On Lebesgue null sets

TL;DR

This paper characterizes Lebesgue null sets in Euclidean space by examining their intersections with Lipschitz-varying affine subspaces and establishing an equivalence involving Hausdorff measure.

Contribution

It provides a new criterion for Lebesgue null sets based on their intersection properties with affine subspaces that vary Lipschitz continuously.

Findings

A Borel set is Lebesgue null iff its intersection with almost every affine subspace has zero Hausdorff measure.

The result links measure-theoretic nullity to geometric intersection properties.

The approach uses Lipschitz variation of affine subspaces to characterize null sets.

Abstract

Letting A be a Borel subset of n dimensional Euclidean space, and W(x) be an m dimensional affine subspace containing x and varying in a Lipschitz way according to x, we establish that A is Lebesgue null if and only if $A \cap W(x)$ has m dimensional Hausdorff measure vanishing for almost every x.

Equations

$${\int }_{A}Jf(x)d{\mathcal{L}}^n(x)={\int }_{{\mathbf{R}}^{n-m}}{\mathcal{H}}^m\left(A\cap f^{-1}\{y\}\right)d{\mathcal{L}}^{n-m}(y).$$

$$Jf(x)=\sqrt{\left|\det \left(Df(x)\circ Df(x{)}^{\ast }\right)\right|}=\Vert {\wedge }_{n-m}Df(x)\Vert$$

$$\text{If }{\mathcal{H}}^m(A\cap \mathbf{W}(x))=0\text{ for all }x\in A\text{ then }{\mathcal{L}}^n(A)=0,$$

$$\Sigma ={\mathbf{R}}^n\times {\mathbf{R}}^n\cap \{(x,u):x\in E\text{ and }u\in \mathbf{W}(x)\},$$

$${\varphi }_E(B)={\int }_E{\mathcal{H}}^m\left(B\cap \mathbf{W}(x)\right)d{\mathcal{L}}^n(x),$$

$${\mathcal{H}}^m\left(\Sigma \cap {\pi }_2^{-1}(B)\cap {\pi }_1^{-1}\{x\}\right)={\mathcal{H}}^m\left(B\cap \mathbf{W}(x)\right),$$

$${\varphi }_E(B)={\int }_{{\Sigma }_B}{J}_{\Sigma }{\pi }_{1}d{\mathcal{H}}^{n+m}.$$

$${\int }_{{\Sigma }_B}{J}_{\Sigma }{\pi }_{2}d{\mathcal{H}}^{n+m}={\int }_B{\mathcal{H}}^m\left({\Sigma }_B\cap {\pi }_2^{-1}\{u\}\right)d{\mathcal{L}}^n(u)$$

$${\mathcal{H}}^m\left({\Sigma }_B\cap {\pi }_2^{-1}\{u\}\right)={\mathcal{H}}^m\left(E\cap g_u^{-1}\{0\}\right).$$

$${\varphi }_E(B)={\int }_B{\mathcal{Z}}_E\mathbf{W}d{\mathcal{L}}^n$$

$$({\mathcal{Z}}_E\mathbf{W})(u)⩾{\beta }_n\underset{j}{\mathrm{liminf}}\text{vbox}-{\int }_{-j^{-1}}^{j^{-1}}{\mathcal{H}}^m\left(E\cap g_u^{-1}\{y\}\right)d{\mathcal{L}}^1(y)={\beta }_n({\mathcal{Y}}_E\mathbf{W})(u),$$

$${\int }_{Z\cap {\mathbf{C}}_{\mathbf{W}}(x_0,r)}\left({\mathcal{Y}}_{Z\cap {\mathbf{C}}_{\mathbf{W}}(x_0,r)}\right)(u)d{\mathcal{L}}^n(u).$$

$${\int }_{V_z}\left({\mathcal{Y}}_{Z\cap {\mathbf{C}}_{\mathbf{W}}(x_0,r)}\right)(u)d{\mathcal{H}}^1(u).$$

$$\begin{gathered} \sum _k{\mathcal{H}}^1(I_k)\text{vbox}-{\int }_{I_k}{\mathcal{H}}^m\left(Z\cap {\mathbf{C}}_{\mathbf{W}}(x_0,r)\cap g_{u_k}^{-1}\{y\}\right)d{\mathcal{L}}^1(y) \\ \cong \sum _k{\int }_{Z\cap {\mathbf{C}}_{\mathbf{W}}(x_0,r)\cap g_{u_k}^{-1}(I_k)}\left|\nabla g_{u_k}(x)\right|d{\mathcal{L}}^n(x)\cong {\mathcal{L}}^n\left(Z\cap {\mathbf{C}}_{\mathbf{W}}(x_0,r)\right) \end{gathered}$$

$${\mathcal{H}}_{\delta }^m(A)=\inf \left\{\sum _{j\in J}{\zeta }^m(S_j):(S_j{)}_{j\in J}\text{ is a δ cover of A }\right\}$$

