Proof of Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs

Each result cited below is universally quantified over the data in its own statement and is applied with the data named at the point of citation.

Throughout, $d(x,x')=|x-x'|$ for $x,x'\in X$ by Real Inner Product Space §distance, as fixed in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space; for $k\in\mathbb{N}$ the coordinate function $x\mapsto x_{k}$ on $X$ is written $c_{k}$; and $\mathcal{G}$ denotes the family of the sets $\{x\in X:x_{k}<s\}$ with $k\in\mathbb{N}$ and $s\in\mathbb{R}$. Generated $\sigma$-algebras occur in the cited items under Generated Sigma-Algebra and under Generated Sigma-Algebra; both definitions describe the same family, namely the subsets of the ambient set that belong to every $\sigma$-algebra containing the given family. By claim 2 of Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra this family is a $\sigma$-algebra, contains the given family, and is contained in every $\sigma$-algebra containing the given family; we refer to this as minimality. The $n$-tuple $(e_{1},\dots,e_{n})$ is orthonormal for every $n\in\mathbb{N}$, because the sequence $(e_{k})_{k\in\mathbb{N}}$ is orthonormal by Orthonormal Basis of a Real Hilbert Space §basis and the conditions of Orthogonality, Orthogonal Complement and Orthonormal Families in a Real Inner Product Space §orthonormal for the tuple are among those for the sequence.

Step 1 (the identities of claim 1). Fix $n\in\mathbb{N}$, $x,x'\in X$ and $y\in\mathbb{R}^{n}$. Since $p_{n}^{*}(y)=\sum_{k=1}^{n}y_{k}e_{k}$, Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §coefficients, applied to the orthonormal tuple $(e_{1},\dots,e_{n})$ and the coefficients $y_{1},\dots,y_{n}$, gives $\langle p_{n}^{*}(y),e_{j}\rangle=y_{j}$ for every $j\in[n]$, that is, $p_{n}(p_{n}^{*}(y))=y$. By Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §norm, applied to the same data, $|p_{n}^{*}(y)|^{2}=\sum_{k=1}^{n}y_{k}^{2}$; since $|p_{n}^{*}(y)|\ge0$ and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square characterises $\lVert y\rVert$ as the unique real $r\ge0$ with $r^{2}=\sum_{k=1}^{n}y_{k}^{2}$, we get $|p_{n}^{*}(y)|=\lVert y\rVert$. Since $P_{n}=p_{n}^{*}\circ p_{n}$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, the last identity with $p_{n}(x)$ in place of $y$ gives $|P_{n}x|=\lVert p_{n}(x)\rVert$. The set $X_{n}$ is a closed linear subspace of $X$ by The Subspaces Spanned by an Orthonormal Sequence and Exhausting Sequences §subspaces, and $P_{n}$ is the orthogonal projection onto it, so Orthogonal Projection onto a Closed Linear Subspace of a Real Hilbert Space §pythagoras, applied with $M=X_{n}$, gives $|x|^{2}=|P_{n}x|^{2}+|x-P_{n}x|^{2}=|P_{n}x|^{2}+|Q_{n}x|^{2}$. By Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates the sequence $(X_{n})_{n\in\mathbb{N}}$ is exhausting, with projections $P_{n}$ and tails $Q_{n}$, so Exhausting Sequences of Finite-Dimensional Subspaces in a Separable Real Hilbert Space, and Their Projections §projections, applied to this exhausting sequence, shows that $P_{n}$ and $Q_{n}$ are linear and that $|P_{n}z|\le|z|$ and $|Q_{n}z|\le|z|$ for every $z\in X$. By additivity and homogeneity of the inner product in its first argument (Real Inner Product Space §inner-product), $\langle x-x',e_{k}\rangle=x_{k}-x'_{k}$ for every $k\in\mathbb{N}$; as the difference of points of $\mathbb{R}^{n}$ is formed componentwise by Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, this says $p_{n}(x)-p_{n}(x')=p_{n}(x-x')$. Hence, by what was just shown with $x-x'$ in place of $x$,

$$\lVert p_{n}(x)-p_{n}(x')\rVert=\lVert p_{n}(x-x')\rVert=|P_{n}(x-x')|\le|x-x'|.$$

