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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
and
are
complete measure spaces. Suppose
is
measurable. If
-
| f (x, y) | d (x, y) < ∞, |
where the integral is taken with respect to a product measure on the product space
, then
-
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
is
continuous on the rectangular region
R ∈ A × B : a ≤ x ≤ b, c ≤ y ≤ d, |
then the equality
-
holds. [1]
Notes
External links