$${\mathfrak{H}}_{\delta }^m(A)=\inf \left\{\sum _{j\in J}{\zeta }^m(S_j):(S_j{)}_{j\in J}\text{ is a δ cover of A and Sj⊆W for all j∈J}\right\},$$

$${\int }_{A}Jf(x)d{\mathcal{L}}^n(x)={\int }_{{\mathbf{R}}^{n-m}}{\mathcal{H}}^m\left(A\cap f^{-1}\{y\}\right)d{\mathcal{L}}^{n-m}(y).$$

$$Jf(x)=\sqrt{\left|\det \left(Df(x)\circ Df(x{)}^{\ast }\right)\right|}=\Vert {\wedge }_{n-m}Df(x)\Vert ,$$

$${\int }_A{J}_{A}f(x)d{\mathcal{H}}^n(x)={\int }_{{\mathbf{R}}^{n-m}}{\mathcal{H}}^m\left(A\cap f^{-1}\{y\}\right)d{\mathcal{L}}^{n-m}(y).$$

$$J_{A}f(x)=\sqrt{\left|\det \left(L\circ L^{\ast }\right)\right|}=\Vert {\wedge }_{n-m}L\Vert ,$$

$$\Vert {\wedge }_{k}L\Vert =\sup \left\{⟨{\wedge }_{k}L,\xi ⟩:\xi \in {\wedge }_{k}V\text{ and }|\xi |=1\right\}.$$

$$\Vert {\wedge }_{k}L\Vert ⩾\frac{|L(v_1)\wedge \dots \wedge L(v_k)|}{|v_1\wedge \dots \wedge v_k|}.$$

$${\overline{w}}_1(W)=w_1(W)\text{ and }{\overline{w}}_i(W)=w_i(W)-\sum _{j=1}^{i-1}⟨w_i(W),{\overline{w}}_j(W)⟩{\overline{w}}_j(W),i=2,\dots ,m,$$

$${\mathbf{W}}_0(x)=\mathrm{span}\{{\mathbf{w}}_1(x),\dots ,{\mathbf{w}}_m(x)\}$$

$${\mathbf{W}}_0(x{)}^{\perp }=\mathrm{span}\{{\mathbf{v}}_1(x),\dots ,{\mathbf{v}}_{n-m}(x)\}.$$

$${\mathbf{W}}_0(x)=\mathrm{span}\{{\mathbf{w}}_1(x),\dots ,{\mathbf{w}}_m(x)\}$$

$${\mathbf{W}}_0(x{)}^{\perp }=\mathrm{span}\{{\mathbf{v}}_1(x),\dots ,{\mathbf{v}}_{n-m}(x)\}.$$

$$g_{\mathbf{v},u}(x)=⟨\mathbf{v}(x),x-u⟩.$$

$$D{g}_{\mathbf{v},u}(x)(h)=⟨\nabla g_{\mathbf{v},u}(x),h⟩=⟨D\mathbf{v}(x)(h),x-u⟩+⟨\mathbf{v}(x),h⟩.$$

$$g_{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_{n-m},u}(x)=\left(g_{{\mathbf{v}}_1,u}(x),\dots ,g_{{\mathbf{v}}_{n-m},u}(x)\right).$$

Full text

The first author was partially supported by the Science and Technology Commission of Shanghai (No. 18dz2271000).

1. 1 Foreword
2. 2 Preliminaries
3. 3 Common setting
4. 4 Two fibrations
5. 5 Upper bound for $\mathscr{Y}_{E}\mathbf{W}$ and $\mathscr{Z}_{E}\mathbf{W}$
6. 6 Lower bound for $\mathscr{Y}_{E}\mathbf{W}$ and $\mathscr{Z}_{E}\mathbf{W}$
7. 7 Proof of the theorem

