Leveraging Borel-Cantelli

Question

Suppose $\{A_k\}_{k=1}^\infty$ and $\{B_k\}_{k=1}^\infty$ are measurable sets in the same probabiilty space such that $P(A_k)=P(B_k)$ for all $k$. If you can show that $P(A_k \text{ i.o.})=1$ (for example, the $A_k$'s are independent and $\sum_{k=1}^\infty P(A_k)=\infty$) then is it necessarily true that $P(B_k \text{ i.o.})=1$? (Obviously if I used Borel-Cantelli for the $A_k$'s, I am assuming that the $B_k$'s are not necessarily independent.) I don't think this is so, but I can't come up with a counterexample yet.

Answer 1

The big point is this: we have no information about how the $A_k$ correlate with each other or how the $B_k$ correlate with each other. So, we can build a scenario like this:

Suppose we have a fair coin, which we flip over and over again.

For $k\in\mathbb{N}$, let $A_k$ be the event that the $k$th flip yields heads; let $B_k$ be the event that the first flip yields heads.

Then $P(A_k)=P(B_k)=\frac{1}{2}$ for all $k$, and $P(A_k\text{ i.o.})=1$, but $$ P(B_k\text{ i.o})=P(\text{first flip gives heads})=\frac{1}{2}. $$