Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation

This setting fixes the standing notation for Borel probability measures on a real Hilbert space carrying a fixed orthonormal basis. It introduces no new concepts.

1. (Background)¶ The notation and background of The Real Numbers: Standing Notation and Background, Real Hilbert Spaces: Standing Notation and Background, Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus and Probability Measures on Euclidean Space and Random Vectors: Standing Notation are in force, and so is that of Measure Spaces and the Lebesgue Integral: Standing Notation, whose measure space is instantiated at each use by the measure space named there, so that the letter $X$ is free for the use fixed in clause 2.

2. (The space)¶ $X$ is a real Hilbert space, playing the role of $H$ in Real Hilbert Spaces: Standing Notation and Background §space, with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$, distance $d$ and zero vector $0_{X}$ as there (the notation $Y^{n}$ for $n$-tuples of Real Hilbert Spaces: Standing Notation and Background §numbers is read for arbitrary sets $Y$), and $(e_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $X$. The metric space $(X,d)$ is complete by Real Hilbert Space §hilbert and separable by A Real Hilbert Space with an Orthonormal Basis is Separable §separable.

3. (Coordinates)¶ For $x\in X$ and $k\in\mathbb{N}$, $x_{k}=\langle x,e_{k}\rangle$ is the $k$-th coordinate of $x$. For $n\in\mathbb{N}$, the space $\mathbb{R}^{n}$ carries the Euclidean norm $\lVert\cdot\rVert$ and the Euclidean distance $d_{E}$ of Euclidean Space and Lebesgue Measure: Standing Notation §space; $p_{n}:X\to\mathbb{R}^{n}$ denotes the coordinate map $p_{n}(x)=(x_{1},\dots,x_{n})$, and $p_{n}^{*}:\mathbb{R}^{n}\to X$ the synthesis map $p_{n}^{*}(y)=\sum_{k=1}^{n}y_{k}e_{k}$. $X_{n}$ is the span of $e_{1},\dots,e_{n}$; by The Subspaces Spanned by an Orthonormal Sequence and Exhausting Sequences §subspaces the orthogonal projection onto $X_{n}$ is $P_{n}=p_{n}^{*}\circ p_{n}$, by The Subspaces Spanned by an Orthonormal Sequence and Exhausting Sequences §exhausting the sequence $(X_{n})_{n\in\mathbb{N}}$ is exhausting, and $Q_{n}x=x-P_{n}x$, in agreement with Real Hilbert Spaces: Standing Notation and Background §separable.

4. (Borel sets and measures)¶ For a metric space $(Y,d_{Y})$, $\mathcal{B}(Y)$ is its Borel $\sigma$-algebra; this applies to $(X,d)$, to $X\times X$ with the distance of clause 6, and to $\mathbb{R}^{n}$ with the distance $d_{E}$. A map between two such spaces is called Borel if it is measurable with respect to their Borel $\sigma$-algebras. $\mathcal{P}(X)$ denotes the set of Borel measures $\mu$ on $(X,d)$ with $\mu(X)=1$, and likewise $\mathcal{P}(X\times X)$. Integrals against a Borel measure are those of Measure Spaces and the Lebesgue Integral: Standing Notation §integral; the integral of $f$ against $\mu$ is also written $\int_{Y}f(y)\,\mu(dy)$.

5. (Push-forwards)¶ For a Borel map $T$ between two of the spaces of clause 4 and a Borel measure $\mu$ on its domain, $T_{\#}\mu$ is the image measure $B\mapsto\mu(T^{-1}(B))$ of claim 1 of that lemma, with the change of variables formula of its claim 2.

6. (Pairs)¶ $X\times X$ is the product of $X$ with itself, a real Hilbert space by Properties of the Product of Two Real Inner Product Spaces §hilbert, with norm $|\cdot|$, distance $d$ and coordinate maps $\pi_{1},\pi_{2}:X\times X\to X$. For maps $S,T$ from a set $\Omega$ into $X$, $(S,T):\Omega\to X\times X$ is the map $\omega\mapsto(S(\omega),T(\omega))$.

7. (Weak convergence and tightness)¶ For finite Borel measures on $X$, on $X\times X$ or on $\mathbb{R}^{n}$, $\mu_{j}\Rightarrow\mu$ is weak convergence; a set of Borel measures is tight and a sequence of Borel measures is tight as defined there.