Null subset of a differential manifold

Definition

Let $M$ be a differential manifold. A subset $S$ of $M$ has measure zero if the following holds: for any open subset $U$ of $M$ and any diffeomorphism between $U$ and $\mathbb {R} ^{n}$, the image of $U\cap S$ under the diffeomorphism has measure zero in $\mathbb {R} ^{n}$.

The well-definedness of this notion rests on the fact that any diffeomorphism from $\mathbb {R} ^{n}$ to itself maps measure zero subsets to measure zero subsets (this is not true for arbitrary homeomorphisms, and hence the notion of a measure zero subset does not make sense for a topological manifold).

Facts

• Any submanifold of codimension at least 1, is a measure zero subset
• Sard's theorem is a generalization of the above