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Fubini's theorem (Theorem)

Fubini's theorem Let $ I \subset \mathbb{R}^N$ and $ J \subset \mathbb{R}^M$ be compact intervals, and let $ f : I \times J \to \mathbb{R}^K$ be a Riemann integrable function such that, for each $ x \in I$ the integral

$\displaystyle F(x) := \int_J f(x, y)\, d\mu_J(y) $
exists. Then $ F:I\to\mathbb{R}^K$ is Riemann integrable, and
$\displaystyle \int_I F = \int_{I\times J} f. $

This theorem effectively states that, given a function of $ N$ variables, you may integrate it one variable at a time, and that the order of integration does not affect the result.

Example Let $ I := [0, \pi/2]\times[0,\pi/2]$, and let $ f : I \to \mathbb{R}, x \mapsto \sin(x)\cos(y)$ be a function. Then

\begin{displaymath}\begin{split}\int_I f &= \iint_{[0, \pi/2]\times[0,\pi/2]} \s... ...\pi/2} \sin(x)\left(1 - 0\right)\,dx =(0 - -1) = 1. \end{split}\end{displaymath}    

Note that it is often simpler (and no less correct) to write $ \idotsint_I f$ as $ \int_I f$.



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See Also: Tonelli's theorem, Fubini's theorem for the Lebesgue integral

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Cross-references: variables, integral, function, Riemann integrable, intervals, compact
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This is version 8 of Fubini's theorem, born on 2003-05-26, modified 2005-01-08.
Object id is 4307, canonical name is FubinisTheorem.
Accessed 8948 times total.

Classification:
AMS MSC26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)

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