If $X,Y$ are defined on the same probability space and have finite expectations, show that $$E(Y)-E(X)=\int_{-\infty}^\infty[P(X<t\le Y)-P(Y<t\le X)]\lambda(dt).$$ My attempt: \begin{align} E(Y)-E(X) &= \int_{-\infty}^\infty P(Y>t)\lambda(dt) - \int_{-\infty}^\infty P(X>t)\lambda(dt) \\&=\int_{-\infty}^\infty P(Y>t)\left(\int_{-\infty}^\infty P_X(dx)\right)\lambda(dt) - \int_{-\infty}^\infty P(X>t)\lambda(dt) \\&=\int_{-\infty}^\infty\int_{-\infty}^\infty P(Y>t) P_X(dx)\lambda(dt) - \int_{-\infty}^\infty P(X>t)\lambda(dt) \end{align}
I was hoping this would somehow lead me to the required by using Fubini's but I change of integral order doesn't seem to be of any help. Any help?