Let Ω be any set and let F be the collection of all subsets of Ω...

Let Ω be any set and let F be the collection of all subsets of Ω that are either countable or have a countable complement. (Recall that a set is countable if it is either finite or can be placed in one-to-one correspondence with the natural numbers N = {1, 2, . . .}.)

(a) Show that F is a σ-algebra.

(b) Show that the set function given by

μ(E)= 0 if E is countable ;

μ(E) = ∞ otherwise

is a measure, where E ∈ F.

Expert Solution

Colby Messinger answered 1 year ago

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