Distribution Functions

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, for $x\in\mathbb{R}$ let $(-\infty,x]$ be the ray $\{t\in\mathbb{R}:t\le x\}$, a Borel subset of $\mathbb{R}$ (Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line (ii)).

(i) Distribution function of a measure. Let $\mu$ be a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. The distribution function of $\mu$ is the function

$$F_{\mu}:\mathbb{R}\to[0,1],\qquad F_{\mu}(x)=\mu\bigl((-\infty,x]\bigr)$$

(Basic Properties of a Measure (iii)).

(ii) Distribution function of a random variable. Let $X$ be a random variable on a probability space $(\Omega,\mathcal{F},P)$, with law $P_{X}$ as a random element of $\mathbb{R}$ (Probability Spaces, Events and Random Elements (iv)). The distribution function of $X$ is $F_{X}=F_{P_{X}}$, so that

$$F_{X}(x)=P\bigl(\{\omega\in\Omega:X(\omega)\le x\}\bigr)\qquad(x\in\mathbb{R}).$$