If set sequences $A_1, A_2, \cdots$, $B_1, B_2, \cdots$ consist of measurable sets in the sigma-algebra $Q$. Suppose $P(A_k \text{ infinitely often }) = 1$, and $P(B_k^c \text{ infinitely often }) = 0$. What is the probability that infinitely many of joint events $A_k \cap B_k$ will occur?
I can see $P(\cap_{n=1}^{\infty}\cup_{k=n}^{\infty}A_k) = 1$, $\displaystyle P(\cup_{n=1}^{\infty}\cap_{k=n}^{\infty}B_k) = 1$ by De Morgan Law, and it's asking $P(\cap_{n=1}^\infty\cup_{k=n}^\infty (A_k \cap B_k))$, however no idea how to proceed. Can anyone help?
It is $1$. If $A_k$ occurs infinitely often and $B_k^{c}$ occurs at most finitely many times $A_k \cap B_k$ occurs infinitely often.
Note that $P(B_k^{c} \, \text {infinitely often })=0$ is same as the statement that all but finitely many $B_k$'s occur with probability $1$.
Let $A$ be the event that $A_k$ occurs infinitely often and $B$ the event that $B_k^{c}$ occurs only finitely many times. It follows from the hypothesis that $P(A)=1$ and $P(B)=1$. Hence $P(A\cap B)=1$. But if $A\cap B$ occurs then $A_k$ occurs infinitely often and $B_k^{c}$ occurs only finitely many times so $A_k\cap B_k$ must occur infinitely often.