Random variable

Definition
A random variable is a measurable function:

or

where $(\mathrm{\Omega },\mathrm{\Sigma })$ is a measurable space and $\mathcal{B}(-)$ is the corresponding Borel $\sigma$-algebra.
If $(\mathrm{\Omega },\mathrm{\Sigma },\mathbb{P})$ is a probabilistic space, we call $X$ a random variable on $\mathbb{P}$.
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Remarks:

1. Boundedness: The condition $\underset{\omega \in \mathrm{\Omega }}{sup}|X(\omega )|<\mathrm{\infty }$ is often added (e.g., for uniform integrability).
2. A random variable which only takes the values 0 or 1 encodes the same data as an event (more precisely, it is the indicator function ${\chi }_E$ of a unique event, which takes the value $1$ on $E$ and $0$ on its complement). More generally, we can construct events from random variables: for any Borel subset $E\subseteq \mathbb{R}$ the preimage $X^{-1}(E)$ is an event (the event that $X$ lies in $E$), often written $X\in E$, and so we can consider its probability $\mathbb{P}(X\in E)$.
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Probability distribution

Every random variable on a probabilistic space,

induces a probability measure ${\mu }_X$ on $({\mathbb{R}}^n,\mathcal{B}({\mathbb{R}}^n))$ by means of the pushforward measure

It is called the distribution of $X$.