The Cylindrical Sigma-Algebra

Published

22 April 2025

Cylindrical sets and their associated $\sigma$-algebra are a tool for doing measure theory on infinite-dimensional spaces. The basic philosophy is that we can attempt to do everything in terms of a projection onto some finite-dimensional set.

Throughout this note, $X$ is an arbitrary real normed vector space. For a collection of functionals $(x_k^*)_{k=1}^n$ we will use $(x_1^* \times x_2^* \times \dots \times x_n^*)$ to represent the mapping $X \to \R^n$ obtained by applying each functional to $x \in X$ and stacking, i.e., \[ x \mapsto (\langle x_1^*, x \rangle, \dots, \langle x_n^*, x \rangle)^{\sf T}. \]

Cylindrical Sets

Fix $n \in \Z_{\geq 0}$,, a collection $(x_k^*)_{k=1}^n \subset X^*$, and a measurable Borel set $A_n \in \mathcal{B}(\R^n)$. The set \[ \hat{A}_n = \left\{ x \in X : [\langle x_1^*, x \rangle, \dots, \langle x_n^*, x \rangle]^{\sf T} \in A_n \right\} \] is called a cylinder with base $A_n$. The class of all cylinders is denoted by \[ \text{Cyl}(X) = \left\{ A \in X : A = \hat{A}_n \, \text{for some } \, A_n \right\} \]

In other words, a cylinder with base $A_n$ is the preimage of the mapping $(x_1^* \times \dots \times x_n^*): X \to \R^n$. Loosely speaking, we can view this mapping as taking $n$ projections (or coordinates) of $x \in X$. A cylinder set, then, is defined by vectors satisfying finitely many constraints. The canonical example to keep in mind is $X = \ell^2$ and $x_k^* = \langle e_k, \cdot \rangle$ is projection onto the $k$th coordinate.

It is important to note that a given $A \in \text{Cyl}(X)$ generally has many representatives. For instance, if $A = \hat{A}_n$, then we also have that $A = (x_1^* \times \cdots \times x_n^* \times x_{n+1}^*)^{-1}(A_n \times \R)$ for any $x_{n+1}^* \in X^*$.

The Cylindrical $\sigma$-Algebra

In the following proposition, we show that $\text{Cyl}(X)$ is unforunately not a $\sigma$-algebra. Thus, for the purposes of doing measure theory, it will be necessary to work with $\CC = \sigma\{ \text{Cyl}(X) \}$, the cylindrical $\sigma$-algebra.

The class $\text{Cyl}(X)$ is an algebra of sets. If $\text{dim} X = \infty$, then it is not a $\sigma$-algebra.

Proof.

The set $\text{Cyl}(X)$ clearly contains $X$. If $A \in \text{Cyl}(X)$, then $A = \hat{A}_n$ for some base set $A_n \in \R^n$ and linear functionals $(x_k^*)_{k=1}^n$. Similarly, for $B \in \text{Cyl}(X)$, we have $B = \hat{B}_m$ for some $B_m \in \R^m$ and $(y_k^*)_{k=1}^m$. Then, \begin{align} X \setminus A &= \left\{x \in X : (x_1^* \times \dots \times x_n^*)(x) \in A_n^c \right\} \end{align} \begin{align} A \cup B = \left\{ x : (x_1^* \times \dots \times x_n^* \times y_1^* \times \dots \times y_m^*)(x) \in (A_n \times \R^m) \cup (\R^n \times B_m) \right\}. \end{align} This shows that $\text{Cyl}(X)$ is an algebra.

