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单词 SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable
释义

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.∎

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更新时间:2025/5/4 2:58:11