Cylinder Sets
1 Cylinder Sets
Cylinder sets are the basic building blocks of product σ-algebras on infinite products such as \(\mathbb{R}^T\), the space of all functions \(T\to\mathbb{R}\). A cylinder set restricts only finitely many coordinates and leaves the rest free. They are how one builds a measure on a space of paths from finite-dimensional information, as in Stochastic Process.
2 Definition
Let \(\{(X_t,\mathcal{M}_t)\}_{t\in T}\) be measurable spaces, \(X=\prod_{t\in T}X_t\), and for \(S\subseteq T\) write \(\pi_S:X\to X_S:=\prod_{t\in S}X_t\) for the restriction \(\pi_S(x)=(x_t)_{t\in S}\).
For a finite set \(S=\{t_1,\dots,t_n\}\subseteq T\) and \(B\in\bigotimes_{t\in S}\mathcal{M}_t\), the cylinder set with base \(B\) over \(S\) is \[ C_S(B):=\pi_S^{-1}(B)=\{x\in X:(x_{t_1},\dots,x_{t_n})\in B\}. \] If \(B=E_{t_1}\times\cdots\times E_{t_n}\) with \(E_{t_i}\in\mathcal{M}_{t_i}\) it is a measurable rectangle cylinder, and if \(n=1\) it is a thin cylinder \(\pi_t^{-1}(E)=\{x:x_t\in E\}\).
For infinitely many fair coin tosses, \(X=\{0,1\}^{\mathbb{N}}\), the event “the first toss is heads and the third is tails” is the cylinder \(\{x:x_1=1,\ x_3=0\}\). The event “heads occurs infinitely often” is not a cylinder, but is built from them using countable unions and intersections (see Limit Supremum & Limit Infimum).
A cylinder over \(S\) with base \(B\) is also a cylinder over any finite \(S'\supseteq S\), with base \(B\times X_{S'\setminus S}\). This is the “padding” used to compare two cylinders on different coordinates.
3 Cylinders Form an Algebra
The collection \(\mathcal{C}\) of cylinder sets is an algebra of subsets of \(X\).
Proof. \(X=\pi_S^{-1}(X_S)\) is a cylinder, and \(\emptyset\) likewise. For the complement, \(C_S(B)^c=\pi_S^{-1}(B)^c=\pi_S^{-1}(B^c)=C_S(B^c)\) by preimages commuting with complements. For \(C_S(B)\) and \(C_{S'}(B')\) pad both to \(S''=S\cup S'\) with bases \(\tilde B,\tilde B'\). Then \(C_S(B)\cup C_{S'}(B')=\pi_{S''}^{-1}(\tilde B\cup\tilde B')\in\mathcal{C}\), again as preimages commute with unions.
In general \(\mathcal{C}\) is not a σ-algebra when \(T\) is infinite, since countable unions can involve infinitely many coordinates.
4 Generating the Product σ-Algebra
\(\sigma(\mathcal{C})=\bigotimes_{t\in T}\mathcal{M}_t\). In fact the thin cylinders, or the measurable rectangle cylinders, already generate it.
Proof. The product σ-algebra is by definition generated by the thin cylinders \(\pi_t^{-1}(E)\), \(E\in\mathcal{M}_t\), which are cylinders, so \(\bigotimes\mathcal{M}_t\subseteq\sigma(\mathcal{C})\). Conversely \(\pi_S:X\to X_S\) is measurable for the product σ-algebras: the generators of \(\bigotimes_{t\in S}\mathcal{M}_t\) are the coordinate sets \(E\) in a single coordinate \(t\in S\), whose preimage under \(\pi_S\) is the thin cylinder \(\pi_t^{-1}(E)\). By checking measurability on generators (here) every \(\pi_S^{-1}(B)\) lies in \(\bigotimes\mathcal{M}_t\), so \(\mathcal{C}\subseteq\bigotimes\mathcal{M}_t\).
For uncountable \(T\) the product σ-algebra is much smaller than one might expect.
Every set in \(\bigotimes_{t\in T}\mathcal{M}_t\) is determined by countably many coordinates: for each such \(A\) there is a countable \(S\subseteq T\) and \(B\in\bigotimes_{t\in S}\mathcal{M}_t\) with \(A=\pi_S^{-1}(B)\).
Proof. Let \(\mathcal{A}\) be the collection of sets determined by countably many coordinates. It contains every cylinder, is closed under complements (same \(S\)), and under countable unions (take the union of the countably many coordinate sets). So \(\mathcal{A}\) is a σ-algebra containing the generators, hence contains \(\bigotimes\mathcal{M}_t\).
On \(\mathbb{R}^{[0,1]}\) with the product σ-algebra, the sets \(C([0,1])\) of continuous paths, \(\{x:\sup_tx_t<1\}\) and the singleton \(\{x_0\}\) for a fixed path are not measurable, since none can be determined by countably many coordinates. This is why constructing Brownian motion requires a continuity argument beyond the extension theorem (see Kolmogorov's Continuity Theorem).
5 Measures on Cylinders
Cylinder sets are the natural domain for specifying a measure from finite-dimensional distributions.
- Uniqueness. Measurable rectangle cylinders are closed under finite intersections, so they form a \(\pi\)-system generating \(\bigotimes\mathcal{M}_t\). By Dynkin’s π-λ theorem, two probability measures agreeing on all finite-dimensional rectangles agree on the whole product σ-algebra.
- Existence. Given a consistent family of finite-dimensional distributions \(\mu_S\) (consistent meaning that marginalising \(\mu_{S'}\) to \(S\subseteq S'\) gives \(\mu_S\)), define \(\mu_0(C_S(B))=\mu_S(B)\) on the algebra \(\mathcal{C}\). For \(X_t\) complete separable metric spaces (e.g. \(\mathbb{R}\)), the Kolmogorov extension theorem says \(\mu_0\) is a premeasure, and the Carathéodory extension theorem extends it to a unique measure on \(\bigotimes\mathcal{M}_t\).