Step 2 (Lipschitz bounds and Borel measurability in claim 1). We verify the inequality of Lipschitz Map Between Metric Spaces with constant $1$ for each map. For $k\in\mathbb{N}$ and $x,x'\in X$, the real number $x_{k}-x'_{k}$ is the $k$-th component of $p_{k}(x)-p_{k}(x')$, so Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate (in $\mathbb{R}^{k}$, with $i=k$) and Step 1 give $d_{\mathbb{R}}(x_{k},x'_{k})=|x_{k}-x'_{k}|\le\lVert p_{k}(x)-p_{k}(x')\rVert\le d(x,x')$, the metric $d_{\mathbb{R}}$ being that of Real Hilbert Spaces: Standing Notation and Background §numbers. For $n\in\mathbb{N}$, Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §distance and Step 1 give $d_{E}(p_{n}(x),p_{n}(x'))=\lVert p_{n}(x)-p_{n}(x')\rVert\le d(x,x')$. For $y,y'\in\mathbb{R}^{n}$, additivity and homogeneity of finite sums of vectors (claims 2 and 3 of Properties of Finite Sums of Vectors, the latter with $\lambda=-1$) give $p_{n}^{*}(y)-p_{n}^{*}(y')=\sum_{k=1}^{n}(y_{k}-y'_{k})e_{k}=p_{n}^{*}(y-y')$, so Step 1 and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §distance give $d(p_{n}^{*}(y),p_{n}^{*}(y'))=|p_{n}^{*}(y-y')|=\lVert y-y'\rVert=d_{E}(y,y')$. Finally, by the linearity and the bounds of Step 1, $d(P_{n}x,P_{n}x')=|P_{n}(x-x')|\le|x-x'|=d(x,x')$, and likewise $d(Q_{n}x,Q_{n}x')=|Q_{n}(x-x')|\le d(x,x')$. Each of these maps is therefore Lipschitz with constant $1$, hence continuous by A Lipschitz Map is Uniformly Continuous, hence measurable with respect to the Borel $\sigma$-algebras of its domain and target by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. For $c_{k}$ the target $\sigma$-algebra is that of $(\mathbb{R},d_{\mathbb{R}})$, which is $\mathcal{B}(\mathbb{R})$ by claim 2 of Borel Measurability and Bounded Integration on a Metric Space; so each $c_{k}$ is measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(\mathbb{R})$. This proves claim 1.

Step 3 (claim 3, necessity). Let $(\Omega,\mathcal{F})$ be a measurable space and let $Y:\Omega\to X$ be measurable with respect to $\mathcal{F}$ and $\mathcal{B}(X)$. For $k\in\mathbb{N}$ the function $\omega\mapsto\langle Y(\omega),e_{k}\rangle$ is $c_{k}\circ Y$, and $c_{k}$ is measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(\mathbb{R})$ by Step 2, so $c_{k}\circ Y$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R})$ by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, applied with $(Z,\mathcal{G})=(\mathbb{R},\mathcal{B}(\mathbb{R}))$ and $g=c_{k}$.

Step 4 (claim 3, sufficiency). Let $(\Omega,\mathcal{F})$ and $Y:\Omega\to X$ be as in claim 3, and suppose that for every $k\in\mathbb{N}$ the function $f_{k}(\omega)=\langle Y(\omega),e_{k}\rangle$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R})$. Let $D$ be the set of finite rational combinations $\sum_{k=1}^{n}q_{k}e_{k}$ of A Real Hilbert Space with an Orthonormal Basis is Separable, applied with $H=X$; it is countable and dense in $X$ by A Real Hilbert Space with an Orthonormal Basis is Separable §dense-set, and it is nonempty since $e_{1}\in D$. Thus $((X,d),D)$ is a separable metric datum in the sense of Borel Sets and Measurable Maps in a Separable Metric Space, and its $\mathcal{B}(X,d)$ is $\mathcal{B}(X)$. Fix $q\in D$ and $s\in\mathbb{Q}$ with $0<s$, and write $q_{k}=\langle q,e_{k}\rangle$. For $\omega\in\Omega$, Orthonormal Expansions in a Real Hilbert Space §parseval, applied with $H=X$ and with $Y(\omega)-q$ in place of $x$, together with $\langle Y(\omega)-q,e_{k}\rangle=f_{k}(\omega)-q_{k}$ (additivity and homogeneity of the inner product, Real Inner Product Space §inner-product), shows that the series $\sum_{k=1}^{\infty}(f_{k}(\omega)-q_{k})^{2}$ converges with sum $|Y(\omega)-q|^{2}=d(Y(\omega),q)^{2}$. By Series of Real Numbers §convergent this means that the partial sums $g_{N}(\omega)=\sum_{k=1}^{N}(f_{k}(\omega)-q_{k})^{2}$ converge to $g(\omega)=d(Y(\omega),q)^{2}$ for every $\omega\in\Omega$. Each $f_{k}-q_{k}$ is measurable by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions (constants and sums), its square by claim 3 there (products), and each $g_{N}$ by claim 2 there (finite sums); hence $g$ is measurable by claim 5 there (pointwise limits), and so is $-g$ by claim 2 there. Since $d(Y(\omega),q)\ge0$ and $s>0$, we have $d(Y(\omega),q)<s$ if and only if $g(\omega)<s^{2}$, if and only if $-g(\omega)>-s^{2}$; therefore

