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Fubini's theorem

Interchange of iterated integrals for absolutely integrable functions on a product measure space.

Fubini's theorem: Let (X,Σ,μ)(X,\Sigma,\mu) and (Y,T,ν)(Y,\mathcal T,\nu) be σ\sigma-finite measure spaces, and let f:X×Y→Rf:X\times Y\to\mathbb R be (Σ⊗T)(\Sigma\otimes\mathcal T)-measurable. If

∫X×Y∣f∣ d(μ×ν)<∞,\int_{X\times Y} |f|\,d(\mu\times\nu)<\infty,

then for μ\mu-almost every x∈Xx\in X the section y↦f(x,y)y\mapsto f(x,y) is ν\nu-integrable, and for ν\nu-almost every y∈Yy\in Y the section x↦f(x,y)x\mapsto f(x,y) is μ\mu-integrable. Moreover, the iterated integrals exist as finite numbers and satisfy

∫X×Yf d(μ×ν)=∫X(∫Yf(x,y) dν(y)) dμ(x)=∫Y(∫Xf(x,y) dμ(x)) dν(y).\int_{X\times Y} f\,d(\mu\times\nu) = \int_X\Big(\int_Y f(x,y)\,d\nu(y)\Big)\,d\mu(x) = \int_Y\Big(\int_X f(x,y)\,d\mu(x)\Big)\,d\nu(y).

This theorem applies to on a space and justifies computing a by iterated integration. Compare , which gives the same conclusion for nonnegative functions without assuming ∫∣f∣<∞\int|f|<\infty.