Product Sigma-Algebra Generated by Projections

Theorem
Let $\struct {X, \Sigma_1}$ and $\struct {Y, \Sigma_2}$ be measurable spaces.

Let $\Sigma_1 \otimes \Sigma_2$ be the product $\sigma$-algebra on $X \times Y$.

Let $\pr_1: X \times Y \to X$ and $\pr_2: X \times Y \to Y$ be the first and second projections, respectively.

Then:


 * $\Sigma_1 \otimes \Sigma_2 = \map \sigma {\pr_1, \pr_2}$

where $\sigma$ denotes generated $\sigma$-algebra.