support of integrable function with respect to counting measure is countable
Let (X,𝔅,μ) be a measure space with μ the counting measure. If f is an integrable function, ∫Xf𝑑μ<∞, then it has countable
(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 preimage 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 sets
and hence is countable.
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Title | support of integrable function with respect to counting measure is countable |
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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 |