PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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An equidimensional $C^{1}$ map sends null sets to null sets

Statement

Let $F : M^{m} \to N^{m}$ be a $C^{1}$ map between smooth manifolds of the same dimension. If $E \subseteq M$ is null, then $F (E) \subseteq N$ is null.

Facts & Assumptions

Given: A $C^{1}$ map $F : M^{m} \to N^{m}$ and a null subset $E \subseteq M$ .

[L1]

A countable chart cover detects manifold nullity (A countable chart cover detects manifold null sets).

[L2]

On compact coordinate pieces a $C^{1}$ map is locally Lipschitz, and Lipschitz maps send Euclidean null sets to Euclidean null sets (A $C^{1}$ map is locally Lipschitz on compact coordinate subsets, A Lipschitz map $R^{m} \to R^{m}$ sends null sets to null sets).

Proof

technique · direct

1.1

Choose countable smooth atlases on $M$ and $N$ detecting nullity by [L1]. Refine the source atlas so that each relatively compact chart domain lies inside the inverse image of one target chart domain.

L1givenchoose

2.1

For each source chart piece $U_{j}$ , the set $φ_{j} (E \cap U_{j})$ is null in $R^{m}$ . Cover $\overline{U_{j}}$ by finitely many smaller coordinate neighbourhoods on which the coordinate representative of $F$ is Lipschitz by [L2]. Applying [L2] on each such piece shows that the corresponding target-chart image of $F (E \cap U_{j})$ is null.

L2step 1.1algebra

3.1

The set $F (E)$ is the countable union of the sets $F (E \cap U_{j})$ . Since each has null image in every target chart from step 2.1, [L1] implies that $F (E)$ is null in $N$ .

L1step 2.1algebra∎

Depends on

Used by

The image of a lower-dimensional C¹ manifold is null Proposition

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)

An equidimensional C1C1 map sends null sets to null sets