Defining Measures on a Manifold: How To
Solution 1:
Having measure zero is something that can be defined without actually defining a measure on the manifold. A subset $A$ of $N$ can be defined to have measure zero if for every chart $\phi:U\to\mathbb R^n$ of $N$, $\phi(U\cap A)$ has measure zero in $\mathbb R^n$.
You would want this to be something that is invariant under diffeomorphism. That is the case because smooth maps between subsets of $\mathbb R^n$ preserve the property of having Lebesgue measure $0$.