Fubini's Theorem
Theorem
Let $\struct {X, \Sigma_1, \mu}$ and $\struct {Y, \Sigma_2, \nu}$ be $\sigma$-finite measure spaces.
Let $\struct {X \times Y, \Sigma_1 \otimes \Sigma_2, \mu \times \nu}$ be the product measure space of $\struct {X, \Sigma_1, \mu}$ and $\struct {Y, \Sigma_2, \nu}$.
Let $f: X \times Y \to \R$ be a $\Sigma_1 \otimes \Sigma_2$-measurable function.
Suppose that:
- $\displaystyle \int_{X \times Y} \size f \map {\rd} {\mu \times \nu} < \infty$
Then $f$ is $\mu \times \nu$-integrable, and:
- $\displaystyle \int_{X \times Y} f \map {\rd} {\mu \times \nu} = \int_Y \int_X \map f {x, y} \map {\rd \mu} x \map {\rd \nu} y = \int_X \int_Y \map f {x, y} \map {\rd \nu} y \map {\rd \mu} x$
Implement a version of this for real functions
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Proof
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Also see
Source of Name
This entry was named for Guido Fubini.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $13.9$