Hi, I keep seeing this come up A1 ⊇ A2 ⊇ A3 .... is an infinite decreasing sequence of events. Prove from first principles that P(intersection of Ai from i=1 to infinity) = Lim P(An) as n--> infinity All i can think of is that since each is a subset of the preceding, then A1 ∩ A2...∩An = An So clearly P(A1 ∩ A2...∩An) = P(An) and thus the same for limits. I think this is too simplistic though, is it or isnt it ? Thanks a lot
Hi stukbv! Did you already encounter the dual version of this? That is, if [tex]A_0\subseteq A_1\subseteq A_2\subseteq ...[/tex] then [tex]P\left(\bigcup_{i=0}^{+\infty}{A_i}\right)=\lim_{n\rightarrow +\infty}{P(A_n)}[/tex] It is the easiest if we prove this first. Now, this statement actually follows from the [itex]\sigma[/itex]-additivity. However, the [itex]\sigma[/itex]-additivity requires the events to be disjoint, which is not the case here. Is there a way to make the events disjoint, though?
Yeah ive done it that way I think Let B1 = A1 B2 = A2∩A1C B3=A3∩A2C And so on, Then the Bi are disjoint and the union Bi (up to n) = An, but also U Bi up to ∞ = Union of Ai upto ∞ (since preceding ones are subsets!) so P(An) = P(U Bi) (to n) = Sum P(Bi (to n) as all bi are disjoint, then let RHS and LHS n tend to infinity and we get sum to infinity of P(Bi) = P(U Bi) to ∞ = P(U Ai) to ∞. according to my lecturer!
Ah yes, that's very good!! So now for your question. You need to prove that [tex]P\left(\bigcap{A_n}\right)=\lim_{n\rightarrow +\infty}{P(A_n)}[/tex] Now what happens if you take the complements of these events?
SO P(U An^c) is equal to the sum since they are disjoint? But i dont see how you can relate the 2 like we did before
No, they are not disjoint. But you have a formula for [tex]P\left(\bigcup{A_n^c}\right)[/tex] Right? You've proven the formula above...
Well, you know already that the statement [tex]P\left(\bigcup{A_n^c}\right)=1-\lim_{n\rightarrow +\infty}{P(A_n)}[/tex] is true. Now try to evaluate the left side.
surely if they are all subsets of their preceding ones then the union of the complements is just a1^c ?
We have [tex]A_1\supseteq A_2\supseteq ...[/tex] and thus [tex]A_1^c\subseteq A_2^c\subseteq ...[/tex] So that doesn't really work.