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 ▼ Set of measurable cylinder sets
Definition D2154
Product sigma-algebra

Let $M_j = (X_j, \mathcal{F}_j)$ be a D1108: Measurable space for each $j \in J$.
Let $X : = \prod_{j \in J} X_j$ be the D326: Cartesian product of $\{ X_j \}_{j \in J}$ such that
 (i) $\mathfrak{S} : = \mathfrak{S}(X)$ is the D484: Set of sigma-algebras on $X$ (ii) $\mathcal{C} : = \mathcal{C}_M(X)$ is the D3801: Set of measurable cylinder sets in $X$ with respect to $M = \{ M_j \}_{j \in J}$
The product sigma-algebra on $X$ with respect to $M = \{ M_j \}_{j \in J}$ is the D11: Set $$\sigma \langle \mathcal{C} \rangle = \bigcap \{ \mathcal{F} : \mathcal{C} \subseteq \mathcal{F} \in \mathfrak{S} \}$$
Children
 ▶ D2153: Measurable product space