# Math Genius: Inequality for joint probabilities of dependent random vectors

Let $$X$$ and $$Y$$ be two dependent random vectors in $$mathbb{R}^d$$, such that $$Xneq Y$$ with probability 1, whose joint probability measure has density $$mu(x,y)$$ with respect to the Lebesgue measure. For measurable sets $$A$$ and $$B$$, does the inequality
$$mathbb{P}(X-Y in A, Yin B) leq sup _{tin B}mathbb{P}(X in A+t)$$
hold true? Herein, operations are meant elementwise and $$A+t:={x+t:x in A}$$.
$$mathbb{P}(X-Y in A, Yin B)= int_{B}int_{A+y}mu(x,y)dxdy\ leq sup_{tin B}int_{B}int_{A+t}mu(x,y)dxdy\ =sup_{tin B}int_{A+t}int_Bmu(x,y)dydx\ leq sup _{tin B}int_{A+t}mu(x)dx\ =sup _{tin B}mathbb{P}(X in A+t)$$
where $$mu(x)$$ denote the marginal density of $$X$$, but I have doubts about the second and third lines, I’m not sure they’re correct. I pass from the third to the fourth line by using the fact that $$mu(x,y)$$ is nonnegative and $$int_Bmu(x,y)dy leq int_{mathbb{R}^d}mu(x,y)dy=mu(x)$$.