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Fubini Theorem

Fubini's theorem equates an integral on a product measure space with repeated integrals when the integrand is Lebesgue integrable. Let (X,A,mu) and (Y,B,nu) be measure spaces with sigma-finite measures, and let f:X×Y->C be a measurable function for the product sigma-algebra A tensor B. If int_(X×Y)|f|d(mu×nu)<infty, then

int_(X×Y)fd(mu×nu)=int_X(int_Yf(x,y)dnu(y))dmu(x)=int_Y(int_Xf(x,y)dmu(x))dnu(y),

where mu×nu is the product measure. The sections are Lebesgue integrable almost everywhere in the outer parameter. The inner integrals, set to zero on the exceptional measurable sets of measure zero, are measurable and Lebesgue integrable as functions of the outer parameter (Hunter 2011, p. 61).

Tonelli's theorem gives the corresponding equality for nonnegative measurable functions without requiring the integrals to be finite. Applying it to |f| can establish the absolute integrability required by Fubini's theorem (Shapiro 2025, pp. 57-58).

In particular, if f(x,y) is continuous on the rectangular region R:a<=x<=b,c<=y<=d, then the equality

intint_Rf(x,y)d(x,y)=int_a^bint_c^df(x,y)dydx

holds (Thomas and Finney 1996, p. 919).

See also

Definite Integral, Multiple Integral, Product Measure, Repeated Integral, Tonelli's Theorem

Portions of this entry contributed by Ronald M. Aarts

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References

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, p. 18, 2004.Fubine, G. "Sugli integrali multipli." Opere scelte, Vol. 2. Cremonese, pp. 243-249, 1958.Hunter, J. K. "Product Measures." Ch. 5 in Measure Theory. Lecture notes, pp. 55-62, 2011. https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch5.pdf.Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 9, 1993.Shapiro, J. Measure Theory. Princeton University MAT425 lecture notes, May 15, 2025. https://web.math.princeton.edu/~js129/PDFs/teaching/MAT425_spring_2025/MAT425_Lecture_Notes.pdf.Thomas, G. B., Jr. and Finney, R. L. Calculus and Analytic Geometry, 8th ed. Reading, MA: Addison-Wesley, p. 919, 1996.

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Fubini Theorem

Cite this as:

Weisstein, Eric W., with contributions by Ronald M. Aarts. "Fubini Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FubiniTheorem.html

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