I came across the following example for a triple integral:
Find the volume of the region bounded by hyperbolic cylinders: $$ xy = 1 \quad , \quad xy = 9$$ $$ xz = 4 \quad , \quad xz = 36 $$ $$ yz = 25 \quad , \quad yz = 49 $$ In the answer I see, the problem is solved by transforming to variables $u=xy$, $v=xz$, and $w=yz$, where the domain gets transformed to the rectangular prism $ 1 \leq u \leq 9$, $ 4 \leq v \leq 36$, and $ 25 \leq w \leq 49$. Once we are here, it is straightforward to see that Fubini's theorem applies for the volume integral with a function of $f(u,v,w)=1$, and so we can use an iterated integral to directly get the volume.
My question though is that before I transform, it is much less clear that the region defined between those hyperbolic cylinders allows us to use Fubini's theorem - particularly because it appears that the region extends to infinity. Can I infer from the fact that it allows such a transformation that the region is measurable? I can believe that it is possible to use such a transformation to construct a cover of the region and hence prove that it is measurable... but I'm not sure.
Thanks for the help!