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Let $(X,\mu)$ be a measure space. The following two theorems hold:

  1. Let $\{f_n\}$ be a sequence of measurable functions $f_n:(X,\mu) \to [0,\infty]$. Then we have $$\int_X \left(\sum_{n=1}^\infty f_n\right) \ d \mu = \sum_{n=1}^\infty \int_X f_n \ d\mu.$$
  1. Let $\{f_n\}$ be a sequence of measurable functions $f_n:(X,\mu) \to \mathbb{C}$. Suppose $\sum_{n=1}^\infty \int_X |f_n|d\mu < \infty$. Then $\sum_{n=1}^\infty f_n(x)$ converges almost everywhere and belongs to $L^1$ and we have $$\int_X \left(\sum_{n=1}^\infty f_n \right) \ d\mu = \sum_{n=1}^\infty \int_X f_n \ d \mu.$$

The first one is a consequence of the monotone convergence theorem, the second one is a consequence of the dominated convergence theorem.

What are some counterexamples that show that the hypotheses of those theorems are necessary?

For example: what counterexample shows that the first result does not hold for functions that are not non-negative? What counterexample shows that the second result does not hold if we don't assume $\sum_{n=1}^\infty \int_X |f_n|d\mu < \infty$?

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    For both parts I think you can take $f_1 = 1$ and $f_n = -\chi_{[n-2,n-1]}$ for $n\geq 2$ on $X=\mathbb{R}$ with the Lebesgue measure. – 2017-02-02
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    @Matt Can you turn your comment in an answer by also adding some details? – 2017-02-04

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