1. Foreword

Let $A$ be a subset of Euclidean space $\mathbf{R}^{n}$, $n\geqslant 2$, and let $\mathscr{L}^{n}$ denote the Lebesgue outer measure. We concern ourselves with the following question: Can one tell whether $A$ is Lebesgue negligible from the knowledge only of its trace on each member of some given collection of <<lower dimensional>> subsets $\Gamma_{i}\subseteq\mathbf{R}^{n}$, $i\in I$. Thus one expects that if $A\cap\Gamma_{i}$ is <<negligible in the dimension of $\Gamma_{i}$>>, for each $i\in I$ then $\mathscr{L}^{n}(A)=0$. Of course a necessary condition is that the sets $\Gamma_{i}$ cover almost all of $A$, i.e. $\mathscr{L}^{n}(A\thicksim\cup_{i\in I}\Gamma_{i})=0$. Consider for instance $n=2$, $I=\mathbf{R}$ and $\Gamma_{t}=\{t\}\times\mathbf{R}$, $t\in\mathbf{R}$, the collection of all vertical lines in the plane. It is not true in general that if $A\subseteq\mathbf{R}^{2}$ and $A\cap\Gamma_{t}$ is a singleton for each $t\in\mathbf{R}$ then $\mathscr{L}^{2}(A)=0$. There exist indeed functions $f:\mathbf{R}\to\mathbf{R}$ whose graph $A=\operatorname{\mathrm{graph}}f$ has $\mathscr{L}^{2}(A)>0$, see e.g. [8, Chapter 2 Theorem 4] for an example due to W. Sierpiński. In order to rule out such examples we will henceforth assume that $A\subseteq\mathbf{R}^{n}$ be Borel measurable. In that case the Theorem of G. Fubini, together with the invariance of the Lebesgue measure under orthogonal transformations imply the following. Given an integer $1\leqslant m\leqslant n-1$, if $(\Gamma_{i})_{i\in I}$ is the collection of all $m$ dimensional affine subspaces of $\mathbf{R}^{n}$ of some fixed direction, and if $\mathscr{H}^{m}(A\cap\Gamma_{i})=0$ for all $i\in I$ then $\mathscr{L}^{n}(A)=0$. Here $\mathscr{H}^{m}$ denotes the $m$ dimensional Hausdorff measure. A special feature of this collection $(\Gamma_{i})_{i\in I}$ is that it partitions $\mathbf{R}^{n}$, its members being the level sets $f^{-1}\{y\}$, $y\in\mathbf{R}^{n-m}$, of a <<nice map>> $f:\mathbf{R}^{n}\to\mathbf{R}^{n-m}$, indeed an orthogonal projection. This is an occurrence of the following more general situation when $f$ and its leaves $f^{-1}\{y\}$ are allowed to be nonlinear. The coarea formula due to H. Federer in [6] asserts that if $f:\mathbf{R}^{n}\to\mathbf{R}^{n-m}$ is Lipschitz and if $A\subseteq\mathbf{R}^{n}$ is Borel then

[TABLE]

Thus if the Jacobian coarea factor $Jf$ is positive $\mathscr{L}^{n}$ almost everywhere in $A$ then the collection $\left(f^{-1}\{y\}\right)_{y\in\mathbf{R}^{n-m}}$ is suitable for detecting whether or not $A$ is Lebesgue null. At $\mathscr{L}^{n}$ almost all $x\in\mathbf{R}^{n}$ the map $f$ is differentiable according to H. Rademacher, and

[TABLE]

see [4, Chapter 3 §4] and [7, 3.2.1 and 3.2.11].

In this paper we focus on the case when $\Gamma_{i}$, $i\in I$, are affine subspaces of $\mathbf{R}^{n}$, but not necessarily members of a partition of the ambient space. Specifically, we assume that with each $x\in\mathbf{R}^{n}$ is associated an $m$ dimensional affine subspace $\mathbf{W}(x)$ of $\mathbf{R}^{n}$ containing $x$. Given a Borel set $A\in\mathbf{R}^{n}$, the question whether

[TABLE]

has a negative answer: O. Nikodým [9] exhibited a Borel subset $A\subseteq\mathbf{R}^{2}$ of the unit square, such that $\mathscr{L}^{2}(A)=1$ and for each $x\in A$ there exists a line $\mathbf{W}(x)\subseteq\mathbf{R}^{2}$ with the property that $A\cap\mathbf{W}(x)=\{x\}$. In this context a selection Theorem due to J. von Neumann implies that (possibly considering a smaller, non Lebesgue null Borel subset of $A$) the correspondence $x\mapsto\mathbf{W}(x)$ can be chosen to be Borel measurable (see 2.19) and in turn, it can be chosen to be continuous according to a result of N. Lusin. This was noted by A. Zygmund in connection with multiparameter Fourier analysis.

Our result assumes that $\mathbf{W}$ be Lipschitz. Below $\mathbf{G}(n,m)$ denotes the Grassmannian manifold of $m$ dimensional linear subspaces of $\mathbf{R}^{n}$.

Theorem. —

Assume $\mathbf{W}_{0}:\mathbf{R}^{n}\to\mathbf{G}(n,m)$ is Lipschitz and $A\subseteq\mathbf{R}^{n}$ is Borel. The following are equivalent.

1. (A)

$\mathscr{L}^{n}(A)=0$; 2. (B)

For $\mathscr{L}^{n}$ almost every $x\in A$, $\mathscr{H}^{m}\left(A\cap(x+\mathbf{W}_{0}(x))\right)=0$; 3. (C)

For $\mathscr{L}^{n}$ almost every $x\in\mathbf{R}^{n}$, $\mathscr{H}^{m}\left(A\cap(x+\mathbf{W}_{0}(x))\right)=0$.

