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'Fubini's theorem'
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| Title of object: |
Fubini's theorem |
| Canonical Name: |
FubinisTheorem |
| Type: |
Theorem |
| Created on: |
2003-05-26 20:33:44 |
| Modified on: |
2005-01-08 05:54:23 |
| Classification: |
msc:26B12 |
Revision comment (for changes between this and next version):
Preamble:
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Content:
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\textbf{Fubini's theorem}
Let $I \subset \R^N$ and $J \subset \R^M$ be compact intervals, and let $f : I \times J \to \R^K$ be a Riemann integrable function such that, for each $x \in I$ the integral
\[
F(x) := \int_J f(x, y)\, d\mu_J(y)
\]
exists. Then $F:I\to\R^K$ is Riemann integrable, and
\[
\int_I F = \int_{I\times J} f.
\]
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.
\textbf{Example} Let $I := [0, \pi/2]\times[0,\pi/2]$, and let $f : I \to \R, x \mapsto \sin(x)\cos(y)$ be a function.
Then
\begin{equation*}
\begin{split}
\int_I f &= \iint_{[0, \pi/2]\times[0,\pi/2]} \sin(x)\cos(y) \\
&=\int_0^{\pi/2} \left( \int_0^{\pi/2} \sin(x)\cos(y)\,dy\right)\,dx \\
&=\int_0^{\pi/2} \sin(x)\left(1 - 0\right)\,dx =(0 - -1) = 1.
\end{split}
\end{equation*}
Note that it is often simpler (and no less correct) to write $\idotsint_I f$ as $\int_I f$. |
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