The change of variables for multiple integral can be formulated like:
$\int_D(f \circ g) (x) |det~J_g(x)|dx = \int_{D'}fdx'$
for a continous $f: D'\rightarrow X$ and two-times continous, injective $g: D \rightarrow D'$.
I wonder what happens with the function: $$l(n) = \int_D(f \circ g) (x) \sqrt[\textbf{n}]{|det~J_g(d)|}dx$$ assuming $\int_{D'}fdx$ is finite (which is $l(1)$). Or maybe: are there any (interesting) constraints that one could put upon $f$ or $g$ so that $l(n)$ is (for example) continous?