This seems to be new. As should be apparent from the discussion above, the difficulty stands with the fact that the affine $m$ planes $\mathbf{W}(x)=x+\mathbf{W}_{0}(x)$ need not be disjointed. The natural route is to reduce the problem to applying the coarea formula by spreading out the $\mathbf{W}(x)$’s in a disjointed way, in a higher dimensional space, i.e. adding a variable $u\in\mathbf{W}(x)$ to the given $x\in\mathbf{R}^{n}$ and considering $\mathbf{W}(x)$ as a fiber above the base space $\mathbf{R}^{n}$. We thus define

[TABLE]

where $E\subseteq\mathbf{R}^{n}$ is Borel. This set is $n+m$ rectifiable owing to the Lipschitz continuity of $\mathbf{W}$. It is convenient to assume that $\mathscr{L}^{n}(E)<\infty$ so that

[TABLE]

$B\subseteq\mathbf{R}^{n}$, is a locally finite Borel measure, 2.16. Now $\Sigma$ was precisely set up so that for each $x\in E$

[TABLE]

where $\pi_{1}$ and $\pi_{2}$ denote the projections of $\mathbf{R}^{n}\times\mathbf{R}^{n}$ to the $x$ and $u$ variable, respectively. Abbreviating $\Sigma_{B}=\Sigma\cap\pi_{2}^{-1}(B)$ the coarea formula yields

[TABLE]

A simple calculation shows that $J_{\Sigma}\pi_{2}>0$ $\mathscr{H}^{n+m}$ almost everywhere on $\Sigma_{B}$, 4.4. Since also

[TABLE]

the implication $(A)\Rightarrow(C)$ above should now be clear: Letting $B=A$ and $E=\mathbf{B}(0,R)$ one infers from hypothesis (A) and (4) that $\mathscr{H}^{n+m}(\Sigma_{B})=0$, thus $\phi_{\mathbf{B}(0,R)}(A)=0$ from (3) and in turn conclusion (C) from (2).

In order to establish that $(B)\Rightarrow(A)$ we need to observe that $J_{\Sigma}\pi_{1}>0$ $\mathscr{H}^{n+m}$ almost everywhere, 4.2 and ideally to show that $\mathscr{H}^{m}\left(\Sigma_{B}\cap\pi_{2}^{-1}\{u\}\right)>0$ for almost every $u\in B$. This last part offers some difficulty. To understand this we let $m=n-1$ in order to keep the notations short. Now $u\in\mathbf{W}(x)$ iff $u-x\in\mathbf{W}_{0}(x)$ iff $\langle\mathbf{v}_{0}(x),x-u\rangle=0$ where $\mathbf{v}_{0}(x)\in\mathbf{W}_{0}(x)^{\perp}$ is, say a unit vector. Abbreviating $g_{u}(x)=\langle\mathbf{v}_{0}(x),x-u\rangle$ we infer that

[TABLE]

The problem remains that two of the nonlinear $m$ sets $E\cap g_{u}^{-1}\{0\}$ and $E\cap g_{u^{\prime}}^{-1}\{0\}$ may intersect, thereby preventing another application of the coarea formula to look out for their lower bound. Yet we already know that

[TABLE]

where $\mathscr{Z}_{E}\mathbf{W}$ is a Radon-Nikodým derivative, 4.6 and also that $(\mathscr{Z}_{E}\mathbf{W})(u)$ is comparable to $\mathscr{H}^{m}\left(E\cap g_{u}^{-1}\{0\}\right)$, 4.8. Adding an extra variable $y$ to the fibered space $\Sigma$, 4.10 we improve on this by showing that

[TABLE]

where the last equality defines $\mathscr{Y}_{E}\mathbf{W}$, and $\beta_{n}>0$, 4.12 and 4.16. We are reduced to showing that $\mathscr{Y}_{E}\mathbf{W}>0$ almost everywhere. The reason why this holds is the following. Fix a Borel set $Z\subseteq\mathbf{R}^{n}$, $x_{0}\in\mathbf{R}^{n}$ and $r>0$. Let $\mathbf{C}_{\mathbf{W}}(x_{0},r)$ denote the cylindrical box consisting of those $x\in\mathbf{R}^{n}$ such that $\left|P_{\mathbf{W}_{0}(x_{0})}(x-x_{0})\right|\leqslant r$ and $\left|P_{\mathbf{W}_{0}(x_{0})^{\perp}}(x-x_{0})\right|\leqslant r$. We want to find a lower bound for

[TABLE]

To this end we fix $z\in\mathbf{W}_{0}(x_{0})\cap\mathbf{B}(0,r)$ and we let $V_{z}=\mathbf{R}^{n}\cap\{x_{0}+z+s\mathbf{v}_{0}(x_{0}):-r\leqslant s\leqslant r\}$ denote the corresponding vertical line segment. According to Fubini’s Theorem we are reduced to estimating

[TABLE]

According to Vitali’s Covering Theorem we can find a disjointed family of line segments $I_{1},I_{2},\ldots$ covering almost all $V_{z}$ such that the above integral nearly equals

[TABLE]

where there first near equality follows from the coarea formula, the second one because $\left|\nabla g_{u_{k}}\right|\cong 1$ at small scales, 2.12 and the <<nonlinear horizontal stripes>> $g_{u_{k}}^{-1}(I_{k})$ are nearly pairwise disjoint. Verification of these claims takes up sections 5 and 6. Now we reach a contradiction if $Z=\mathbf{R}^{n}\cap\{\mathscr{Y}_{E}\mathbf{W}=0\}$ is assumed to have $\mathscr{L}^{n}(Z)>0$ and $x_{0}$ is a point of density of $Z$.

