When dealing with integration in respect to a product measure, I have noticed many difficulties, so that I came about to think about an algorithm in order to use appropriately the theorems of Fubini and Tonelli:
If the function $f(x,y)$ is positive over the integration range belonging to some measure space, then you can use one of the iterated integrals to determine the integral in respect to the product measure. The integral might eventually diverge, but that is not an issue;
If the function $f(x,y)$ changes the sign, i.e. it is not postive, I would compute both of the iterated integrals: only in the case they are different one can come to a conclusion, which is the integral with respect to the product measure would diverge and thus one can not use the theorem of Fubini. Otherwise, i.e. if both the iterated integrals are equal , one can not conclude since there are counterexamples in which both iterated integrals are equal (and finite) but the integral in respect to the product measure is divergent;
If the previous point does not lead to a conclusion I will then majorate the function $f$ ( in the best case by the least upper bound sup $f(x,y)$ over the defined region ). One can then only conclude if the majorant or the sup $f$ is integrable, in which case $f$ must be integrable and thus one can use the Tonelli's theorem for positve functions and eventually the Fubini's theorem so that computing one of the iterated integrals will suffice; In the case sup $f$ is not integrable one can not conclude since both cases are possible: $f$ can be integrable or not.
In the case the previous point(s) dont lead to a conclusion, one has to compute directly one of the iterated integrals of $|f(x,y)|$, which I dont know how. Can you give me a suggestion how to compute an integral if the integrand is in absolute value ? I guess one must find the measurable sets in which $f\geq 0$, and the measurable sets in which $f<0.$ It seems complicated; If the value of that inegral is finite, i.e. $f$ is integrable then one can state that one can use Fubini's theorem;
Otherwise one can compute directly the integral with respect to the product measure, which I dont know how to do it; Can somebodw explain how one can compute the integral of $f(x,y)$ with respect to the product measure without using the Fubini's theorem ?
Is the above algorithm reasonable and necessary ?