Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X:(\Omega,\mathcal F)\to([0,\infty),\mathcal B([0,\infty)))$ be a nonnegative real-valued [random variable](/page/Random%20Variable). Let $F_X:\mathbb R\to[0,1]$ be the distribution function of $X$, defined by
where the supremum is allowed to be $\infty$. Then $Q_X$ is $\mathcal B([0,1])$-measurable as a map into $[0,\infty]$ with its extended Borel $\sigma$-algebra, and