One counterargument is that space is finite, and so your choice of n is greater than the volume of the universe. (And so sigma-additivity doesn't apply, since y

One counterargument is that space is finite, and so your choice of n is greater than the volume of the universe. (And so sigma-additivity doesn't apply, since your choices of cubic inches are not disjoint.) But sure, if you're assuming an unbounded space with finite measure, a uniform density across that space must be identically zero everywhere.