富比尼定理

定理

若 ∫ _(A× B)|f(x,y)| d(x,y)<∞，

其中A和B都是σ-有限测度空间， A× B 是 A 和 B 的积可测空间， （公式略） 是可测函数，那么 ∫ _(A)(∫ _(B)f(x,y) dy) dx=∫ _(B)(∫ _(A)f(x,y) dx) dy=∫ _(A× B)f(x,y) d(x,y)，

前二者是在两个测度空间上的逐次积分，但积分次序不同；第三个是在乘积空间上关于乘积测度的积分。

特别地，如果 f(x,y)=h(x)g(y)，则 ∫ _(A)h(x) dx∫ _(B)g(y) dy=∫ _(A× B)f(x,y) d(x,y)。

如果条件中绝对值积分值不是有限，那么上述两个逐次积分的值可能不同。

非负函数的Tonelli定理

托内利定理延续了富比尼定理。托内利定理的结论与富比尼定理一样，但是条件从 |f| 积分有限改为了 f 非负。

应用

富比尼定理一个应用是计算高斯积分。高斯积分是很多概率论结果的基础： ∫ _(-∞)^(∞)e⁻α x² dx=√((π)/(α))

Fubini's theorem gives the conditions under which a double integral can be computed as an iterated integral, i.e. by integrating in one variable at a time. Intuitively, just as the volume of a loaf of bread is the same whether one sums over standard slices or over long thin slices, the value of a double integral does not depend on the order of integration when the hypotheses of the theorem are satisfied. The theorem is named after Guido Fubini, who proved a general result in 1907; special cases were known earlier through results such as Cavalieri's principle, which was used by Leonhard Euler.

More formally, the theorem states that if a function is Lebesgue integrable on a rectangle X× Y, then one can evaluate the double integral as an iterated integral: （公式略）

This formula is generally not true for the Riemann integral (however, it is true if the function is continuous on the rectangle; in multivariable calculus, this weaker result is sometimes also called Fubini's theorem).

Tonelli's theorem, introduced by Leonida Tonelli in 1909, is similar but is applied to a non-negative measurable function rather than to an integrable function over its domain. The Fubini and Tonelli theorems are usually combined and form the Fubini–Tonelli theorem, which gives the conditions under which it is possible to switch the order of integration in an iterated integral.

A related result is often called Fubini's theorem for infinite series, although it is due to Alfred Pringsheim. It states that if （公式略） is a double-indexed sequence of real numbers, and if ∑ _((m,n)∈ N × N)a_(m,n) is absolutely convergent, then （公式略）