Product sigma-algebra

Summary

The product σ-algebra on a Cartesian product of measurable spaces is the smallest σ-algebra that makes every coordinate projection measurable. For two factors it is generated by the measurable rectangles $A \times B$. It is the σ-algebra on which product measures live and on which Fubini's theorem is stated. For infinite products it is generated by the cylinder sets, which constrain only finitely many coordinates.