Lebesgue Integral

Context

Every measurable function may be approximated by simple functions. A function $s:A\subseteq {\mathbb{R}}^n\to \mathbb{R}$ is said to be simple if its range is constituted by a finite number of values $s_1,\dots ,s_N$, attained respectively on measurable sets $A_1,\dots ,A_N$, contained in $A$. Introducing the characteristic functions ${\chi }_{A_j}$, we may write

$$s=\sum _{j=1}^N{s}_j{\chi }_{A_j}.$$

Statements

Definition. Let $(\Omega ,\mathcal{F},P)$ be a measurable space, $A\subset \mathcal{F}$ be measurable set and $f:A\to \mathbb{R}$ be a measurable function. The Lebesgue integral of $f$ is defined as

$${\int }_{A}f=\sum _{j=1}^N{s}_{j}P(A_j),$$

with the convention that, if $s_j=0$ and $|A_j|=+\infty$, then $s_{j}P(A_j)=0$.

References

@Sal08, pp. 541