Conversely, suppose that $\text{dim}X = \infty$. It follows that the dual space is also infinite dimensional, and thus we can find a linearly independent sequence $(x_k^*)_{k=1}^\infty \subset X^*$. Let $S = \cap_{k=1}^\infty \text{ker}(x_k^*)$. Observe that each of the $\text{ker}(x_k^*)$'s is a cylindrical set. However, $S$ is not cylindrical. To show this, suppose for the sake of contradiction that $S$ takes the form \begin{equation} S = \left\{ x \in X : (z_1^* \times \dots \times z_m^*)(x) \in B_m \right\}. \end{equation} Let $K = \cap_{k=1}^m \text{ker}(z_k^*)$. We have that $K \subseteq S$. This is because if $x \in S$ and $z \in K$, then $x+z \in S$, and the fact that $0 \in S$. Thus, for any $n$, we see that $K \subseteq \text{ker}(x_n^*)$. A well-known theorem of linear algebra then implies that $x_n^* \in \text{span}(z_1^*, \dots, z_m^*)$. Once $n > m$, this contradicts the assumption that $(x_k^*)_{k=1}^\infty$ is linearly independent. Thus, $S$ cannot be cylindrical.

In fact, one of the main reasons to work with $\CC$ is that it coincides with the Borel $\sigma$-algebra when $X$ is separable. It is worth appreciating how powerful this is – studying the Borel sets on a separable space can be reduced to studying the much simpler cylindrical sets.

Suppose $(x_n) \subset X$ is a dense set in $X$. Then, there exists a collection $(x_n^*) \subset X^*$ with $|x_n^*| = 1$ and $\langle x_n , x_n^* \rangle = |x_n|$. Moreover, for any $x \in X$, we have \[ |x| = \sup_{n} | \langle x, x_n^* \rangle |. \]

Proof.

The existence of such a sequence $(x_n^*)_{n=1}^\infty$ is a well-known consequence of the Hahn-Banach theorem. For the second claim, note that $| \langle x_n^*, x \rangle| \leq |x|$ and so the supremum is at most $|x|$. On the other hand, thanks to density, we can find a sequence in $(x_n)_{n=1}^\infty$ converging to $x$. By the reverse triangle inequality, \begin{align} | \langle x_n^*, x \rangle| &\geq |\langle x_n, x_n^* \rangle| - |\langle x_n - x, x_n^* \rangle| \\ &\geq |x_n| - |x_n - x|. \end{align} Since the supremum is at most the limit, we conclude.

Let $X$ be a separable real normed vector space. Then, the cylindrical $\sigma$-algebra $\CC$ coincides with the Borel $\sigma$-algebra $\BB$.

Proof.

We first show $\CC \subseteq \BB$. Fix $A \in \CC$, so that $A = (x_1^* \times \dots \times x_n^*)^{-1}(A_n)$ for some Borel $A_n \in \R^n$. It is clear enough that the mapping $(x_1^* \times \dots \times x_n^*): X \to \R^n$ is continuous. In particular, it is then Borel measurable, and thus $A \in \BB$.

Conversely, recall that since $X$ is separable, $\BB$ is generated by the closed balls in $X$. Moreover, there exists a countable dense subset $(x_n)_{n=1}^\infty \subset X$. By the previous lemma, we obtain a corresponding collection of norming functionals $(x_n^*)_{n=1}^\infty$. For some fixed $x_0 \in X$ and $r > 0$, note that the previous lemma shows that \[ |x - x_0| \leq r \iff |\langle x - x_0, x_n^* \rangle| \leq r \quad \forall n. \] In other words, we have that \[ \overline{B}(x_0, r) = \bigcap_{n=1}^\infty \left\{ x \in X : \langle x - x_0, x_n^* \rangle \leq r \right\}. \] Thus we see that $\overline{B}(x_0, r) \in \CC$ is actually cylindrical. It follows that $\BB \subseteq \CC$.

The key lesson to remember in the above proof is that we’re trying to show that closed balls are cylindrical, i.e., expressed in terms of some finite collection of functionals. In separable spaces, we can take a “norming net” of functionals to accomplish this.

Cylindrical sets give us a tool for studying measures through their finite-dimensional projections, as the following lemma illustrates.

Suppose $X$ is separable and $\mu, \nu \in \P(X)$ are Borel probability measures. If $\mu$ and $\nu$ agree on $\text{Cyl}(X)$, then $\mu = \nu$.

Proof.

Since $X$ is separable, $\BB(X)$ is generated by the cylinder sets, and thus the result follows immediately from the Caratheodory extension theorem.

