Probability Spaces, Events and Random Elements

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(\Omega,\mathcal{F})$ be a measurable space and let $P$ be a measure on $(\Omega,\mathcal{F})$.

(i) Probability space. $(\Omega,\mathcal{F},P)$ is a probability space if it is a measure space whose measure $P$ is a probability measure, that is,

$$P(\Omega)=1.$$

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

(ii) Events. The elements of $\mathcal{F}$ are the events; for an event $A$, the number $P(A)$ is the probability of $A$.

(iii) Random elements. Let $(S,d)$ be a metric space with Borel $\sigma$-algebra $\mathcal{B}(S)$. A random element of $S$ is an $\mathcal{F}/\mathcal{B}(S)$-measurable function $X:\Omega\to S$ (an element of $\mathrm{Meas}(\Omega,\mathcal{F};S,\mathcal{B}(S))$, Measurable Map, and the Set of Measurable Maps (ii)).

(iv) Random variables. A random variable is a function $X:\Omega\to\mathbb{R}$ that is measurable as a function into $\mathbb{R}$ (an element of $\mathrm{Meas}(\Omega,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$, Measurable Map, and the Set of Measurable Maps (ii); as $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra of the metric space $(\mathbb{R},d_{\mathbb{R}})$, Borel Sigma-Algebra of a Metric Space (ii), a random variable is a random element of $\mathbb{R}$).

(v) Random vectors. Let $n\in\mathbb{N}$. A random vector in $\mathbb{R}^{n}$ is a function $X:\Omega\to\mathbb{R}^{n}$ that is measurable as a function into $\mathbb{R}^{n}$ (an element of $\mathrm{Meas}(\Omega,\mathcal{F};\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$, Measurable Map, and the Set of Measurable Maps (ii); as $\mathcal{B}(\mathbb{R}^{n})$ is the Borel $\sigma$-algebra of the metric space $(\mathbb{R}^{n},d_{E})$, Borel Sigma-Algebra of a Metric Space (iii), a random vector in $\mathbb{R}^{n}$ is a random element of $\mathbb{R}^{n}$).