Here's a question I'm sturglling with:
Show that for an increasing sequence of events $$A_1\subset A_2\subset
A_3\subset …$$ the next equation holds
$$P\big(\bigcup_nA_n\big)=\lim_{n\rightarrow\infty}P(A_n)$$
Here's what I have so far: let $(B_n)_{n=1}^{\infty}$ be a sequence of events such that
$$B_{1}=A_{1},B_{2}=A_{2}\backslash A_{1},B_{3}=A_{3}\backslash A_{2}\cap A_{1},…,B_{n}=A_{n}\backslash(\cap_{i=1}^{n-1}A_{i})$$
So obviously every two elements in $B_n$ are disjoint and $\bigcup B_n = \bigcup A_n$ so
$$P\big(\bigcup A_n\big) = P\big(\bigcup B_n \big) = \sum_n P(B_n)$$
And I'm stuck here. Really I'm not sure what a limit of a sequence of events here so I was just following the little I do know. I know sequence limits, but what does it mean when the elements are sets? Does the sequence converges to a set? What is that set?
Would appreciate any motivation of the limit issue, and on solving the question.
Thanks!
Best Answer
If $$A_1\subset A_2\subset A_3\subset ...$$ then $$\bigcup_n A_n=\lim_{n\rightarrow \infty }A_n$$
Try with $$B_{1}=A_{1},B_{2}=A_{2}\backslash A_{1},B_{3}=A_{3}\backslash A_2,...,B_{n}=A_{n}\backslash(A_{n-1})$$
$$P\big(\bigcup_n A_n\big) = P\big(\bigcup_n B_n \big) = \sum_{n=1}^\infty P(B_n)=\lim_{n\rightarrow \infty} P(A_n)$$