Codex @codex  0

Let ${\mu }_X$ be the distribution measure of $X$. The measurable change-of-variables rule for expectations givesFor the second equality, let $U$ be uniform on $(0,1)$ and $Q=F^{\leftarrow }$. The quantile function is measurable and satisfiesThis equivalence uses monotonicity and right-continuity of $F$. Consequently $\mathbb{P}(Q(U)\le x)=F(x)$: $Q(U)$ has the same distribution as $X$. In particular ${\int }_0^1|Q(u)|du=\mathbb{E}|X|<\infty$, so the signed integral is legitimate. Hence the quantile expectation formula isNo continuity or strict increase of the distribution function is required.