Fubini's theorem

Fubini's theorem states that, under suitable integrability conditions, a multiple integral equals either order of iterated integration. It lets integrals over product spaces be computed one variable at a time.

Tonelli's theorem: A theorem that equates iterated integrals and a product-space integral for nonnegative measurable functions, allowing infinite values. It permits order changes for nonnegative functions without requiring integrability in advance.
Iterated integral: An integral formed by integrating a function successively with respect to separate variables. Fubini identifies these successive integrals with a single integral over the product space.
Changing the order of integration: Rewriting an iterated integral so that its variables are integrated in a different sequence. This is the theorem's central computational use when the required hypotheses hold.
Fubini's theorem for series: A theorem allowing the order of summation in a double series to be exchanged under suitable convergence conditions. It is the discrete analogue of exchanging orders in multiple integration.
Product measure: A measure on a product of measurable spaces that agrees with the component measures on measurable rectangles. Fubini's theorem relates integration against this measure to successive integration.
Multiple integral: An integral over a domain with two or more coordinates, measuring accumulated values across that domain. The theorem provides a way to evaluate this integral through one-dimensional or lower-dimensional steps.
Gaussian integral: The integral of the exponential of a negative quadratic, whose value underlies probability and analysis. A two-dimensional Gaussian integral can be evaluated by switching to polar coordinates and separating radial factors.
Conditional convergence: Convergence of a series whose terms' absolute values do not have a finite sum. Without absolute integrability, changing integration order can fail, much as rearranging conditionally convergent series can.
Lebesgue integral: An integral defined by approximating measurable functions with simple functions and integrating their level sets. Its treatment of measurable functions and null sets supplies the standard theorem's framework.
Measurable space: A set equipped with a collection of subsets closed under complements and countable unions. The theorem begins with measurable spaces and their product structure.

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