A theorem in measure theory ensuring that for a nonnegative measurable function on a product measure space the multiple integral equals each iterated integral, possibly with value +∞, without requiring integrability hypotheses on the function.

Principle

If f:X×Y→[0,∞] is measurable, then the function x ↦ ∫_Y f(x,y) dy is measurable, similarly y ↦ ∫_X f(x,y) dx is measurable, and ∫_{X×Y} f = ∫_X (∫_Y f(x,y) dy) dx = ∫_Y (∫_X f(x,y) dx) dy, where the integrals may be infinite.

Demonstration

Example: let f(x,y)=χ_A(x)χ_B(y) where A⊂X and B⊂Y are measurable sets. Tonelli gives ∫_{X×Y} χ_{A×B} = μ(A)μ(B) and allows computing the product measure by iterated integrals even when measures are infinite.

Misapplication

Using Tonelli for signed or non-nonnegative functions is incorrect; applying it to functions that take both signs can hide cancellation and produce false equalities unless further conditions (e.g., absolute integrability) hold.

Consequence

Tonelli justifies Fubini in the nonnegative case and enables use of monotone convergence on product spaces; it is the foundational tool for constructing product measures and for interchanging limits and integrals when nonnegativity applies.

Reversal

Reversing the sign hypothesis yields Fubini-type requirements: if functions are not nonnegative one needs absolute integrability to guarantee equality of iterated integrals; the nonnegative hypothesis cannot be dropped without extra integrability.

Boundary

Applies only to measurable functions taking values in [0,∞]; it does not assert finiteness of the integral and typically assumes standard product-measure constructions and σ-finiteness for measure-theoretic convenience.

Semantic Tension

Tension exists between Tonelli (nonnegative, possibly infinite integrals) and Fubini (requires absolute integrability); another competing notion is conditional integrability where neither theorem applies directly.

Synthesis

Tonelli's theorem states that nonnegative measurable functions on product spaces admit iterated integrals equal to the multiple integral, possibly infinite, providing the measure-theoretic basis for product measures and monotone convergence on products.