I extend my warm thanks to Jean-Christophe Léger for carefully reading the manuscript.

2. Preliminaries

2.1. —

In this paper $1\leqslant m\leqslant n-1$ are integers. The ambient space is $\mathbf{R}^{n}$. The canonical inner product of $x,x^{\prime}\in\mathbf{R}^{n}$ is denoted $\langle x,x^{\prime}\rangle$ and the corresponding Euclidean norm of $x$ is $|x|$. If $S\subseteq\mathbf{R}^{n}$ we let $\mathscr{B}(S)$ denote the $\sigma$ algebra of Borel subsets of $S$.

2.2Hausdorff measure. —

We let $\mathscr{L}^{n}$ denote the Lebesgue outer measure in $\mathbf{R}^{n}$ and $\boldsymbol{\alpha}(n)=\mathscr{L}^{n}(\mathbf{B}(0,1))$. For $S\subseteq\mathbf{R}^{n}$ we abbreviate $\zeta^{m}(S)=\boldsymbol{\alpha}(m)2^{-m}(\operatorname{\mathrm{diam}}S)^{m}$. Given $0<\delta\leqslant\infty$ we call $\delta$ cover of $A\subseteq\mathbf{R}^{n}$ a finite or countable family $(S_{j})_{j\in J}$ of subsets of $\mathbf{R}^{n}$ such that $A\subseteq\cup_{j\in J}S_{j}$ and $\operatorname{\mathrm{diam}}S_{j}\leqslant\delta$ for every $j\in J$. We define

[TABLE]

and $\mathscr{H}^{m}(A)=\lim_{\delta\to 0^{+}}\mathscr{H}^{m}_{\delta}(A)=\sup_{\delta>0}\mathscr{H}^{m}_{\delta}(A)$. Thus $\mathscr{H}^{m}$ is the $m$ dimensional Hausdorff outer measure in $\mathbf{R}^{n}$.

1. (1)

If $(K_{k})_{k}$ is a sequence of nonempty compact subsets of $\mathbf{R}^{n}$ converging in Hausdorff distance to $K$ then $\mathscr{H}^{m}_{\infty}(K)\geqslant\limsup_{k}\mathscr{H}^{m}_{\infty}(K_{k})$.

Given $\varepsilon>0$ choose a cover $(S_{j})_{j=1,2,\ldots}$ of $K$ such that $\mathscr{H}^{m}_{\infty}(K)+\varepsilon\geqslant\sum_{j}\zeta^{m}(S_{j})$. Since $\lim_{r\to 0^{+}}\zeta^{m}(\mathbf{U}(S_{j},r))=\zeta^{m}(S_{j})$ for each $j=1,2,\ldots$ we can choose an open set $U_{j}$ containing $S_{j}$ such that $\zeta^{m}(U_{j})\leqslant\varepsilon 2^{-j}+\zeta^{m}(S_{j})$. Since $U=\cup_{j}U_{j}$ is open there exists an integer $k_{0}$ such that $K_{k}\subseteq U$ whenever $k\geqslant k_{0}$. Thus in that case $(U_{j})_{j}$ is a cover of $K_{k}$ and therefore $\mathscr{H}^{m}_{\infty}(K_{k})\leqslant\sum_{j}\zeta^{m}(U_{j})\leqslant 2\varepsilon+\mathscr{H}^{m}_{\infty}(K)$. Take the $\limsup$ of the left hand side as $k\to\infty$, and then let $\varepsilon\to 0$.

1. (2)

For all $A\subseteq\mathbf{R}^{m}$ one has $\mathscr{L}^{m}(A)=\mathscr{H}^{m}(A)=\mathscr{H}^{m}_{\infty}(A)$.

It suffices to note that $\mathscr{H}^{m}(A)\geqslant\mathscr{H}^{m}_{\infty}(A)\geqslant\mathscr{L}^{m}(A)\geqslant\mathscr{H}^{m}(A)$. The first inequality is trivial; the second one follows from the isodiametric inequality [7, 2.10.33]; the last one is a consequence of the Vitali Covering Theorem [4, Chapter 2 §2 Theorem 2].

1. (3)

If $W\subseteq\mathbf{R}^{n}$ is an $m$ dimensional affine subspace and $A\subseteq W$ then $\mathscr{H}^{m}(A)=\mathscr{H}^{m}_{\infty}(A)$.

