer
If you are allowed to use Fatou's lemma$^{(1)}$, then $$\int f = \int \liminf f_n \leq \liminf \int f_n \leq \int f,$$ where the middle inequality is Fatou's lemma. This implies that $\liminf \int f_n=\int f$. It is obvious that $\limsup \int f_n \leq \int f$. Since the $\limsup$ is bigger than the $\liminf$, it is also equal to $\int f$, and it follows that $\lim \int f_n=\int f.$
$^{(1)}$ Using Fatou's lemma is not a big deal, since it is just the monotone convergence theorem applied to $g_i:=\inf\{f_i,f_{i+1},\cdots\}$.