Null subset of a differential manifold

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Let M be a differential manifold. A subset S of M is said to have measure zero or to be a null subset if the following holds: for any open subset U of M and any diffeomorphism between U and \R^n, the image of U \cap S under the diffeomorphism has measure zero in \R^n.

The well-definedness of this notion rests on the fact that any diffeomorphism between open subsets of \R^n 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