Let $\mathfrak{H}^{m}$ denote the $m$ dimensional Hausdorff outer measures in the metric space $W$. In other words

[TABLE]

and $\mathfrak{H}^{m}(A)=\sup_{\delta>0}\mathfrak{H}^{m}_{\delta}(A)$. It is elementary to observe that $\mathfrak{H}^{m}(A)=\mathscr{H}^{m}(A)$ and that $\mathfrak{H}^{m}_{\infty}(A)=\mathscr{H}^{m}_{\infty}(A)$. Now if $f:W\to\mathbf{R}^{m}$ is an isometry then $\mathscr{H}^{m}(A)=\mathfrak{H}^{m}(A)=\mathscr{H}^{m}(f(A))=\mathscr{H}^{m}_{\infty}(f(A))=\mathfrak{H}^{m}_{\infty}(A)=\mathscr{H}^{m}_{\infty}(A)$, where the third equality follows from claim (2) above.

2.3Coarea formula. —

Here we recall two versions of the coarea formula. First if $A\subseteq\mathbf{R}^{n}$ is $\mathscr{L}^{n}$ measurable and $f:A\to\mathbf{R}^{n-m}$ is Lipschitz then $\mathbf{R}^{n-m}\to[0,\infty]:y\mapsto\mathscr{H}^{m}\left(A\cap f^{-1}\{y\}\right)$ is $\mathscr{L}^{n-m}$ measurable and

[TABLE]

Here the coarea Jacobian factor is well defined $\mathscr{L}^{n}$ almost everywhere according to Rademacher’s Theorem and equals

[TABLE]

see for instance [4, Chapter 3 §4].

Secondly if $A\subseteq\mathbf{R}^{p}$ is $\mathscr{H}^{n}$ measurable and countably $(\mathscr{H}^{n},n)$ rectifiable, and if $f:A\to\mathbf{R}^{n-m}$ is Lipschitz then $\mathbf{R}^{n-m}\to[0,\infty]:y\mapsto\mathscr{H}^{m}\left(A\cap f^{-1}\{y\}\right)$ is $\mathscr{L}^{n-m}$ measurable and

[TABLE]

To give a formula for the coarea Jacobian factor $J_{A}f(x)$ of $f$ relative to $A$ we consider a point $x\in A$ where $A$ admits an approximate $n$ dimensional tangent space $T_{x}A$ and where $f$ is differentiable along $A$. Letting $L:T_{x}A\to\mathbf{R}^{n-m}$ denote the derivative of $f$ at $x$ we have

[TABLE]

see for instance [7, 3.2.22].

In both cases it is useful to recall the following. If $L:V\to V^{\prime}$ is a linear map between two inner product spaces $V$ and $V^{\prime}$ then

[TABLE]

On the one hand $\|\wedge_{k}L\|\leqslant\|L\|^{k}$ [7, 1.7.6], and $\|L\|\leqslant\operatorname{\mathrm{Lip}}f$ with $L$ as above. On the other hand if $v_{1},\ldots,v_{k}$ are linearly independent vectors of $V$ then

[TABLE]

Finally we observe that both coarea formulæ hold true when $f$ is merely locally Lipschitz, according to the Monotone Convergence Theorem.

2.4Grassmannian. —

We let $\mathbf{G}(n,m)$ denote the set whose members are the $m$ dimensional linear subspaces of $\mathbf{R}^{n}$. With $W\in\mathbf{G}(n,m)$ we associate $P_{W}:\mathbf{R}^{n}\to\mathbf{R}^{n}$ the orthogonal projection onto $W$. We give $\mathbf{G}(n,m)$ the structure of a compact metric space by letting $d(W_{1},W_{2})=\|P_{W_{1}}-P_{W_{2}}\|$. If $W\in\mathbf{G}(n,m)$ then $W^{\perp}\in\mathbf{G}(n-m)$ is so that $P_{W}+P_{W^{\perp}}=\operatorname{\mathrm{id}}_{\mathbf{R}^{n}}$, therefore $\mathbf{G}(n,m)\to\mathbf{G}(n,n-m):W\mapsto W^{\perp}$ is an isometry. The bijective correspondence $\varphi:\mathbf{G}(n,m)\to\operatorname{\mathrm{Hom}}(\mathbf{R}^{n},\mathbf{R}^{n}):W\mapsto P_{W}$ identifies $\mathbf{G}(n,m)$ with the submanifold $M_{n,m}=\operatorname{\mathrm{Hom}}(\mathbf{R}^{n},\mathbf{R}^{n})\cap\{L:L\circ L=L\,,\,L^{*}=L\text{ and }\operatorname{\mathrm{trace}}L=m\}$. There exists an open neighborhood $V$ of $M_{n,m}$ in $\operatorname{\mathrm{Hom}}(\mathbf{R}^{n},\mathbf{R}^{n})$ and a Lipschitz retraction $\rho:V\to M_{n,m}$, according for instance to [7, 3.1.20]. Therefore if $S\subseteq\mathbf{R}^{n}$ and if $\mathbf{W}_{0}:S\to\mathbf{G}(n,m)$ is Lipschitz then there exist an open neighborhood $U$ of $E$ in $\mathbf{R}^{n}$ and a Lipschitz extension $\widehat{\mathbf{W}}_{0}:U\to\mathbf{G}(n,m)$ of $\mathbf{W}_{0}$. Indeed $\varphi\circ\mathbf{W}_{0}$ admits a Lipschitz extension $\mathbf{Y}:\mathbf{R}^{n}\to\operatorname{\mathrm{Hom}}(\mathbf{R}^{n},\mathbf{R}^{n})$, see e.g. [7, 2.10.43], and it suffices to let $U=\mathbf{Y}^{-1}(V)$ and $\widehat{\mathbf{W}}_{0}=\rho\circ\left(\mathbf{Y}|_{U}\right)$.