$$\{\omega\in\Omega:d(Y(\omega),q)<s\}=\{\omega\in\Omega:-g(\omega)>-s^{2}\}\in\mathcal{F}$$

by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line, applied to the measurable function $-g$. As $q\in D$ and $s$ were arbitrary, claim 3 of Borel Sets and Measurable Maps in a Separable Metric Space, applied to the datum $((X,d),D)$, shows that $Y$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(X)$. Together with Step 3 this proves claim 3.

Step 5 (claim 2). Write $\sigma(\mathcal{G})$ for the $\sigma$-algebra generated by $\mathcal{G}$. First, every member of $\mathcal{G}$ belongs to $\mathcal{B}(X)$: for $k\in\mathbb{N}$ and $s\in\mathbb{R}$ the function $-c_{k}$ is measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(\mathbb{R})$ by Step 2 and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and $\{x\in X:x_{k}<s\}=\{x\in X:-x_{k}>-s\}\in\mathcal{B}(X)$ by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line. Since $\mathcal{B}(X)$ is a $\sigma$-algebra (it is a generated $\sigma$-algebra by Borel Sigma-Algebra of a Metric Space), minimality gives $\sigma(\mathcal{G})\subseteq\mathcal{B}(X)$. Conversely, $\sigma(\mathcal{G})$ is a $\sigma$-algebra by minimality, and each $c_{k}$ is measurable with respect to $\sigma(\mathcal{G})$ and $\mathcal{B}(\mathbb{R})$: for real $a$, a real $t$ satisfies $t>a$ if and only if $t\ge a+\frac{1}{m}$ for some $m\in\mathbb{N}$, so

$$\{x\in X:x_{k}>a\}=\bigcup_{m\in\mathbb{N}}\Bigl(X\setminus\bigl\{x\in X:x_{k}<a+\tfrac{1}{m}\bigr\}\Bigr),$$

which belongs to $\sigma(\mathcal{G})$ by properties 2 and 3 of Sigma-Algebra and Measurable Space, and claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line applies. Now apply Step 4 with $(\Omega,\mathcal{F})=(X,\sigma(\mathcal{G}))$ and $Y$ the identity map of $X$, for which $\langle Y(x),e_{k}\rangle=c_{k}(x)$: the identity map is measurable with respect to $\sigma(\mathcal{G})$ and $\mathcal{B}(X)$, that is, every $B\in\mathcal{B}(X)$ belongs to $\sigma(\mathcal{G})$. Hence $\mathcal{B}(X)=\sigma(\mathcal{G})$, which is claim 2.

Step 6 (claim 4). Let $\mathcal{C}$ be the family of the sets $p_{n}^{-1}(B)$ with $n\in\mathbb{N}$ and $B\in\mathcal{B}(\mathbb{R}^{n})$, the Borel $\sigma$-algebra of $(\mathbb{R}^{n},d_{E})$ fixed in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures. Each member of $\mathcal{C}$ belongs to $\mathcal{B}(X)$, since $p_{n}$ is Borel by Step 2.

First, $\mathcal{C}$ is a $\pi$-system in the sense of Dynkin's Pi-Lambda Theorem. It is nonempty, as $X=p_{1}^{-1}(\mathbb{R}^{1})$. Let $n,m\in\mathbb{N}$ with $n\le m$ and put $r=p_{n}\circ p_{m}^{*}:\mathbb{R}^{m}\to\mathbb{R}^{n}$. The map $p_{m}^{*}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{m})$ and $\mathcal{B}(X)$, and $p_{n}$ with respect to $\mathcal{B}(X)$ and $\mathcal{B}(\mathbb{R}^{n})$, by Step 2, so $r$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{m})$ and $\mathcal{B}(\mathbb{R}^{n})$ by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, applied with $(\Omega,\mathcal{F})=(\mathbb{R}^{m},\mathcal{B}(\mathbb{R}^{m}))$, $Y=p_{m}^{*}$ and $g=p_{n}$. For $x\in X$, $p_{m}^{*}(p_{m}(x))=\sum_{k=1}^{m}x_{k}e_{k}$, whose $j$-th coordinate is $x_{j}$ for every $j\in[m]$ by Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §coefficients; as $[n]\subseteq[m]$, this gives $r(p_{m}(x))=p_{n}(x)$. Hence $p_{n}^{-1}(B)=p_{m}^{-1}(r^{-1}(B))$ for $B\in\mathcal{B}(\mathbb{R}^{n})$, and for $B'\in\mathcal{B}(\mathbb{R}^{m})$

