Lebesgue integral

Integral of a measurable function defined from its positive and negative parts.

Lebesgue integral

A Lebesgue integral on a measure space $(X,\Sigma,\mu)$ assigns a value to a measurable function $f:X\to[-\infty,\infty]$ by reducing to the nonnegative case. Define the positive and negative parts

f^+ := \max(f,0), \qquad f^- := \max(-f,0),

so that $f=f^+-f^-$ and $f^+,f^-$ are nonnegative measurable functions. If at least one of $\int_X f^+\,d\mu$ or $\int_X f^-\,d\mu$ is finite, define

\int_X f\,d\mu := \int_X f^+\,d\mu - \int_X f^-\,d\mu,

where the integrals on the right are understood via the nonnegative Lebesgue integral . If both $\int_X f^+\,d\mu$ and $\int_X f^-\,d\mu$ are $+\infty$ , the Lebesgue integral of $f$ is left undefined.

When $f$ is Lebesgue integrable , the integral is a finite real number. Moreover, if $f$ and $g$ satisfy a.e. equality , then (whenever defined) their Lebesgue integrals agree.

Examples:

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