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Critical images of Euclidean Lipschitz maps are null

Statement

Assume the Axiom of Choice. Let $g : R^{n} \to R^{n}$ be $L$ -Lipschitz in the Euclidean norm (Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction), where $n \geq 1$ . Write $B (x, r)$ for the Euclidean ball (Open ball, closed ball and sphere in a metric space). Let $C = {p \in R^{n} : g (p + h) = g (p) + D g (p) h + o (∣ h ∣) as h \to 0, \det D g (p) = 0} .$ Then $g (C)$ is null in $R^{n}$ .

Facts & Assumptions

Given: The Axiom of Choice, an integer $n \geq 1$ , and an $L$ -Lipschitz map $g : R^{n} \to R^{n}$ .

[A1]

Every family of nonempty sets has a choice function (The Axiom of Choice); in particular this supplies Countable Choice.

[F1]

A set in $R^{n}$ is null when it has closed-cube covers with arbitrarily small total volume (Measure zero and content zero in $R^{m}$ by countable and finite cube covers).

[F2]

Countable unions and subsets of null subsets of $R^{n}$ are null under Countable Choice (Subsets and countable unions of null subsets of $R^{m}$ are null).

[F3]

Zorn's lemma follows from the Axiom of Choice (Zorn's lemma).

[F4]

The Euclidean unit ball is measurable and has measure $v_{n}$ with $0 < v_{n} < \infty$ (Open ball, closed ball and sphere in a metric space, Euclidean balls have positive finite Lebesgue measure).

[F5]

Lebesgue measure is translation invariant and scales under positive dilations by the $n$ th power (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, For a nonzero real $c$ , dilation by $c$ multiplies Lebesgue outer measure by $∣ c ∣^{n}$ , and reflection in the origin preserves it).

[F6]

Lebesgue measure is countably additive on pairwise disjoint measurable sets and monotone (Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume, Measures are monotone).

[F7]

The product $N \times N$ is countable ( $N \times N \approx N$ ).

[F8]

The rational vectors $Q^{n}$ form a countable dense subset of $R^{n}$ by coordinatewise rational approximation, and every nonempty subset of the enumerating index set has a least element ( $Q$ is countably infinite, A product of two at most countable sets is at most countable, The rationals embed densely in the reals, The well-ordering principle).

[F9]

Finite sums of nonnegative terms are monotone, and a nonnegative series whose partial sums are bounded by $M$ converges with sum at most $M$ (Finite sums and finite products, by recursion, Laws of finite sums and finite products, Series in the nonnegative extended real line, A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum).

Proof

1.1A1F3F8givenconstruct

Fix $R \geq 1$ and $0 < ε < 1$ . For each $p \in C \cap B (0, R)$ , differentiability gives a nonempty set of radii $0 < r_{p} < 1$ such that $∣ g (p + h) - g (p) - D g (p) h ∣ \leq ε ∣ h ∣$ whenever $∣ h ∣ \leq 5 r_{p}$ . By [A1], choose one such radius for each $p$ . Divide this ball family into dyadic radius classes $[2^{- j - 1}, 2^{- j})$ , largest first. In each class use [F3] on the set of subfamilies disjoint from all previously selected balls; the union of any chain is an upper bound, so a maximal disjoint subfamily exists. The union of selected subfamilies is disjoint. It is countable by [F8]: enumerate $Q^{n}$ and assign to each ball the first rational point it contains; disjoint balls get distinct points. If a rejected ball has radius $r$ and meets a selected ball of radius $R_{s} > r / 2$ , then $r < 2 R_{s}$ and every point of the rejected ball is within $2 r + R_{s} < 5 R_{s}$ of the selected center. Thus the fivefold dilates of a countable disjoint subfamily $B (p_{i}, r_{i})$ cover $C \cap B (0, R)$ .

1.2givenconstructalgebra

Put $v_{n} = λ (B (0, 1))$ . For any selected ball $B (p_{i}, r_{i})$ , the image $g (B (p_{i}, 5 r_{i}))$ lies within distance $5 ε r_{i}$ of $g (p_{i}) + D g (p_{i}) B (0, 5 r_{i})$ . This affine image lies in a disk of radius $5 L r_{i}$ centered at $g (p_{i})$ in an affine subspace of dimension $q \leq n - 1$ . In orthonormal coordinates centered at $g (p_{i})$ and adapted to that subspace, the tube lies in a box with $q$ side lengths at most $10 L r_{i} + 10 ε r_{i}$ and $n - q$ side lengths $10 ε r_{i}$ . Subdivide each side into intervals of length at most $ε r_{i}$ . The number of resulting boxes is at most $c_{n} (1 + L / ε)^{q}$ ; each is contained in an axis-parallel cube of side at most $c_{n} ε r_{i}$ . Since $q \leq n - 1$ and $ε < 1$ , their total cube volume is at most $C_{n} ε (L + 1)^{n - 1} r_{i}^{n}$ for a finite constant $C_{n} \geq 1$ depending only on $n$ . This constructs actual null covers without a rotation-invariance theorem or an area formula.

2.1A1F4F5F6step 1.1algebra

The selected balls lie in $B (0, R + 1)$ and are pairwise disjoint. By [F5], each has measure $v_{n} r_{i}^{n}$ , and the containing ball has measure $v_{n} (R + 1)^{n}$ . Countable additivity and monotonicity [F6] give $\sum_{i} r_{i}^{n} \leq (R + 1)^{n}$ , using $v_{n} > 0$ . The Countable Choice hypotheses of [F5] and [F6] follow from [A1].

3.1A1F1F7F9step 1.1step 1.2step 2.1algebra

The fivefold balls in step 1.1 cover $C \cap B (0, R)$ , and step 1.2 supplies a finite cube cover of each corresponding image with total volume at most $C_{n} ε (L + 1)^{n - 1} r_{i}^{n}$ . By [A1], choose one such finite cover for each selected ball. Index these cubes by $(i, j)$ ; [F7] enumerates the resulting subset of $N^{2}$ . Every finite partial sum is at most $C_{n} ε (L + 1)^{n - 1} \sum_{i} r_{i}^{n} \leq C_{n} ε (L + 1)^{n - 1} (R + 1)^{n}$ by [F9] and step 2.1. For any cover budget $δ > 0$ , take $ε = \min {1 / 2, δ / [C_{n} (L + 1)^{n - 1} (R + 1)^{n}]} > 0$ ; the bound is then at most $δ$ . By [F1], $g (C \cap B (0, R))$ is null.

4.1A1F2step 3.1∎

The set $C$ is the union of $C \cap B (0, R)$ over integers $R \geq 1$ . Each image in this countable union is null by step 3.1; [F2] therefore gives that $g (C)$ is null. Its Countable Choice hypothesis follows from [A1].

Depends on

Used by

Alexandrov second differentiability of locally semiconvex functions Theorem

Dependency tree · two levels

124 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

Juha Kinnunen, Sobolev Spaces, complete 168-page notes; Sections 2.6 and 3.3-3.4 (standard reference, not scraped)
Crandall, Ishii and Lions, User's guide to viscosity solutions, Appendix A, Theorem A.2 and pp. 58-60, notes pp. 62-63 (standard reference, not scraped)