* 18. Show that the axiom that P is countably additive is equivalent to the axiom that P is finitely additive and continuous. That is to say, let  be a set and F an event space of subsets of . If P is a mapping from F into [0, 1] satisfying

(i) P() = 1, P(∅) = 0,

(ii) if A, B ∈ F and A ∩ B = ∅ then P(A ∪ B) = P(A) + P(B),

(iii) if A1, A2, . . . ∈ F and Ai ⊆ Ai+1 for i = 1, 2, . . . , then P(A) = lim i→∞

P(Ai ), where A =

S

∞i

=1 Ai , then P satisfies P

????S i Ai

=

P