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After encountering many different formulations of Fubini's theorem, I'm really confused and I'm not sure of which hypotheses are really necessary in order to be able to apply the theorem to a calculation.

Just to give the example that is the easiest to find, Wikipedia's article on this theorem states that if:

$$\displaystyle\int\int_{A\times B}|f(x,y)|d(x,y)<\infty$$

then we can separate the integral as:

$$\displaystyle\int_B\bigg(\int_A f(x,y)dx\bigg)dy$$

or as the product of two independent integrals if $f(x,y)=g(x)h(y)$, where A and B are intervals in $\mathbb{R}$. However, is this hypothesis of the integral of the absolute value being finite really necessary? I want to apply this theorem to an integral that has this form, encountered while doing quantum mechanics calculations:

$$\displaystyle\int D\psi \ e^{\sum_{j=1}^n a_j(\psi_j-b_j)^2}$$

where we use the following notation:

$$\displaystyle\int D\psi=\int^\infty_{-\infty}d\psi_1...\int^\infty_{-\infty}d\psi_n$$

However, since we are talking about the integral of an exponential over the entire space, it isn't finite, right? So I wouldn't be able to apply the theorem in this form? However, Fubini's theorem is indeed used to calculate the Gaussian integral, where we also have an exponential in the integrand. So I find this formulation of the theorem very confusing. I know that in physics we usually just assume we can apply the theorem, separate the integrals and carry on with our lives, but I want to do things right and know why it's correct to do what I'm doing.

Any advice would be greatly appreciated!

Wild Feather
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    See here and here for example – Mr.Gandalf Sauron Sep 15 '22 at 12:12
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    But even in simpler cases. For example summing over a two dimensional infinite array (an infinite matrix) you might see that if you sum over rows first and then add over columns , the result is not the same as summing over columns first and then rows. Well summation is itself an integral wrt counting measure so Fubini/Tonelli theorems still carry over . If the integrand is positive however(regardless whether the absolute integral is finite or not), you can work in an iterated manner. That is guaranteed by Tonelli's theorem . – Mr.Gandalf Sauron Sep 15 '22 at 12:17
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    here is another example. – Mr.Gandalf Sauron Sep 15 '22 at 12:19
  • @Mr.GandalfSauron Thank you! So, if I've understood correctly, $f(x,y)$ being integrable and positive is enough to guarantee that we can calculate the $n$ integrals separately, in an iterated way? And if $f(x,y)$ is not positive, then we would need for the hypothesis I mentioned to be satisfied, that is, for the integral of its absolute value to be finite? – Wild Feather Sep 15 '22 at 16:40
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    Not exactly. If $|f(x,y)|$ is integrable(that is the integral is finite) then you can apply the theorem of Fubini and do the iteration stuff. Or if f(x,y) is positive then regardless of whether $|f|=f$ is integrable or not, you can always do the iterated thingy due to the theorem of Tonelli . – Mr.Gandalf Sauron Sep 15 '22 at 18:17
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    Note that in your specific example, the integrand is positive, so the condition for Fubini is exactly that the integral you want is not infinite. If each of the iterated integrals is finite, it would not be possible for the full integral to be infinite. Thus if the Fubini test fails, the full integral is $\infty$ and the iterated integrals are also $\infty$. So they are still equal. When the integrand can be negative, the situation is much more complicated, but at least in this case, physicists ignoring the Fubini conditions is fully justified. – Paul Sinclair Sep 16 '22 at 11:08

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