2

I really need help to show that:

Let {$f_n$} be a sequence of nonnegative measurable functions that converges to $f$ pointwise on E. Let $M>0$ be such that $\int_Ef_n\le M$ for all n. Show that $\int_E f\le M$.

Any help I really appreciate. Thanks

Vui Tinh
  • 285
  • 1
    Hint, from Apostol's book: Let $g_n=\inf(f_n,f_{n+1},\dots)$. Then $g_n\nearrow f$ a.e. on $E$ and $\int_E g_n\leq \int_E f_n\leq M$. So $\lim_{n\to\infty}\int_E g_n\leq M$. Now apply Levi's Monotone convergence theorem. – LeviathanTheEsper Apr 17 '17 at 04:26
  • Thank you so much for your help. – Vui Tinh Apr 17 '17 at 04:32

1 Answers1

2

You can use Fatou's lemma: $\liminf_{n} f_n = f$ since $f_n \to f$ pointwise, and $$ \liminf_{n} \int_E f_n \leq M $$ since all the $\int_E f_n$ are, and then Fatou says $$ \int_E f = \int_E \liminf_n f_n \leq \liminf_n \int_E f_n \leq M. $$

Chappers
  • 67,606
  • Note that the question, stated that way, is as an exercise 10.8 of Tom Apostol's Book, and is called Fatou's Lemma there. It comes with a hint and is easy to prove following that hint. This answer looks good to see an equivalence between both Fatou's Lemmas. – LeviathanTheEsper Apr 17 '17 at 04:16