FubiniTonelli

Fubini–Tonelli refers to two fundamental results in measure theory about interchanging the order of integration on product spaces. They describe when the integral over a product measure μ×ν can be computed as iterated integrals, and under what hypotheses this interchange is valid.

Tonelli's theorem provides the nonnegative case. Let (X, A, μ) and (Y, B, ν) be σ-finite measure spaces and f: X×Y → [0, ∞] be measurable. Then the double integral equals the iterated integrals: ∫_{X×Y} f dμ×ν = ∫_X (∫_Y f(x,y) dν(y)) dμ(x) = ∫_Y (∫_X f(x,y) dμ(x)) dν(y). The value may be infinite. Tonelli requires only nonnegativity and measurability, and it allows swapping the order of integration without assuming absolute integrability.

Fubini's theorem covers the general integrable case. If f is integrable on X×Y, meaning ∫_{X×Y} |f| dμ×ν < ∞ (with μ and ν σ-finite), then for almost every x, the function y ↦ f(x,y) is integrable, the function x ↦ ∫_Y f(x,y) dν(y) is integrable, and ∫_{X×Y} f dμ×ν = ∫_X (∫_Y f(x,y) dν(y)) dμ(x) = ∫_Y (∫_X f(x,y) dμ(x)) dν(y). Thus, under absolute integrability, the order of integration can be swapped.

Both theorems have broad applications in analysis and probability, notably for evaluating double integrals and justifying the interchange of limits and integrations.