2.5Orthonormal frames. —

We let $\mathbf{V}(n,m)$ denote the set orthonormal $m$ frames in $\mathbf{R}^{n}$, i.e. $\mathbf{V}(n,m)=(\mathbf{R}^{n})^{m}\cap\{(w_{1},\ldots,w_{m}):\text{ the family }w_{1},\ldots,w_{m}\text{ is orthonormal}\}$. We will consider it as a metric space with its structure inherited from $(\mathbf{R}^{n})^{m}$.

2.6. —

*Let $\mathscr{V}\subseteq\mathbf{G}(n,m)$ be a nonempty closed set such that $\operatorname{\mathrm{diam}}\mathscr{V}<1$. There exists a Lipschitz map $\Xi:\mathscr{V}\to\mathbf{V}(n,m)$ such that $W=\operatorname{\mathrm{span}}\{\Xi_{1}(W),\ldots,\Xi_{m}(W)\}$ for every $W\in\mathscr{V}$. *

Proof.

Pick arbitrarily $W_{0}\in\mathscr{V}$. If $W\in\mathscr{V}$ then the map $W_{0}\to W:w\mapsto P_{W}(w)$ is bijective: if $w\in W_{0}\thicksim\{0\}$ then $\left|P_{W}(w)-w\right|=\left|P_{W}(w)-P_{W_{0}}(w)\right|<|w|$ thus $P_{W}(w)\neq 0$. Letting $w_{1},\ldots,w_{m}$ be an arbitrary basis of $W_{0}$ it follows that for each $W\in\mathscr{V}$ the vectors $w_{i}(W)=P_{W}(w_{i})$, $i=1,\ldots,m$, constitute a basis of $W$. Furthermore the maps $w_{i}:\mathscr{V}\to\mathbf{R}^{n}$ are Lipschitz: $\left|w_{i}(W)-w_{i}(W^{\prime})\right|=\left|P_{W}(w_{i})-P_{W^{\prime}}(w_{i})\right|\leqslant d(W,W^{\prime})|w_{i}|$. We apply the Gram-Schmidt process:

[TABLE]

so that $\overline{w}_{1}(W),\ldots,\overline{w}_{m}(W)$ is readily an orthogonal basis of $W$ depending upon $W$ in a Lipschitz way. Since each $\left|\overline{w}_{i}\right|$ is bounded away from zero on $\mathscr{V}$ the formula $\Xi_{i}(W)=\left|\overline{w}_{i}(W)\right|^{-1}\overline{w}_{i}(W)$, $i=1,\ldots,m$, defines $\Xi$ with the required property. ∎

2.7. —

*There exists a Borel measurable map $\Xi:\mathbf{G}(n,m)\to\mathbf{V}(n,m)$ with the property that $W=\operatorname{\mathrm{span}}\{\Xi_{1}(W),\ldots,\Xi_{m}(W)\}$ for every $W\in\mathbf{G}(n,m)$. *

Proof.

Since $\mathbf{G}(n,m)$ is compact it can partitioned into finitely many Borel sets $\mathscr{V}_{1},\ldots,\mathscr{V}_{J}$ each having diameter bounded by $1/2$. Define $\Xi$ piecewise to coincide on $\mathscr{V}_{j}$ with a $\Xi_{j}$ associated with $\operatorname{\mathrm{Clos}}\mathscr{V}_{j}$ in 2.6, $j=1,\ldots,J$. ∎

2.8. —

Assume $S\subseteq\mathbf{R}^{n}$, $x_{0}\in S$ and $\mathbf{W}_{0}:S\to\mathbf{G}(n,m)$ is Lipschitz. There then exist an open neighbordhood $U$ of $x_{0}$ in $\mathbf{R}^{n}$ and Lipschitz maps $\mathbf{w}_{1},\ldots,\mathbf{w}_{m},\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:U\to\mathbf{R}^{n}$ such that:

1. (1)

For every $x\in U$ the family $\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x),\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$ is an orthonormal basis of $\mathbf{R}^{n}$; 2. (2)

For every $x\in S\cap U$ one has

[TABLE]

and

[TABLE]

Proof.

