On $n$ dimensional manifolds and lattices
Here is a measure-free proof which works for affine manifolds over an arbitrary infinite field $\mathbb F$ (the result is false for finite fields).

By induction on $n\ge0$, we will show that an affine manifold $A\subseteq\mathbb F^m$ of dimension $n$ is not a finite union of affine manifolds of dimension less than $n$.

The statement is clear for $n=0$: a point is not a (finite) union of empty sets.

Assume the statement holds for $n$, we will show it for $n+1$. Let $A=\bigcup_{i&lt;k}A_i$, where $\dim(A)=n+1$ and $\dim(A_i)\le n$. Consider an arbitrary affine submanifold $B\subset A$ of dimension $n$. Since $B=\bigcup_i(B\cap A_i)$, the induction hypothesis implies that $\dim(B\cap A_i)=n$ for some $i&lt;k$, i.e., $B=A_i$. Since there are only $k$ sets $A_i$, and $B$ was arbitrary, it follows that $A$ has only finitely many submanifolds of dimension $n$. However, this is a contradiction: if we fix any such submanifold $B_0$ and a vector $v$ parallel to $A$ but not to $B_0$, there are infinitely many affine submanifolds of $A$ of the form $B_0+av$, where $a\in\mathbb F$.