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Tonelli's Theorem

Tonelli's theorem equates the integral of a nonnegative measurable function with respect to a product measure with its repeated integrals, allowing the common value to be infinite. Let (X,A,mu) and (Y,B,nu) be measure spaces with sigma-finite measures, and let f:X×Y->[0,infty] be measurable for the product sigma-algebra A tensor B. 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. All sections are measurable, and the inner integrals are measurable functions with values in [0,infty] (Shapiro 2025, pp. 57-58).

Fubini's theorem applies to absolutely integrable real- or complex-valued functions. Tonelli's theorem applied to |f| shows that finiteness of either repeated integral of |f| establishes that hypothesis (Hunter 2011, p. 61).

See also

Fubini Theorem, Product Measure, Repeated Integral

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References

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.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.

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Tonelli's Theorem

Cite this as:

Weisstein, Eric W. "Tonelli's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TonellisTheorem.html

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