$$p_{n}^{-1}(B)\cap p_{m}^{-1}(B')=p_{m}^{-1}\bigl(r^{-1}(B)\cap B'\bigr),$$

where $r^{-1}(B)\cap B'\in\mathcal{B}(\mathbb{R}^{m})$ because $\mathcal{B}(\mathbb{R}^{m})$ is closed under finite intersections by Sigma-Algebra and Measurable Space. Since intersection is symmetric, the intersection of any two members of $\mathcal{C}$ lies in $\mathcal{C}$.

Second, the $\sigma$-algebra generated by $\mathcal{C}$ is $\mathcal{B}(X)$. It is contained in $\mathcal{B}(X)$ by minimality. Conversely, let $k\in\mathbb{N}$ and $s\in\mathbb{R}$, and let $U=\{y\in\mathbb{R}^{k}:y_{k}<s\}$. If $y\in U$ and $t=s-y_{k}$, then $t>0$, and every $y'\in\mathbb{R}^{k}$ with $d_{E}(y,y')<t$ satisfies $y'_{k}-y_{k}\le\lVert y'-y\rVert=d_{E}(y,y')<t$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §distance, hence $y'_{k}<s$; so $B_{d_{E}}(y,t)\subseteq U$, $U$ is open in $(\mathbb{R}^{k},d_{E})$ by Open Subset of a Metric Space, and $U\in\mathcal{B}(\mathbb{R}^{k})$ by claim 1 of Borel Measurability and Bounded Integration on a Metric Space. Since $\{x\in X:x_{k}<s\}=p_{k}^{-1}(U)$, every member of $\mathcal{G}$ lies in $\mathcal{C}$; by Step 5 and minimality, $\mathcal{B}(X)=\sigma(\mathcal{G})$ is contained in the $\sigma$-algebra generated by $\mathcal{C}$.

Third, let $\mu,\nu$ be as in claim 4. They are measures on $(X,\mathcal{B}(X))$ by Borel Measure on a Metric Space, and for $n\in\mathbb{N}$ and $B\in\mathcal{B}(\mathbb{R}^{n})$, by the definition of the push-forward in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward,

$$\mu(p_{n}^{-1}(B))=((p_{n})_{\#}\mu)(B)=((p_{n})_{\#}\nu)(B)=\nu(p_{n}^{-1}(B)).$$

So $\mu$ and $\nu$ agree on the $\pi$-system $\mathcal{C}$, which generates $\mathcal{B}(X)$, and $\mu(X)=\nu(X)<\infty$; claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law, applied with $(X,\mathcal{S})=(X,\mathcal{B}(X))$ and $\mathcal{P}=\mathcal{C}$, gives $\mu=\nu$. This proves claim 4.

Step 7 (claim 5). Let $\rho$ be the product metric on $X\times X$, $\rho((x,x'),(z,z'))=\max\{d(x,z),d(x',z')\}$. By Properties of the Product of Two Real Inner Product Spaces §metric, applied with $E_{1}=E_{2}=X$, the metric spaces $(X\times X,d)$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs and $(X\times X,\rho)$ have the same open subsets, so by Borel Sigma-Algebra of a Metric Space they have the same Borel $\sigma$-algebra $\mathcal{B}(X\times X)$. Let $D$ be as in Step 4. By claim 4 of Borel Sets and Measurable Maps in a Separable Metric Space, applied to the data $((X,d),D)$ and $((X,d),D)$, the pair $((X\times X,\rho),D\times D)$ is a separable metric datum, so by claim 2 of that lemma $\mathcal{B}(X\times X)$ is generated by the open balls $B_{\rho}((q,q'),s)$ with $q,q'\in D$ and $s\in\mathbb{Q}$, $s>0$. By Open Ball in a Metric Space and Product Metric on the Cartesian Product of Two Metric Spaces, and since a maximum of two reals is less than $s$ exactly when both are (by claims 1 and 2 of Elementary Properties of the Maximum of Two Elements: the maximum is at least each of the two and equals one of them),

