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单词 FubinisTheoremForTheLebesgueIntegral
释义

Fubini’s theorem for the Lebesgue integral


This is the version of the Fubini’s Theorem for the Lebesgue integralMathworldPlanetmath. Forthe Riemann integral, see the standard calculus version (http://planetmath.org/FubinisTheorem).

In the following suppose we will by convention define Xf𝑑μ:=0 in case fis not integrable. This simplifies notation and does not affect the results since it will turn out that such cases happen on a set of measureMathworldPlanetmath 0.

Also if we have a function f:X×Y𝔽 then define fx(y):=f(x,y) and fy(x):=f(x,y).

Theorem (Fubini).

Suppose (X,M,μ) and (Y,N,ν) are σ-finite (http://planetmath.org/SigmaFinite)measure spaces. If fL1(X×Y)then fxL1(ν) for μ-almost every x andfyL1(μ) for ν-almost every y. Furtherthe functionsxYfx𝑑ν and yXfy𝑑ν arein L1(μ) and L1(ν) respectively and

X×Yfd(μ×ν)=X[Yf(x,y)𝑑ν(y)]𝑑μ(x)=Y[Xf(x,y)𝑑μ(x)]𝑑ν(y).

You can now see the reason for defining the integral even where fx andfy are not integrable since the functionsxYfx𝑑ν and yXfy𝑑νare normally only almost everywhere defined, and we’d like to define them everywhere. Since we have changed the definition only on a set of measurezeroMathworldPlanetmath, this does not change the final result and we can interchange theintegrals freely without having to worry about where the functions areactually defined.

Note the of this theorem and Tonelli’s theorem for non-negative functions. Here you actually need to check some integrability before switchingthe integral . A application of Tonelli’s theorem actually shows that you can prove any one of these equations to show that fL1(X×Y)

X×Y|f|d(μ×ν)<,
X[Y|f(x,y)|𝑑ν(y)]𝑑μ(x)<, or 
Y[X|f(x,y)|𝑑μ(x)]𝑑ν(y)<.

If we take the counting measure on , then one can the Fubini theoremPlanetmathPlanetmath for sums.

Theorem (Fubini for sums).

Suppose that fij is absolutely summable, that isi,jN|fij|<, then

i,jfij=i=1j=1fij=j=1i=1fij.

In the above theorem we have used as our set for simplicity and familiarity of notation. Any summable function fij will have only a countable number of non-zero elements and thus the theorem for arbitrary sets just reduces to the above case.

References

  • 1 Gerald B. Folland. . John Wiley & Sons, Inc., New York, New York, 1999
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