support of integrable function with respect to counting measure is countable


Let (X,𝔅,μ) be a measure spaceMathworldPlanetmath with μ the counting measure. If f is an integrable function, Xf𝑑μ<, then it has countableMathworldPlanetmath (http://planetmath.org/Countable) support (http://planetmath.org/Support6).

Proof.

WLOG, we assume that f is real valued and is nonnegative. Let S0 denote the preimageMathworldPlanetmath of the interval [1,) and, for every positive integer n, let Sn denote the preimage of the interval [1n+1,1n). Since the integral of f is bounded, each Sn can be at most finite. Taking the union of all the Sn, we get the support suppf=n=0Sn. Thus, suppf is a union of countably many finite setsMathworldPlanetmath and hence is countable. ∎

Title support of integrable function with respect to counting measure is countable
Canonical name SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable
Date of creation 2013-03-22 14:59:34
Last modified on 2013-03-22 14:59:34
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 11
Author Wkbj79 (1863)
Entry type Result
Classification msc 28A12
Related topic UncountableSumsOfPositiveNumbers
Related topic SupportOfIntegrableFunctionIsSigmaFinite