Monotone coupling of random variables.

Question

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?

Answer 1

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 $Y(\omega) \leq t$.