# Null subset of a differential manifold

## Definition

Let be a differential manifold. A subset of is said to have **measure zero** or to be a **null subset** if the following holds: for any open subset of and any diffeomorphism between and , the image of under the diffeomorphism has measure zero in .

The well-definedness of this notion rests on the fact that any diffeomorphism between open subsets of 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). `Further information: Diffeomorphism implies nullset-preserving`

## Facts

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