Definition:Integrable Function/Lebesgue

Definition

Let $\lambda^n$ be a Lebesgue measure on $\R^n$ for some $n > 0$.

Let $f: \R^n \to \overline \R$ be an extended real-valued function.

Then $f$ is said to be Lebesgue integrable if and only if it is $\lambda^n$-integrable.

Similarly, for all real numbers $p \ge 1$, $f$ is said to be Lebesgue $p$-integrable if and only if it is $p$-integrable under $\lambda^n$.

Also see

• Definition:Lebesgue Integral

• Results about Lebesgue integrable functions can be found here.

Source of Name

This entry was named for Henri Léon Lebesgue.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 10$