I recently looked at the proof of Minkowski's Bound given in Number Rings, and it appeared to implicitly rely on the interesting (and somewhat non-trivial) fact that convex subsets of $\mathbb{R}^n$ are Lebesgue-measurable. (By way of comparison, this is false for the Borel measure.)
My question: is anyone aware of a proof of Minkowski's Bound that doesn't rely on any non-trivial measure theory? It seems a little funny to me that measure theory would be required to prove a purely number-theoretic statement.