---
tag: R80G20B2ZW
kind: definition
name: Cylinder set measure
msc: [28C20, 60B11, 46G12]
defines: G95YDNFXQN
uses: [0NRYHJPQG2, 059TJN0NZ1, 0WQXY32Q8D, 3N66TM3TSQ, KFWQ24D9M6, 05AQE0QR0D]
sources:
  - url: https://en.wikipedia.org/wiki/Cylinder_set_measure
---

## Definition

Let $E$ be a real topological vector space.
For $n \ge 0$, let $\mathcal{T}_n$ be the set of continuous [[linear maps|0NRYHJPQG2]] $T \colon E \to \R^n$ that are surjective, and let $\mathcal{T} = \bigcup_n \mathcal{T}_n$.
A *cylinder set measure* on $E$ is a family $(\mu_T)_{T \in \mathcal{T}}$ with these two properties.

1. For each $T \in \mathcal{T}_n$, $\mu_T$ is a [[probability measure|059TJN0NZ1]] on the [[Borel σ-algebra|0WQXY32Q8D]] of $\R^n$.
2. If $S \in \mathcal{T}_m$, $T \in \mathcal{T}_n$ and $\pi \colon \R^m \to \R^n$ is linear with $T = \pi \circ S$, then $\mu_T$ is the [[pushforward|3N66TM3TSQ]] $\pi_* \mu_S$.

## Remarks

A *cylinder set* is a set $T^{-1}(B)$ with $T \in \mathcal{T}_n$ and $B \subseteq \R^n$ Borel.
Condition 2 makes the rule $\mu(T^{-1}(B)) = \mu_T(B)$ well defined, and $\mu$ is a [[finitely additive measure|KFWQ24D9M6]] on the algebra of cylinder sets.

Example: a Borel probability measure $\nu$ on $E$ gives the cylinder set measure $\mu_T = T_* \nu$.
Condition 2 holds since $\pi_*(S_* \nu) = (\pi \circ S)_* \nu$.

Example: let $E = H$ be a real Hilbert space and let $\mu_T$ be the centered normal law on $\R^n$ with covariance matrix $T T^*$.
Condition 2 holds, because $\pi_*$ of the centered normal law with covariance $S S^*$ is centered normal with covariance $\pi S S^* \pi^*$.
This is the canonical Gaussian cylinder set measure on $H$.

Non-example: on $E = \R^2$, let every $\mu_T$ with $T \in \mathcal{T}_1$ be the standard [[normal distribution|05AQE0QR0D]] $N(0, 1)$.
For $T(x) = x_1$ and $T'(x) = 2 x_1$, one has $T' = \pi \circ T$ with $\pi(t) = 2t$, and $\pi_* N(0, 1) = N(0, 4) \ne N(0, 1)$, so condition 2 fails.

## Explanations

### L1 human

A cylinder set measure gives the law of every finite set of linear readings of a random point, in a consistent way.
It is a measure that is only known on sets described by finitely many coordinates.