I need help with this question:
Let ${f_n}$ be a sequence of nonnegative and measurable functions on a measurable set E. Prov that:
$\int_E \sum_{n=1}^\infty f_n =\sum_{n=1}^\infty\int_E f_n $
If $\sum_{n=1}^\infty\int_E f_n \lt \infty$ then $\sum_{n=1}^\infty f_n(x)$ converges pointwise a.e on E.
Any help I really appreciate.
For 1) Consider the sequence of partial sums $\{s_k=\sum\limits_{n=1}^{k}\}$. This is a sequence of non-negative measurable functions which is clearly increasing. Moreover, $ \lim_{k \to \infty}{s_k}=\sum\limits_{n=1}^{\infty}{f_n}$ i.e it converges p.w. Thus, by the Monotone Convergence Theorem we have that $\int{\sum\limits_{n=1}^{\infty}{f_n}}=\sum\limits_{n=1}^{\infty}\int{f_n}$.
2) Think Beppo Levi.
