Does the following property valid for Lebesgue integration:
If $f$ is integrable on $E$ and $E = \bigcup_{i=1}^{\infty} E_n$ then
$$\sum_{i=1}^{\infty} \int_{E_n} |f|\le \int_E |f|. $$
If it is true, can you give a reference where the proof is written, and if it is not true, can you give a counterexample?
The inequality that you wrote is clearly false if we take $E_n=E$ for all $n$. Actually, the reverse inequality is true. To see this, note that:
$|f|\cdot 1_E\leq\sum\limits_{n=1}^{\infty} |f|\cdot 1_{E_n}$.
Now take an integral on both sides.