Do you understand the proof of the R^1 case? Have you tried to generalize the argument to R^n? If not, that's a good place to start. If so, where did you get stuck?
A standard proposition in measure theory is that if is any Lebesgue measurable set in with , then for any there is a finite, nontrivial interval such that .
To generalize this, suppose with . Then why for any is there some box such that ? Thank you.
I tried adapting the argument like so: Assume and . Then take finite intervals such that and . Then and for some
Is this the correct idea? I'm essentially following the approach in the case that I've read, but I'm unsure of some steps. For instance, why is it possible to find such such that ? It seems to be asking a lot from such sets, and I don't see why they should exist.