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.
