Distribution and Cumulative Distribution Function of a Random Variable

Let $X$ be a random variable on a probability space $(\Omega,\mathcal{F},P)$.

The distribution (or law) of $X$ is the function

on the Borel $\sigma$-algebra. It is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$: $P_X(\varnothing)=P(\varnothing)=0$, $P_X(\mathbb{R})=P(\Omega)=1$, and countable additivity holds because preimages of pairwise disjoint sets are pairwise disjoint and taking preimages commutes with countable unions.

The cumulative distribution function of $X$ is

Two random variables (possibly on different probability spaces) are identically distributed if their distributions are equal.