User:Caliburn/s/mt/Borel Sigma-Algebra of R2 is Product of Borel-Sigma Algebra of R with Itself

Theorem

We have:

$\map {\mathcal B} {\R^2} = \map {\mathcal B} \R \otimes \map {\mathcal B} \R$

where:

$\map {\mathcal B} {\R^2}$ is the Borel $\sigma$-algebra on $\R^2$

$\map {\mathcal B} {\R}$ is the Borel $\sigma$-algebra on $\R$

$\map {\mathcal B} \R \otimes \map {\mathcal B} \R$ is the product $\sigma$-algebra of $\map {\mathcal B} \R$ with $\map {\mathcal B} \R$.