Let $(X,d_X)$ and $(Y,d_Y)$ be separable metric spaces, and equip $X\times Y$ with the [product topology](/page/Product%20Topology). Then
\begin{align*}
\mathcal B(X\times Y)=\mathcal B(X)\otimes\mathcal B(Y).
\end{align*}
AnalysisMeasure Theory
Discussion
For separable metric spaces, the Borel sigma-algebra of the product topology agrees with the product of the Borel sigma-algebras. This identifies topological and measurable products in common analytic settings.
Proof
No proof available for this theorem.
Prerequisites
(0/4 completed)
Prerequisites Graph
Interactive dependency map showing how this theorem builds on foundational concepts