If $E[X1_{X > Y}]=E[Y1_{X > Y}]$ and $E[X1_{X < Y}]=E[Y1_{X < Y}]$ then $X=Y$ a.s.

Question

If $E[X1_{X > Y}]=E[Y1_{X > Y}]$ and $E[X1_{X < Y}]=E[Y1_{X < Y}]$ then $X=Y$ a.s.

This fact was used in a proof I saw, and although it seems intuitive I cannot find a way to prove it. My attempt:

if $E[X1_{X > Y}]=E[Y1_{X > Y}]\Rightarrow E[(X-Y)1_{X > Y}]=0$

Now it would help if I knew if $X,Y$ are positive, because if they are then I could simply state that $X=Y$ a.s. but I don't, so what should I do?

Answer 1

The two equalities rewrite as $E((X-Y)1_{X>Y})=0$ and $E((Y-X)1_{Y>X})=0$. Summing them yields $$E(|X-Y|)=0$$ Since $|X-Y|\geq 0$, that implies $X-Y=0$ a.s., that is $X=Y$ a.s.