How does one check the integral condition of Fubini's?
How does one check the integral condition of Fubini's?
I.e. that
$$\int_{A \times B} |f(x,y)|d(x,y) < \infty$$
Isn't the use of Fubini's to avoid having to calculate this?
I don't understand how to calculate such integral w/o iterated.
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$\begingroup$Suppose we have two $\sigma$-finite measure spaces $(A,\mathcal{M},\mu)$ and $(B,\mathcal{N},\nu)$, and a measurable function $f:A\times B \to \Bbb{C}$. To check the hypothesis of Fubini's theorem, we usually apply Tonelli's theorem to guarantee that $f$ is integrable (with repsect to the product measure).
Tonelli's theorem tells us that if $\phi:A\times B \to [0,\infty]$ is measurable (relative to the appropriate $\sigma$-algebras) then you can always iterate the integrals and the result is always the same:\begin{align} \int_{A\times B}\phi(x,y)\, d(x,y) = \int_A \int_B \phi(x,y)\, dy \, dx = \int_B \int_A \phi(x,y)\, dx \, dy \end{align}Just to be more explicit, this says all three terms are finite and equal, or they are all equal to $\infty$.
So, with this in mind, to check the hypothesis $\int_{A\times B}|f(x,y)| \,d(x,y)< \infty$ which is required for Fubini's theorem, we first apply Tonelli's theorem:\begin{align} \int_{A\times B}|f(x,y)|\, d(x,y) = \int_A \int_B |f(x,y)|\, dy \, dx = \int_B \int_A |f(x,y)|\, dx \, dy \tag{$*$} \end{align}Out of these three, calculate whichever integral is the easiest to do (after all, they are all equal). If it is finite, great! Fubini's theorem's hypotheses are satisfied so you can remove absolute values and say that\begin{align} \int_{A\times B}f(x,y)\, d(x,y) = \int_A \int_B f(x,y)\, dy \, dx = \int_B \int_A f(x,y)\, dx \, dy \end{align}If however $(*)$ results in $\infty$, then that is very unfortunate, and you have to be very careful in your subsequent analysis.
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