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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!

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    I think you must have a bounded event space , ie no events happen outside this event space. Otherwise it limit doesn't seem to make sense. But i am not sure.2012-11-15
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    From the question it looks like we have countably many events, so I'm not sure if it is bounded2012-11-15
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    Note that you're not considering a limit of a sequence of events here at all. You're considering the limit of an ordinary sequence of numbers - $P(A_n)$ is a real number for each $n$.2012-11-15
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    Yes if it is countably many events then you can at least cover all the events by countably infinit disjoint sets . so that makes sense to me .2012-11-15
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    Also, the sets $B_n$ you've described are *not* pairwise disjoint. You may want to define $B_3 \setminus A_1 \cup A_2 = B_3 \setminus A_2$ instead (note that $A_2 \cap A_1 = A_1$, and so on...)2012-11-15

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