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The product measure is the measure on the product sigma-algebra generated by the measurable rectangles, such that the measure of each rectangle is the product of the measures of its sides.

A sigma-algebra is a collection of subsets closed under complementation and countable unions. Required structure for defining a measure.

A measurable rectangle is the Cartesian product of two measurable sets, each from a space equipped with a sigma-algebra.

Fubini's Theorem states that if two spaces are sigma-finite with respect to their measures, then the integral of the product measure is equal to the iterated integral of the functions over each space.

Tonelli's Theorem is a relaxation of Fubini's Theorem which allows one to integrate non-negative measurable functions even if the measure spaces are not sigma-finite, by using iterated integrals.

The product sigma-algebra on a product of two measurable spaces is the smallest sigma-algebra containing all measurable rectangles.

A measure space is sigma-finite if it can be decomposed into a countable union of sets with finite measure.

An iterated integral is the repeated application of the integral operator to functions of several variables, integrating one variable at a time.

Carathéodory’s Extension Theorem provides a method to extend a pre-measure to a measure on the sigma-algebra generated by the pre-measure’s domain.

The Lebesgue Measure is the standard way of assigning a volume to subsets of $n$-dimensional Euclidean space, which generalizes the concept of length, area, and volume.