Theorem Statement
Suppose A and B are complete measure spaces. Suppose f(x,y) is A × B measurable. If
where the integral is taken with respect to a product measure on the space over A × B, 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. See below for an illustration of this possibility.
Read more about this topic: Fubini's Theorem
Famous quotes containing the words theorem and/or statement:
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
—Ralph Waldo Emerson (1803–1882)