Probability Spaces: Notation

(i) Base. This setting extends Lᵖ Spaces: Notation: its conventions and notation are in force. $(\Omega,\mathcal{F},P)$ denotes a probability space, and the integration notation of the adopted settings applies with $(\Omega,\mathcal{F},P)$ in place of $(X,\mathcal{F},\mu)$. Events and their probabilities are as in Probability Spaces, Events and Random Elements (ii).

(ii) Random elements, laws and distribution functions. Random elements of a metric space, random variables and random vectors are as in Probability Spaces, Events and Random Elements (iii), Probability Spaces, Events and Random Elements (iv) and Probability Spaces, Events and Random Elements (v). For a random element $X$, $P_{X}$ is its law, and identically distributed random elements are as in The Law of a Random Element (i) and The Law of a Random Element (ii). For a probability measure $\mu$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ and a random variable $X$, $F_{\mu}$ and $F_{X}$ are the distribution functions of Distribution Functions (i) and Distribution Functions (ii).

(iii) Events given by random elements. For functions $X,Y,\dots$ on $\Omega$, each a random element of a metric space or an element of $\mathcal{M}^{+}(\Omega,\mathcal{F})$, and a condition on their values, $\{\text{condition}\}$ denotes the set

$$\{\omega\in\Omega:\ \text{the condition holds at }X(\omega),Y(\omega),\dots\},$$

and $P(\text{condition})$ denotes $P(\{\text{condition}\})$ when this set is an event. For $n\in\mathbb{N}$ and events $A_{1},\dots,A_{n}$, $\bigcap_{k=1}^{n}A_{k}$ is the intersection of the family $(A_{k})_{k\in[n]}$, an event (Unions and Intersections over a Countable Index Set in a σ-Algebra (ii)), and for real numbers $a_{1},\dots,a_{n}$, $\prod_{k=1}^{n}a_{k}$ is their finite product. For a sequence $(A_{k})_{k\in\mathbb{N}}$ of events and $n\in\mathbb{N}$, $\bigcup_{k\ge n}A_{k}$ is the union of the family $(A_{k})_{k\in\mathbb{N},\,k\ge n}$, and

$$\{A_{k}\text{ infinitely often}\}=\bigcap_{n\in\mathbb{N}}\ \bigcup_{k\ge n}A_{k},$$

an event (Unions and Intersections over a Countable Index Set in a σ-Algebra (i), Unions and Intersections over a Countable Index Set in a σ-Algebra (ii)).

(iv) Functions of random elements. Let $m,n,r\in\mathbb{N}$. For a function $X$ on $\Omega$ with values in a set $S$ and a function $g$ on $S$, $g(X)$ denotes the composite $g\circ X$, $\omega\mapsto g(X(\omega))$. For functions $X$ and $Y$ on $\Omega$ with values in sets $S$ and $T$ and a function $g$ on $S\times T$, $(X,Y)$ denotes the function $\omega\mapsto(X(\omega),Y(\omega))$ and $g(X,Y)$ the function $\omega\mapsto g(X(\omega),Y(\omega))$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (ii), Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii)). A real number, a point of $\mathbb{R}^{n}$ or a real matrix also denotes the constant function on $\Omega$ with that value, and sums, differences and products of functions on $\Omega$ with values in $\mathbb{R}$, in $\mathbb{R}^{n}$ or in a set of real matrices, such constants included, are taken pointwise (Pointwise Operations on Real-Valued Maps (ii), Pointwise Operations on Real-Valued Maps (i), Maps Formed Pointwise from Vector and Matrix Operations (i)). For functions $X_{1},\dots,X_{n}$ on $\Omega$ with values in $\mathbb{R}$, or all with values in $\mathbb{R}^{m}$, $X_{1}+\dots+X_{n}=\sum_{k=1}^{n}X_{k}$ is their finite sum for this pointwise sum, the function $\omega\mapsto\sum_{k=1}^{n}X_{k}(\omega)$ (Iterated Operations: Recursion, Splitting, Reordering, Termwise Combination and Homomorphisms (ii)). For $a,b\in\mathbb{R}$ and a random variable $X$, $aX+b$ is the function $\omega\mapsto aX(\omega)+b$. For a random vector $X$ in $\mathbb{R}^{n}$, $A\in\mathbb{R}^{m\times n}$ and $b\in\mathbb{R}^{m}$, $AX+b$ is the function $\omega\mapsto AX(\omega)+b$ (Maps Formed Pointwise from Vector and Matrix Operations (i)). For $x\in\mathbb{R}^{n}$ and $y\in\mathbb{R}^{r}$, $xy^{\top}\in\mathbb{R}^{n\times r}$ is the matrix with entries

$$(xy^{\top})_{ij}=x_{i}y_{j}\qquad(i\in[n],\ j\in[r])$$

(Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i)), and for random vectors $X$ in $\mathbb{R}^{n}$ and $Y$ in $\mathbb{R}^{r}$, $XY^{\top}$ is the function $\omega\mapsto X(\omega)Y(\omega)^{\top}$.

(v) Expectation. $\mathbb{E}X$, also written $\mathbb{E}[X]$, is the expectation of Expectation (i), Expectation (ii), Expectation (iii) or Expectation (iv), according to whether $X$ is an element of $\mathcal{M}^{+}(\Omega,\mathcal{F})$, an integrable random variable, an integrable random vector or an element of $\mathcal{L}^{1}(P;\mathbb{R}^{k\times l})$ for some $k,l\in\mathbb{N}$. For a random variable $X$, $\mathbb{E}|X|$ denotes the expectation of $|X|\in\mathcal{M}^{+}(\Omega,\mathcal{F})$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (iv)).

(vi) Variance and covariance. For $X,Y\in\mathcal{L}^{2}(P)$, $\operatorname{Cov}(X,Y)$ and $\operatorname{Var}X$ are the covariance and variance of Variance and Covariance (i) and Variance and Covariance (ii), and uncorrelated random variables are as in Variance and Covariance (iii). For $m,n\in\mathbb{N}$, $X\in\mathcal{L}^{2}(P;\mathbb{R}^{n})$ and $Y\in\mathcal{L}^{2}(P;\mathbb{R}^{m})$, $\operatorname{Cov}(X)$ is the covariance matrix of $X$ and $\operatorname{Cov}(X,Y)$ the cross-covariance matrix of $X$ and $Y$, as in Variance and Covariance (iv) and Variance and Covariance (v). Background: Lᵖ Inclusions and Constants on a Finite Measure Space.

(vii) Almost surely. "Almost surely" (a.s.) stands for $P$-almost everywhere.