Alphabeta MathHow statement and proof provenance work
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A Lipschitz map sends null sets to null sets
Statement
If is Lipschitz and is null, then is null.
Facts & Assumptions
Given: A Lipschitz constant and null .
Lipschitz means (Lipschitz map, -Hölder map for rational , and contraction).
Norm comparisons on bound Euclidean diameter of a side- cube by a fixed dimension multiple of (The -norms for rational , and , Each is a norm on , and the induced metrics are exactly , and of the published metric-spaces page, The finite and reverse triangle inequalities for a norm; and for every norm on satisfies and is Lipschitz, hence continuous, for ).
Proof
If , is empty or a singleton, covered by cubes of arbitrarily small side.
Suppose . The image of a side- cube lies in a cube of side , where is the fixed norm-comparison factor. Its volume is .
Given an output budget , cover by cubes with total volume below . Replacing each by its image-containing cube gives a cover of with total volume below .
Both cases prove nullity. Equal domain and codomain dimensions are used in the volume scaling.
Depends on
- Measure zero and content zero in $\mathbb{R}^m$ by countable and finite cube covers
- Lipschitz map, $\alpha$-Hölder map for rational $0 < \alpha \le 1$, and contraction
- The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$
- Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page
- The finite and reverse triangle inequalities for a norm; and for $n \ge 1$ every norm $N$ on $\mathbb{R}^n$ satisfies $N(x) \le C\lVert x\rVert_1$ and is Lipschitz, hence continuous, for $d_2$
- Axis-parallel rectangles in $\mathbb{R}^m$ and their volume
- Integer powers $a^m$
- The canonical natural $\iota(n) = n \cdot 1_F$ of a field
- Laws of finite sums and finite products
Used by
- A C¹ diffeomorphism maps Lebesgue null sets to Lebesgue null sets Lemma
- A C¹ map sends a compact set of content zero to a set of content zero Lemma
- A Lipschitz self-map of ℝⁿ carries Lebesgue null sets to Lebesgue null sets Lemma
- C¹ local diffeomorphisms preserve null sets locally Lemma
- On a small cube, a C¹ diffeomorphism distorts Jordan content by factors arbitrarily close to its linearized absolute determinant Lemma
- An equidimensional C¹ map sends null sets to null sets Proposition
- Conventions and proved scope for the Riemann integral in ℝᵐ and Jordan content Remark
- A linear endomorphism of ℝⁿ sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant Theorem
- An injective C¹ map with invertible derivative sends compact Jordan sets to compact Jordan sets Theorem
Dependency tree · two levels
67 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
- J. Lebl, Basic Analysis, Jordan Measurable Sets (standard reference, not scraped)
- J. Lebl, Basic Analysis, Outer Measure and Null Sets (standard reference, not scraped)