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.∎