# Too large to measure

Let $X$ be an infinite dimensional normed space. There is no non-trivial translation invariant Borel measure on $X$ which is finite on open balls.

Show that if $K$ is a compact subset of $\Bbb R^n$, and if every intersection of $K$ with an hyperplane $x_n=a$ has measure zero in $\Bbb R^{n-1}$, then $K$ has measure zero.
Note This is valid for any measurable set in any $\sigma$-compact measure space by virtue of Tonelli’s theorem. The point is to prove this without using this. The fact that $K$ is compact will come in handy.