Product Sigma Algebras

Why

We want to generalize the construction of cover area as generated as a product of two cover lengths, and more generally for arbitrary measure spaces.¹

Definition

Consider two measurable spaces. The product base set is the cartesian product of the first base set with the second base set. A first distinguished set is a distinguished set of the first measurable space, and likewise for a second distinguished set.

A rectangle with measurable sides is a set in product base set which is a product of a first distinguished set with a second distinguished set. The product sigma algebra is the sigma algebra generated by the rectangles with measurable sides.

The product measurable space is the measurable space whose base set is the product base set and whose sigma algebra is the product sigma algebra.

Notation

Let $(X, \mathcal{A} )$ and $(Y, \mathcal{B} )$ be measurable spaces. The product base set is $X \times Y$. A set $R \in X \times Y$ is a rectangle with measurable sides if $R = A \times B$ for $A \in \mathcal{A} $ and $B \in \mathcal{B} $. We denote the product sigma algebra of $\mathcal{A} $ and $\mathcal{B} $ by $\mathcal{A} \times \mathcal{B} $. $(X \times Y, \mathcal{A} \times \mathcal{B} )$ is the product measurable space.

1. Future editions will expand. ↩︎