FubiniTheorem

Fubini's theorem is a fundamental result in measure theory that provides conditions under which the order of integration in a double integral can be interchanged. It concerns measurable functions f on the product of two measure spaces (X, μ) and (Y, ν) and relates the integral over X×Y to iterated integrals over X and Y. A closely related result is Tonelli's theorem, which handles nonnegative functions.

Statement (integrable case). Let (X, μ) and (Y, ν) be σ-finite measure spaces. If f is integrable on X×Y, i.e., ∫∫ |f(x,y)| dμ×ν < ∞, then for almost every x, the function y ↦ f(x,y) is ν-integrable and the function F(x) := ∫ f(x,y) dν(y) is μ-integrable. Moreover, ∫∫ f dμ×ν = ∫ F(x) dμ(x) = ∫ (∫ f(x,y) dμ(x)) dν(y). In particular, the double integral equals the two iterated integrals, and the order of integration can be interchanged.

Tonelli's theorem (nonnegative case). If f ≥ 0 is measurable on X×Y, then the iterated integrals exist (possibly taking the value ∞) and ∫∫ f dμ×ν = ∫ [∫ f(x,y) dν(y)] dμ(x) = ∫ [∫ f(x,y) dμ(x)] dν(y). The nonnegativity assumption removes the need for absolute integrability.

Remarks. Fubini's theorem typically requires the underlying measure spaces to be σ-finite. The theorem underpins many analyses in probability, statistics, and applied mathematics by justifying the interchange of integration order in multiple integrals.