Product Sigma-Algebra Generated by Projections

Theorem
Let $\left({X, \Sigma_1}\right)$ and $\left({Y, \Sigma_2}\right)$ be measurable spaces.

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

Let $\operatorname{pr}_1: X \times Y \to X$ and $\operatorname{pr}_2: X \times Y \to Y$ be the first and second projections, respectively.

Then:


 * $\Sigma_1 \otimes \Sigma_2 = \sigma \left({\operatorname{pr}_1, \operatorname{pr}_2}\right)$

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