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

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Advance Math

Let Ω be any set and let F be the collection of all subsets of Ωthat are either countable or have a countable complement. (Recallthat a set is countable if it is either finite or can be placed inone-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.

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