$$B_{\rho}((q,q'),s)=B_{d}(q,s)\times B_{d}(q',s).$$

By claim 2 of Borel Sets and Measurable Maps in a Separable Metric Space, applied to $((X,d),D)$, the balls $B_{d}(q,s)$ and $B_{d}(q',s)$ belong to $\mathcal{B}(X)$; hence each generating ball is a measurable rectangle in the sense of Product Sigma-Algebra and lies in $\mathcal{B}(X)\otimes\mathcal{B}(X)$. By minimality, $\mathcal{B}(X\times X)\subseteq\mathcal{B}(X)\otimes\mathcal{B}(X)$.

For the converse and for the Borel measurability of $\pi_{1},\pi_{2}$: by Properties of the Product of Two Real Inner Product Spaces §coordinates, $\pi_{i}$ is linear with $|\pi_{i}w|\le|w|$ for $w\in X\times X$, so $d(\pi_{i}z,\pi_{i}z')=|\pi_{i}(z-z')|\le|z-z'|=d(z,z')$ for $z,z'\in X\times X$. Thus $\pi_{i}$ is Lipschitz with constant $1$ (Lipschitz Map Between Metric Spaces), continuous by A Lipschitz Map is Uniformly Continuous, and Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. For $A,B\in\mathcal{B}(X)$ we have $A\times B=\pi_{1}^{-1}(A)\cap\pi_{2}^{-1}(B)\in\mathcal{B}(X\times X)$, by closure of the $\sigma$-algebra $\mathcal{B}(X\times X)$ under finite intersections (Sigma-Algebra and Measurable Space). By minimality, $\mathcal{B}(X)\otimes\mathcal{B}(X)\subseteq\mathcal{B}(X\times X)$. This proves claim 5.

Step 8 (claim 6). Let $(\Omega,\mathcal{F})$ and $S,T$ be as in claim 6. By claim 4 of Borel Sets and Measurable Maps in a Separable Metric Space, applied to the data $((X,d),D)$ and $((X,d),D)$ of Step 7 with $Y=S$ and $Y'=T$, the map $\omega\mapsto(S(\omega),T(\omega))$, which is $(S,T)$, is measurable with respect to $\mathcal{F}$ and the Borel $\sigma$-algebra of $(X\times X,\rho)$, which equals $\mathcal{B}(X\times X)$ by Step 7. This proves claim 6.

Step 9 (claim 7). Let $\mu,\nu$ be Borel measures on $(X,d)$ with $\mu(X)<\infty$ and $\nu(X)<\infty$. Each is $\sigma$-finite in the sense of Measure, Measure Space, and Probability Measure, as witnessed by the constant sequence with every term $X$. By Existence and Uniqueness of the Product Measure, applied to the measure spaces $(X,\mathcal{B}(X),\mu)$ and $(X,\mathcal{B}(X),\nu)$, the product measure $\mu\otimes\nu$ is a measure on $\mathcal{B}(X)\otimes\mathcal{B}(X)$ with $(\mu\otimes\nu)(A\times B)=\mu(A)\nu(B)$ for all $A,B\in\mathcal{B}(X)$. By Step 7, $\mathcal{B}(X)\otimes\mathcal{B}(X)=\mathcal{B}(X\times X)$, so $\mu\otimes\nu$ is a Borel measure on $X\times X$ by Borel Measure on a Metric Space. Since $\pi_{1},\pi_{2}$ are Borel by Step 7, the push-forwards of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward are defined, and for $A\in\mathcal{B}(X)$, using $\pi_{1}^{-1}(A)=A\times X$ and $\pi_{2}^{-1}(A)=X\times A$,

$$((\pi_{1})_{\#}(\mu\otimes\nu))(A)=(\mu\otimes\nu)(A\times X)=\mu(A)\,\nu(X),\qquad((\pi_{2})_{\#}(\mu\otimes\nu))(A)=(\mu\otimes\nu)(X\times A)=\mu(X)\,\nu(A).$$

As $\nu(X)$ and $\mu(X)$ are real, these identities say $(\pi_{1})_{\#}(\mu\otimes\nu)=\nu(X)\,\mu$ and $(\pi_{2})_{\#}(\mu\otimes\nu)=\mu(X)\,\nu$. This proves claim 7.