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
Map
Cartesian product
Cylinder set
Measurable cylinder set
Definition D3801
Set of measurable cylinder sets
Formulation 3
Let $M_j = (X_j, \mathcal{F}_j)$ be a D1108: Measurable space for each $j \in J$ such that
(i) $X = \prod_{j \in J} X_j$ and $\mathcal{F} = \prod_{j \in J} \mathcal{F}_j$ are each a D326: Cartesian product
(ii) $\mathcal{P}_{\mathsf{cofinite}}(J)$ is the D2200: Set of cofinite sets in $J$
The set of measurable cylinder sets in $X$ with respect to $M = \{ M_j \}_{j \in J}$ is the D11: Set \begin{equation} \left\{ \prod_{j \in J} E_j \subseteq \mathcal{F} \mid \exists \, I \in \mathcal{P}_{\mathsf{cofinite}}(J) : \forall \, i \in I : E_i = X_i \right\} \end{equation}
Children
Product sigma-algebra