locally integrable function

Primary tabs

View(active tab)
Coauthors
PDF
Source

locally integrable function

Definition Suppose that $U$ is an open set in $\mathbb{R}^{n}$ , and $f\colon U\to\mathbb{C}$ is a Lebesgue measurable function. If the Lebesgue integral

\int_{K}|f|dx

is finite for all compact subsets $K$ in $U$ , then $f$ is locally integrable. The set of all such functions is denoted by $L^{1}_{{\scriptsize{\mbox{loc}}}}(U)$ .

Example

1.

$L^{1}(U)\subset L^{1}_{{\scriptsize{\mbox{loc}}}}(U)$ , where $L^{1}(U)$ is the set of (globally) integrable functions.
2.

Continuous functions are locally integrable.
3.

The function $f(x)=1/x$ for $x\neq 0$ and $f(0)=0$ is not locally integrable.

Type of Math Object:

Definition

Major Section:

Reference

Parent:

$L^p$-space

Groups audience:

World Writable

Mathematics Subject Classification

28B15 Set functions, measures and integrals with values in ordered spaces

Add a correction
Attach a problem
Ask a question

User menu

locally integrable function

Primary tabs

locally integrable function

Example

Mathematics Subject Classification

Recent Activity

Info

Attached Articles

Corrections

Versions

User menu

User login

You are here

locally integrable function

Primary tabs

locally integrable function

Example

Mathematics Subject Classification

Search form

Recent Activity

Info

Attached Articles

Corrections

Versions