I recently started some courses on Measure Theory, which are becoming more complex than I thought at the beginning. Since my academic background is is not in-depth in mathematics, I am finding it difficult to solve problems that involve "proving that..."
Turning to the reason for my question, I have not been able to complete the answer to the following problem:
Let $(\Omega,\mathcal F,P)$ a probability space. And $A_1, A_2 ,A_3...$ a succession of independent events. Prove that:
a) $ P (\bigcap_{k=1}^{\infty}{A_k}) = \prod_{k=1}^{\infty}{P(A_k)}$
At this point, I have tried to apply the independence property for $n$ events (finite case) and take the limit. Something like that:
- We know that: $ P (\bigcap_{k=1}^{n}{A_k}) = \prod_{k=1}^{n}{P(A_k)}$ with $k=1,2,...,n$
- And applying limits to both sides of the equation, we have: $\lim_{n\to \infty} P (\bigcap_{k=1}^{n}{A_k}) = P (\bigcap_{k=1}^{\infty}{A_k})$ and $\lim_{n\to \infty} \prod_{k=1}^{n}{P(A_k)} = \prod_{k=1}^{\infty}{P(A_k)}$
So, we can conclude what was proposed in the question (Although I guess we need to develop the idea some more)
b) $ P (\bigcup_{k=1}^{\infty}{A_k}) = 1-\prod_{k=1}^{\infty}({1-P(A_k)})$
In this item, I haven't had any ideas yet....
I appreciate any help the community can give me.
Thanks,
Cristian.
I think your approach to (a) is fine. You might want to think about why the limits exist, however.
For (b), use $1-P(A_k) = P(A_k^c)$. Next, use the fact that if $A_1,...,A_k$ are independent then so are their complements. Finally, use (a) and De Morgan's identity: $P(\cup_k A_k) = P((\cap_k A_k^c)^c)$.