I want to compute a geometric double integral. Does the Fubini theorem apply to a geometric integral?
Fubini's theorem allows a sum double integral to be computed as an iterated integral.
And a product (or geometric) integral can be expressed in terms of a sum integral as follows: $$ \prod_a^bf(x)^{dx}=\exp\bigg(\int_a^b\ln f(x)dx\bigg) $$
So, if I want to compute a geometric double integral, how can I apply Fubini's theorem?
For example, are any of these statements valid (geometric double integral = $\prod_{X \times Y}f(x,y)^{d(x,y)}$): $$ \begin{aligned} & \prod_{X \times Y}f(x,y)^{d(x,y)}=\prod_X \big(\prod_Yf(x,y)^{dy}\big)^{dx}&\\ & \prod_{X \times Y}f(x,y)^{d(x,y)}=\exp\bigg(\int_X\ln \big(\exp\big(\int_Y\ln f(x,y)dy\big)\big)dx\bigg)&\\ & \prod_{X \times Y}f(x,y)^{d(x,y)}=\exp\bigg(\int_{X \times Y}\ln f(x,y)d(x,y)\bigg) \end{aligned} $$
I would like to (know how to) compute a geometric double integral.
Sure enough you do have:
$$\begin{align}\prod_X\big(\prod_Y f(x,y)^{\mathrm d y}\big)^{\mathrm d x}~&=~ \exp\int_X\ln\prod_Y f(x,y)^{\mathrm d y}\,\mathrm d x \\ &=~ \exp\int_X\ln\exp\int_Y\ln f(x,y)\,\mathrm d y\,\mathrm d x \\ &=~ \exp\int_X\int_Y \ln f(x,y)\,\mathrm d y\,\mathrm d x \end{align}$$
At this point you can apply Fubini's theorem, if the criteria is met.
$$\int_X\int_ Y \lvert \ln f(x,y)\rvert \operatorname d y\operatorname d x<\infty$$
If that is so, then indeed:
$$\prod_X\prod_Y f(x,y)~{}^{\mathrm d y}{\,}^{\mathrm d x}~=~\prod_{X\times Y} f(x,y)^{\mathrm d (x,y)}~=~\prod_Y\prod_X f(x,y){\,}^{\mathrm d x}{\,}^{\mathrm d y} \\ \exp\int_X\int_Y \ln f(x,y)\,\mathrm d y\,\mathrm d x ~=~ \exp\iint_{X\times Y}\ln f(x,y)\operatorname d (x,y)~=~\exp\int_Y\int_X \ln f(x,y)\,\mathrm d x\,\mathrm d y $$