Hilbert Setting

When $X = H$ is a separable Hilbert space, we can give a somewhat more concrete construction. Here, the idea is to explicitly choose an orthonormal basis which will play the role of a coordinate system from which to do projections.

To that end, let $(e_k)_{k=1}^\infty$ be an orthonormal basis in $H$. We may consider the cylinder sets arising from maps of the form $x \mapsto (\langle x, e_1 \rangle, \dots, \langle x, e_n \rangle)^{\sf T}$. Note this is a special case of what we have previously introduced, fixing the collection of projection functionals to be those arising from the first $n$ basis elements. In this way, we obtain a sub-class of cylindrical sets $\text{Cyl}’$. The following result is a straightforward adaptation of what we have already seen.

Let $H$ be a separable Hilbert space with orthonormal basis $(e_k)_{k=1}^\infty$ and $\text{Cyl}'$ be the algebra of cylindrical sets arising from maps of the form $x \mapsto (\langle x, e_1 \rangle, \dots, \langle x, e_n \rangle)^{\sf T}$. Then, \[ \sigma\left( \text{Cyl}' \right) = \BB(H). \]

As above, we can study measures on $H$ via their projections onto a given orthonormal basis. The proof is identical.

Let $H$ be a separable Hilbert space and $\mu, \nu \in \P(H)$ be Borel probability measures. Fix an orthonormal basis $(e_k)_{k=1}^\infty$ for $H$. Let $P_n$ be the orthonormal projection onto $\text{span}\{e_1, \dots, e_n\}$. If $(P_n)_{\#}\mu = (P_n)_{\#}\nu$ for all $n \geq 1$, then $\mu = \nu$.

Characteristic Functionals

As an illustrative application, we will see how to use cylindrical sets to reduce the study of infinite-dimensional characteristic functionals down to the finite-dimensional case. In this way, we can sometimes “lift” finite-dimensional results into the infinite-dimensional setting.

Let $\mu \in \P(X)$ be Borel probability measure. The functional $\varphi_\mu: X^* \to \C$ given by \[ \varphi_\mu(x^*) = \int_X \exp \left( i \langle x^*, x \rangle \right) d \mu(x) \] is called the characteristic functional of $\mu$.

We show that the data encoded by the characteristic functional is equivlant to that of the original measure. The basic structure of the proof is to show that all cylindrical projections of the measures agree, and thus the measures agree.

Suppose $X$ is separable and $\mu, \nu \in \P(X)$ are Borel probability measures such that $\varphi_\mu = \varphi_\nu$ for every $x^* \in X$. Then, $\mu = \nu$.

Proof.

We take it for granted that the result holds for $X = \R^n$. Fix $\{x_1^*, \dots, x_n^* \} \subset X^*$ and $\alpha \in \R^n$. Let $T: X \to \R^n$ be given by $T(x) = (\langle x_1^*, x \rangle, \dots, \langle x_n^*, x \rangle)^{\sf T}$. Consider the pushforward measure $\mu_n \in \BB(\R^n)$ given by $\mu_n = T_{\#}\mu$. Observe further that if $A_n \in \R^n$, the cylindrical set $\hat{A}_n = T^{-1}(A_n)$ satisfies \[ \mu(\hat{A}_n) = \mu(T^{-1} A_n) = T_{\#}\mu(A_n) = \mu_n(A_n). \] Now, observe that \[ \varphi\left(\sum_{k=1}^n \alpha_k x_k^* \right) = \int \exp \left( i \alpha^{\sf T} T(x) \right) d \mu(x) = \varphi_{\mu_n}(\alpha). \] An analogous calculation for $\nu$ shows that $T_{\#} \mu = T_{\#} \nu$ for any such choice of $T$. This is equivalent to saying that $\mu$ and $\nu$ agree on the cylinder sets by our previous observations, and thus $\mu = \nu$.

References.

Kukush, A. (2020). Gaussian measures in Hilbert space: Construction and properties, Chapter 2, 4.