Find borel set such that $P(X \in A, Y \in B) \neq P(X \in A)P(Y \in B)$

Question

I am given the pdf of the uniform distribution on the disk of radius R: $1/\pi(R^2)$ I have shown that X and Y are not independent by showing that $f_X(x)f_Y(y) \neq 1/\pi(R^2)$

However, despite knowing that the variables are not independent, I am unable to find borel sets A and B such that $P(X \in A, Y \in B) \neq P(X \in A)P(Y \in B)$

Any help would be appreciated

Thanks

Answer 1

Just take $A$ and $B$ so that $P(X\in A,Y\in B) = 0$.

For example $A=B= [0.9R,R]$

We clearly have $P(X \in A) > 0$ and $P(Y\in B) > 0$