Quantile expectation formula ($\mathbb{E}X={\int }_0^1{F}^{\leftarrow }(u)du$)

For an integrable real random variable with distribution function $F$, the generalized inverse $F^{\leftarrow }(u)=\inf \{x:F(x)\ge u\}$ sends a uniform random variable on $(0,1)$ to the distribution $F$. This follows from $F^{\leftarrow }(u)\le x$ if and only if $u\le F(x)$. Equality of laws proves both the expectation formula and absolute integrability of the quantile. Atoms and flat pieces of $F$ cause no problem.