Lebesgue Integral

The Lebesgue Integral (over $A\in M$) of measurable $f:X\to [0,\infty ]$ is ${\int }_{A}fd\mu \equiv \underset{s\text{measurable and simple}}{\underbrace{\underset{0\le s\le f}{\sup }}}{\int }_{A}sd\mu \in [0,\infty ]$.