Lebesgue integral of a nonnegative function

Definition of the Lebesgue integral for nonnegative measurable functions.

Lebesgue integral of a nonnegative function

A Lebesgue integral of a nonnegative measurable function on a measure space $(X,\Sigma,\mu)$ is defined as follows. For a nonnegative simple function of the form

s=\sum_{k=1}^n a_k\,\mathbf{1}_{E_k} \quad (a_k\ge 0,\; E_k\in\Sigma),

where $\mathbf{1}_{E_k}$ is the indicator function of $E_k$ , define

\int_X s\,d\mu := \sum_{k=1}^n a_k\,\mu(E_k).

For a nonnegative measurable function $f:X\to[0,\infty]$ , define

\int_X f\,d\mu := \sup\left\{\int_X s\,d\mu : s \text{ is simple and } 0\le s\le f\right\}.

This is the starting point for the Lebesgue integral of general real-valued functions, obtained by decomposing a function into its positive and negative parts.

Examples:

If $E$ is a measurable set , then $\int_X \mathbf{1}_E\,d\mu=\mu(E)$ .
On $\mathbb R$ with Lebesgue measure , if $f(x)=x$ for $x$ in the interval $[0,1]$ and $f(x)=0$ otherwise, then $\int_{\mathbb R} f\,dx=\tfrac12$ .