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Fubini's theorem

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Fubini’s theorem, named after Guido Fubini, is a theorem in mathematical analysis which gives the conditions under which it is possible to compute a double integral using iterated integrals. Under those conditions, it allows the order of integration to be changed when using iterated integrals.

Theorem statement

Suppose
A
and
B
are complete measure spaces. Suppose
f (x, y)
is
A  ×  B
measurable. If
A × B
A × B
|   f (x, y) |
d (x, y) < ∞,
where the integral is taken with respect to a product measure on the product space
A  ×  B
, then
A
A
B
B
f (x, y) dy
dx  = 
B
B
A
A
f (x, y) dx
dy  = 
A × B
A × B
f (x, y) d (x, y),

the first two integrals being iterated integrals with respect to two measures, respectively, and the third being an integral with respect to a product of these two measures.

If the above integral of the absolute value is not finite, then the two iterated integrals may actually have different values.

If
f (x, y)
is continuous on the rectangular region
RA  ×  B : a   ≤   x   ≤   b, c   ≤   y   ≤   d,
then the equality
R
R
f (x, y) d (x, y)  = 
b
a
d
c
f (x, y) dy
dx  = 
d
c
b
a
f (x, y) dx
dy

holds. [1]

Notes

  1. Thomas, G. B., Jr. and Finney, R. L., “Calculus and Analytic Geometry”, 9th ed., Reading, MA: Addison-Wesley, 1996, pp. 1004–1006.

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