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 $A+t:={x+t:x in A}$.

I was thinking to go through something like:

$$

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 $int_Bmu(x,y)dy leq int_{mathbb{R}^d}mu(x,y)dy=mu(x)$.