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Let A1, A2, ... be a infinitely countable collection of events, then P lim P (UA m+00 i=1

Answer 1

$P(\cup_{i=1}^\infty A_i)=P(\lim_{n\rightarrow \infty}B_n)~where,~B_n=\cup_{i=1}^n A_i.\\\\ We~see~that~\{B_n\}~is~sequence~of~monotonically~increasing~events~then\\ from~Continuity~theorem,\\\\ P(\cup_{i=1}^\infty A_i)=P(\lim_{n\rightarrow \infty}B_n)=\lim_{n\rightarrow \infty}P(B_n)=\lim_{n\rightarrow \infty}P\left(\cup_{i=1}^nA_i \right )$