support of integrable function is σ-finite

Theroem - Let (X,,μ) be a measure spaceMathworldPlanetmath and f:X a measurable functionMathworldPlanetmath. If f is integrable, then the support of f is σ-finite (

It follows easily from this result that any function in an Lp-space (, with 1p<, must have σ-finite support.

: Let A0:=[1,[, and for each n let An:=[1n+1,1n[. Since f is integrable, we must necessarily have μ(|f|-1(An))< for each n{0}, because


Since f and |f| have the same support, and the the support of the latter is supp|f|=n=0|f|-1(An), it follows that the support of f is σ-finite.

Title support of integrable function is σ-finite
Canonical name SupportOfIntegrableFunctionIssigmafinite
Date of creation 2013-03-22 18:38:47
Last modified on 2013-03-22 18:38:47
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 4
Author asteroid (17536)
Entry type Theorem
Classification msc 26A42
Classification msc 28A25
Related topic SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable
Defines Lp functions have σ-finite support