Probability Measures on Euclidean Space and Random Vectors: Standing Notation

This setting fixes the standing notation used by results about probability measures on Euclidean space, random vectors and their laws. It is layered on Euclidean Space and Lebesgue Measure: Standing Notation and Measure Spaces and the Lebesgue Integral: Standing Notation, whose notation is in force throughout, the measure space $(X,\mathcal{F},\mu)$ of the latter being instantiated by whichever measure space a clause names, so that the letter $X$ is free for other uses below; it introduces no new concept and asserts nothing beyond the identifications recorded below, each of which is justified by the reference attached to it.

1. (Euclidean spaces and Borel sets)¶ As in Euclidean Space and Lebesgue Measure: Standing Notation §space, $q$ and $p$ denote natural numbers with $1\le q$ and $1\le p$, and the notation below is introduced for all such dimensions simultaneously. Points of $\mathbb{R}^{q}$ are read as $q$-tuples of real numbers by Euclidean Points as Tuples of Real Numbers, the $k$th component of $x$ being written $x_{k}$. $\mathcal{B}(\mathbb{R}^{q})$ is the Borel $\sigma$-algebra of Euclidean Space and Lebesgue Measure: Standing Notation §borel; by The Borel $\sigma$-Algebras of Euclidean Space and of the Euclidean Metric Coincide it is the Borel $\sigma$-algebra of the metric space $(\mathbb{R}^{q},d_{E})$, and by claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets it is the $\sigma$-algebra $\mathcal{B}_{q}$ of that lemma, whose claims are thereby in force for it. For nonnegative real $t$, $\sqrt{t}$ denotes the nonnegative square root of $t$, and $2=1+1$.

2. (Borel maps)¶ A map from $\mathbb{R}^{q}$ to $\mathbb{R}^{p}$, to $\mathbb{R}$ or to $[0,\infty]$ is Borel in the sense fixed in the preamble of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets, namely measurable with respect to $\mathcal{B}(\mathbb{R}^{q})$ and the Borel $\sigma$-algebra of the target, respectively measurable in the sense of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable for the target $[0,\infty]$. As recorded there, a map $\mathbb{R}^{q}\to\mathbb{R}^{p}$ is Borel exactly when each of its components is (claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets), a map into $\mathbb{R}$ that is continuous for the Euclidean distance and the absolute-value metric is Borel (claim 3 there, with The Euclidean Distance on the Real Line is the Absolute Value Metric), and a composition of Borel maps is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space.

3. (Probability measures and integrals)¶ $\mathcal{P}(\mathbb{R}^{q})$ denotes the set of probability measures on $(\mathbb{R}^{q},\mathcal{B}(\mathbb{R}^{q}))$; by clause 1 each $\mu\in\mathcal{P}(\mathbb{R}^{q})$ is a Borel measure on $(\mathbb{R}^{q},d_{E})$ with $\mu(\mathbb{R}^{q})=1$. For $\mu\in\mathcal{P}(\mathbb{R}^{q})$, the integral $\int_{\mathbb{R}^{q}}f\,d\mu$ of a Borel $f:\mathbb{R}^{q}\to[0,\infty]$, and of a Borel $f:\mathbb{R}^{q}\to\mathbb{R}$ that is integrable with respect to $\mu$, is that of Measure Spaces and the Lebesgue Integral: Standing Notation §integral for the measure space $(\mathbb{R}^{q},\mathcal{B}(\mathbb{R}^{q}),\mu)$, and is also written $\int_{\mathbb{R}^{q}}f(x)\,\mu(dx)$; a nonnegative Borel $f:\mathbb{R}^{q}\to\mathbb{R}$ is integrated as a $[0,\infty]$-valued map, the two readings of measurability agreeing by Measure Spaces and the Lebesgue Integral: Standing Notation §measurable. Every bounded Borel $f:\mathbb{R}^{q}\to\mathbb{R}$ is integrable with respect to every $\mu\in\mathcal{P}(\mathbb{R}^{q})$, by claim 6 of Borel Measurability and Bounded Integration on a Metric Space.

