Convergence of improper integral of c.d.f.

Question

Let $F(x),G(x)$ be two cumulative distribution functions. And $$ \int^{+\infty}_{-\infty}|x|dF(x)<\infty,\int^{+\infty}_{-\infty}|x|dG(x)<\infty $$ show that: $$ \int^{+\infty}_{-\infty}|F(x)-G(x)|dx<\infty $$ I try to do this by \begin{aligned} \int^{+\infty}_{-\infty}xdF(x)-\int^{+\infty}_{-\infty}xdG(x) = \int^{+\infty}_{-\infty}G(x)-F(x)dx<\infty \end{aligned} Then I don't know how to continue, since convergence doesn't imply absolute convergence. Maybe this is the wrong direction.

Answer 1

By triangle inequality and linearity of expectation, $$ \int^{+\infty}_{-\infty}|F(x)-G(x)|dx \leq \int^{+\infty}_{-\infty} F(x) + G(x)dx \leq \mathbb{E}_G [|X|] + \mathbb{E}_F [|X|] <\infty, $$ since $$ \int^{+\infty}_{-\infty}|x|dF(x) = \mathbb{E}_F [|X|] <\infty, $$ where $\mathbb{E}_F$ denotes expectation w.r.t. the CDF $F$.