support of integrable function with respect to counting measure is countable
Let be a measure space with the counting measure. If is an integrable function, , then it has countable (http://planetmath.org/Countable) support (http://planetmath.org/Support6).
Proof.
WLOG, we assume that is real valued and is nonnegative. Let denote the preimage of the interval and, for every positive integer , let denote the preimage of the interval . Since the integral of is bounded, each can be at most finite. Taking the union of all the , we get the support . Thus, is a union of countably many finite sets 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 |