Lebesgue integrable function

A Lebesgue integrable function on a measure space $(X,\Sigma,\mu)$ is a measurable function $f:X\to \mathbb R$ (or extended real-valued) such that

where $|f|$ denotes the absolute value applied pointwise.

Lebesgue integrability ensures that the Lebesgue integral of $f$ is a finite real number and depends only on the a.e. equality class of $f$. The collection of such functions (modulo a.e. equality) is the space of L1 functions .

Examples:

• On $\mathbb R$ with Lebesgue measure , the function $f(x)=\frac{1}{1+x^2}$ is Lebesgue integrable.
• If $E$ is a measurable set with $\mu(E)<\infty$, then the indicator function $\mathbf{1}_E$ is Lebesgue integrable.