Countable additivity with respect to integrands in Lebesgue integrals

Question

The following property of Lebesgue integrals is true for nonnegative measurable functions $f_n$ (because it is a consequence of the monotone convergence theorem): $$\int (\sum_{n=1}^\infty f_n) d\mu = \sum_{n=1}^\infty \int f_n d\mu$$

Can anyone give me an example to show it's wrong for a series of functions that are not nonnegative.

Or is there some additional conditions so that the theorem is still true.

Answer 1

It's true if there is a function $F$ such that $\sum_n |f| \le F$ and $\int F \; d\mu < \infty$ (use Lebesgue dominated convergence theorem).

Answer 2

If your measure space is $[0,2\pi]$ and $f_n = sinx$

$\sum_{n=1}^\infty \int f_n d\mu = 0$

and $ \int\sum_{n=1}^\infty f_n d\mu$ is undefined.