Monotone coupling of random variables. Let $F_1,F_2$ be two cumulative distribution functions. Let also the distrubutions have the property that $F_1\leq F_2$.
I want to prove that there exists two random variables $X,Y$ such that:
$F_X=F_1,F_Y=F_2$, but $P(X<Y)=0$.
Any tips on what to use to prove this result?

On $(0,1)$ with Lebesgue measure define $X(\omega) =\inf \{t: F_1(t) \geq \omega \}$ and $Y(\omega) =\inf \{t: F_2(t) \geq \omega \}$. These random variables have the required properties. Hint: $F_1(t) \geq \omega$ iff $X(\omega) \leq t$ and $F_2(t) \geq \omega$ iff $Y(\omega) \leq t$.
