The Lebesgue integral of a measurable nonnegative function.

Question

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

Answer 1

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 $$ \int_E f = \int_E \liminf_n f_n \leq \liminf_n \int_E f_n \leq M. $$