Let be an infinite dimensional normed space. There is no non-trivial translation invariant Borel measure on which is finite on open balls.
Show that if is a compact subset of , and if every intersection of with an hyperplane has measure zero in , then has measure zero.
Note This is valid for any measurable set in any -compact measure space by virtue of Tonelli’s theorem. The point is to prove this without using this. The fact that is compact will come in handy.