Definition:Lebesgue Integral

Definition
Let $\lambda^n$ be Lebesgue measure on $\R^n$.

For any Lebesgue integrable function $f: \R^n \to \overline{\R}$, the $\lambda^n$-integral:


 * $\displaystyle \int f \, \mathrm d\lambda^n$

is called the Lebesgue integral of $f$.

Also known as
Historically, the notations $\displaystyle \int f \left({x}\right) \, \mathrm dx$ and $\displaystyle \int f \, \mathrm dx$ are used in place of the formally correct $\displaystyle \int f \, \mathrm d\lambda^n$.

Also see

 * Integral of Integrable Function, of which the Lebesgue integral is an instance.