Definition
Product sigma-algebra
The sigma-algebra generated by measurable rectangles in a product of measurable spaces.
For measurable spaces and , their product sigma-algebra is
Thus it is generated by measurable rectangles. It is the smallest sigma-algebra making both coordinate projections measurable.
Sections
For every set in this sigma-algebra, fixing one coordinate gives a measurable section in the other space. To see this, observe that the sets with that property form a sigma-algebra containing all rectangles. Even when the factor measures are complete, their product sigma-algebra need not contain every subset of a product-null set.