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  "body": "\n## Definition\n\nLet $E$ be a real topological vector space.\nFor $n \\ge 0$, let $\\mathcal{T}_n$ be the set of continuous [[linear maps|0NRYHJPQG2]] $T \\colon E \\to \\R^n$ that are surjective, and let $\\mathcal{T} = \\bigcup_n \\mathcal{T}_n$.\nA *cylinder set measure* on $E$ is a family $(\\mu_T)_{T \\in \\mathcal{T}}$ with these two properties.\n\n1. For each $T \\in \\mathcal{T}_n$, $\\mu_T$ is a [[probability measure|059TJN0NZ1]] on the [[Borel σ-algebra|0WQXY32Q8D]] of $\\R^n$.\n2. 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]] $\\pi_* \\mu_S$.\n\n## Remarks\n\nA *cylinder set* is a set $T^{-1}(B)$ with $T \\in \\mathcal{T}_n$ and $B \\subseteq \\R^n$ Borel.\nCondition 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.\n\nExample: a Borel probability measure $\\nu$ on $E$ gives the cylinder set measure $\\mu_T = T_* \\nu$.\nCondition 2 holds since $\\pi_*(S_* \\nu) = (\\pi \\circ S)_* \\nu$.\n\nExample: let $E = H$ be a real Hilbert space and let $\\mu_T$ be the centered normal law on $\\R^n$ with covariance matrix $T T^*$.\nCondition 2 holds, because $\\pi_*$ of the centered normal law with covariance $S S^*$ is centered normal with covariance $\\pi S S^* \\pi^*$.\nThis is the canonical Gaussian cylinder set measure on $H$.\n\nNon-example: on $E = \\R^2$, let every $\\mu_T$ with $T \\in \\mathcal{T}_1$ be the standard [[normal distribution|05AQE0QR0D]] $N(0, 1)$.\nFor $T(x) = x_1$ 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.\n\n## Explanations\n\n### L1 human\n\nA cylinder set measure gives the law of every finite set of linear readings of a random point, in a consistent way.\nIt is a measure that is only known on sets described by finitely many coordinates.\n",
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