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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.

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

A measure space is complete if every subset of every null set is measurable (and thus has measure zero).

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

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.

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

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

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

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.