ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
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Cartesian product
Cylinder set
Definition D3800
Measurable cylinder set
Formulation 0
Let $M_j = (X_j, \mathcal{F}_j)$ be a D1108: Measurable space for each $j \in J$.
Let $X = \prod_{j \in J} X_j$ and $\mathcal{F} = \prod_{j \in J} \mathcal{F}_j$ each be a D326: Cartesian product.
Let $\mathcal{P}_{\mathsf{cofinite}}(J)$ be the D2200: Set of cofinite sets in $J$.
A D11: Set $\prod_{j \in J} E_j \subseteq \mathcal{F}$ is a measurable cylinder set in $X$ with respect to $M = \{ M_j \}_{j \in J}$ if and only if \begin{equation} \exists \, I \in \mathcal{P}_{\mathsf{cofinite}}(J) : \forall \, i \in I : E_i = X_i \end{equation}
Children
Set of measurable cylinder sets