4. (Push-forward)¶ For $\mu\in\mathcal{P}(\mathbb{R}^{q})$ and a Borel map $T:\mathbb{R}^{q}\to\mathbb{R}^{p}$, $T_{\#}\mu$ denotes the image measure of $\mu$ under $T$, given by $T_{\#}\mu(B)=\mu(T^{-1}(B))$ for $B\in\mathcal{B}(\mathbb{R}^{p})$ and called the push-forward of $\mu$ by $T$; it belongs to $\mathcal{P}(\mathbb{R}^{p})$ by claim 1 of that lemma, and the change-of-variables formula of its claim 2 is in force: $\int_{\mathbb{R}^{p}}g\,d(T_{\#}\mu)=\int_{\mathbb{R}^{q}}g\circ T\,d\mu$ for every Borel $g:\mathbb{R}^{p}\to[0,\infty]$, and for every Borel $g:\mathbb{R}^{p}\to\mathbb{R}$ that is integrable with respect to $T_{\#}\mu$, which holds exactly when $g\circ T$ is integrable with respect to $\mu$.

5. (Pairs of points, projections and product measures)¶ $\iota^{q,p}:\mathbb{R}^{q}\times\mathbb{R}^{p}\to\mathbb{R}^{q+p}$ is the concatenation map, a bijection; $\mathrm{pr}^{q,p}_{1}:\mathbb{R}^{q+p}\to\mathbb{R}^{q}$ and $\mathrm{pr}^{q,p}_{2}:\mathbb{R}^{q+p}\to\mathbb{R}^{p}$ are the coordinate projections, which are Borel; for a measurable space $(E,\mathcal{E})$ and measurable $u:E\to\mathbb{R}^{q}$, $v:E\to\mathbb{R}^{p}$, $(u,v)$ is their pairing, measurable with respect to $\mathcal{E}$ and $\mathcal{B}(\mathbb{R}^{q+p})$; and for $\mu\in\mathcal{P}(\mathbb{R}^{q})$ and $\nu\in\mathcal{P}(\mathbb{R}^{p})$, $\mu\boxtimes\nu\in\mathcal{P}(\mathbb{R}^{q+p})$ is the product measure on $\mathbb{R}^{q+p}$, the image under $\iota^{q,p}$ of the product measure $\mu\otimes\nu$, with the properties recorded there. When the two summands are equal, $q=p$, we write $\mathbb{R}^{q+q}$, $\iota=\iota^{q,q}$, $\mathrm{pr}_{1}=\mathrm{pr}^{q,q}_{1}$ and $\mathrm{pr}_{2}=\mathrm{pr}^{q,q}_{2}$; the maps $x\mapsto\lVert x\rVert^{2}$ on $\mathbb{R}^{q}$ and $z\mapsto\mathrm{pr}_{1}(z)\cdot\mathrm{pr}_{2}(z)$, $z\mapsto\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert^{2}$ on $\mathbb{R}^{q+q}$ are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, together with the elementary inequalities recorded there.

6. (Probability space)¶ $(\Omega,\mathcal{F},P)$ denotes a probability space; event and random variable (a real-valued one) are as defined there, together with the shorthand $\{V\in B\}$, $P(V\in B)$ fixed there. For a random variable $V$, $\mathbb{E}[V]$ is its expectation, an element of $[0,\infty]$ when $V$ is nonnegative and a real number when $V$ has finite expectation, that is, is integrable with respect to $P$; for a map $V:\Omega\to[0,\infty]$ that is measurable in the sense of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable we also write $\mathbb{E}[V]=\int_{\Omega}V\,dP\in[0,\infty]$, the integral of Measure Spaces and the Lebesgue Integral: Standing Notation §integral, which agrees with the expectation of a nonnegative random variable. An event occurs, and a property of points of $\Omega$ holds, almost surely in the sense fixed there. Square-integrable random variables, their mean-square inner product $\langle V,W\rangle_{2}=\mathbb{E}[VW]$ and norm $\lVert V\rVert_{2}$ are those of Square-Integrable Random Variables and the Mean-Square Inner Product.

7. (Background)¶ The following results are in force by reference: Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets, Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space, The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, Borel Measurability and Bounded Integration on a Metric Space, Image Measures, Measures with Densities, and Change of Variables, Basic Properties of a Measure, Markov's and Chebyshev's Inequalities and Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm.