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Finite disjoint unions of measurable rectangles form an algebra generating the product sigma-algebra

Statement

Let $(X, A)$ and $(Y, B)$ be measurable spaces. The family of finite disjoint unions of measurable rectangles in $X \times Y$ is an algebra of subsets of $X \times Y$ , and it generates $A \otimes B$ .

Facts & Assumptions

Given: Measurable spaces $(X, A)$ and $(Y, B)$ .

[L1]

A measurable rectangle has the form $A \times B$ with $A \in A$ and $B \in B$ . (Measurable rectangles in a product of measurable spaces)

[L2]

The product sigma-algebra $A \otimes B$ is the sigma-algebra generated by the measurable rectangles. (The product sigma-algebra and its finite iterates)

[A1]

For rectangles, $(A_{1} \times B_{1}) \cap (A_{2} \times B_{2}) = (A_{1} \cap A_{2}) \times (B_{1} \cap B_{2}),$ and $(X \times Y) ∖ (A \times B) = (A^{c} \times Y) \cup (A \times B^{c}) .$

[A2]

If $R_{j} = A_{j} \times B_{j}$ for $1 \leq j \leq n$ , take the nonempty Boolean atoms generated by $A_{1}, \dots, A_{n}$ in $X$ and by $B_{1}, \dots, B_{n}$ in $Y$ . The products of an $X$ -atom and a $Y$ -atom are finitely many pairwise disjoint measurable rectangles partitioning $X \times Y$ , and each $R_{j}$ , hence also $⋃_{j = 1}^{n} R_{j}$ , is the union of a subfamily of these product atoms.

Proof

technique · direct

1.1L1A1

By [L1] and [A1], the intersection of two measurable rectangles is again a measurable rectangle, and the complement of a measurable rectangle is a finite union of measurable rectangles.

2.1step 1.1A2

Let $R$ be the family of finite disjoint unions of measurable rectangles. It contains $\emptyset$ and $X \times Y$ . If $E, F \in R$ , then [A2] disjointifies the finite union $E \cup F$ into finitely many pairwise disjoint measurable rectangles, so $E \cup F \in R$ . Likewise $E ∖ F = E \cap F^{c}$ belongs to $R$ by step 1.1. Thus $R$ is an algebra.

3.1L2step 2.1∎

Every measurable rectangle belongs to $R$ , so [L2] gives $A \otimes B = σ (measurable rectangles) \subseteq σ (R) .$ The reverse inclusion holds because every member of $R$ is a finite union of measurable rectangles and hence lies in $A \otimes B$ . Therefore $σ (R) = A \otimes B$ , and $R$ is an algebra generating the product sigma-algebra.

Depends on

Dependency tree · 2 levels

Sources

John K. Hunter, Measure Theory, paragraph before Definition 5.10