Lebesgue integrable function

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

\int_X |f|\,d\mu<\infty,

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

Its Lebesgue integral is finite and depends only on the almost-everywhere equivalence class of $f$ . These equivalence classes form the space $L^1(X,\mu)$ .

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.