Tonelli's theorem

Tonelli's theorem states that a nonnegative measurable function on a product measure space has equal iterated integrals and product-space integral, even when the value is infinite.

Product measure: A measure on a product of measurable spaces that gives measurable rectangles the product of their measures. It supplies the measure used for the theorem's integral over the product space.
Measure space: A set equipped with a sigma-algebra and a measure on that sigma-algebra. Tonelli's theorem applies to measurable spaces equipped with measures.
Fubini's theorem: A theorem equating iterated and product-space integrals under integrability assumptions. Fubini permits signed integrands when absolute integrability replaces Tonelli's nonnegativity condition.
Beppo Levi theorem: Another name for the monotone convergence theorem for nonnegative measurable functions. It supports limit arguments used to derive Tonelli's theorem and evaluate increasing approximations.
Leonida Tonelli: An Italian mathematician known for work in real analysis and the calculus of variations. Tonelli's name is attached to the integration theorem.
Measurable function: A function whose inverse images of measurable sets are measurable. Measurability is one of the theorem's essential hypotheses.
Sigma-algebra: A collection of subsets closed under complements and countable unions. It specifies which subsets in each factor space are measurable.
Monotone convergence theorem: A theorem exchanging limits of increasing nonnegative functions with their integrals. Its approximation principle underlies standard proofs of Tonelli's theorem.
Counting measure: The measure assigning each set its number of elements, possibly infinite. With counting measure, Tonelli's theorem becomes the interchange rule for nonnegative double series.
Guido Fubini: An Italian mathematician whose work includes a theorem on exchanging iterated integrals. Fubini's related theorem is often presented alongside Tonelli's result.

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