We let $\widehat{\mathbf{W}}_{0}:\widehat{U}\to\mathbf{G}(n,m)$ be a Lipschitz extension of $\mathbf{W}_{0}$ where $\widehat{U}$ is an open neighborhood of $S$ in $\mathbf{R}^{n}$ (recall 2.4). Abbreviate $W_{0}:=\mathbf{W}_{0}(x_{0})$. Define $\mathscr{V}=\mathbf{G}(n,m)\cap\left\{W:d(W,W_{0})<1/4\right\}$ and $V=\widehat{U}\cap\widehat{\mathbf{W}}_{0}^{-1}(\mathscr{V})$. Apply 2.6 to $\operatorname{\mathrm{Clos}}\mathscr{V}$ and denote $\Xi$ the resulting Lipschitz map $\mathscr{V}\to(\mathbf{R}^{n})^{m}$. Next define $\mathscr{V}^{\perp}=\mathbf{G}(n,n-m)\cap\left\{W^{\perp}:W\in\mathscr{V}\right\}$, apply 2.6 to $\operatorname{\mathrm{Clos}}\mathscr{V}^{\perp}$ and denote $\Xi^{\perp}$ the resulting Lipschitz map $\mathscr{V}^{\perp}\to(\mathbf{R}^{n})^{n-m}$. Letting $\mathbf{w}_{i}(x)=\left(\Xi_{i}\circ\widehat{\mathbf{W}}_{0}\right)(x)$, $i=1,\ldots,m$, and $\mathbf{v}_{i}(x)=\left(\Xi^{\perp}_{i}\circ\widehat{\mathbf{W}}_{0}\right)(x)$, $i=1,\ldots,n-m$, completes the proof. ∎

2.9. —

Assume $\mathbf{W}_{0}:\mathbf{R}^{n}\to\mathbf{G}(n,m)$ is Borel measurable. There then exist Borel measurable maps $\mathbf{w}_{1},\ldots,\mathbf{w}_{m},\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:\mathbf{R}^{n}\to\mathbf{R}^{n}$ such that:

1. (1)

For every $x\in\mathbf{R}^{n}$ the family $\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x),\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$ is an orthonormal basis of $\mathbf{R}^{n}$; 2. (2)

For every $x\in\mathbf{R}^{n}$ one has

[TABLE]

and

[TABLE]

Proof.

Choose $\Xi:\mathbf{G}(n,m)\to\mathbf{V}(n,m)$ and $\Xi^{\perp}:\mathbf{G}(n,n-m)\to\mathbf{V}(n,n-m)$ be as in 2.7. Letting $\left(\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x)\right)=\left(\Xi\circ\mathbf{W}_{0}\right)(x)$ and $\left(\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)\right)=\left(\Xi^{\perp}\circ\mathbf{W}_{0}^{\perp}\right)(x)$, $x\in\mathbf{R}^{n}$, completes the proof. ∎

2.10Definition of $\mathbf{W}(x)$. —

The typical situation that arises in the remaining part of this paper is that we are given a set $S\subseteq\mathbf{R}^{n}$, a Lipschitz map $\mathbf{W}_{0}:S\to\mathbf{G}(n,m)$ and $x_{0}\in S$. We will represent $\mathbf{W}_{0}(x)$ and $\mathbf{W}_{0}^{\perp}(x)$ in a neighborhood $U$ of $x_{0}$ as in 2.8. We will then further reduce the size of $U$ several times in order that various conditions be met. With no exception we will denote as $\mathbf{W}(x)=x+\mathbf{W}_{0}(x)$ the affine subspace containing $x$, of direction $\mathbf{W}_{0}(x)$, whenever $\mathbf{W}_{0}(x)$ is defined.

2.11Definition of $g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}$ and lower bound of its coarea factor. —

Given an open set $U\subseteq\mathbf{R}^{n}$, a Lipschitz map $\mathbf{v}:U\to\mathbf{R}^{n}$, and $u\in\mathbf{R}^{n}$ we define $g_{\mathbf{v},u}:U\to\mathbf{R}$ by the formula

[TABLE]

Clearly $g_{\mathbf{v},u}$ is locally Lipschitz. If $\mathbf{v}$ is differentiable at $x\in U$ then so is $g_{\mathbf{v},u}$ and for every $h\in\mathbf{R}^{n}$ one has

[TABLE]

Next we assume we are given Lipschitz maps $\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:U\to\mathbf{R}^{n}$. We define $g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}:U\to\mathbf{R}^{n-m}$ by the formula

[TABLE]

It is Lipschitz as well. The relevance of $g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}$ stems from the following observation, assuming that $\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}$ are associated with $\mathbf{W}_{0}$ and $\mathbf{W}$ as in 2.8 and 2.10:

[TABLE]

In fact $|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x)|=\left|P_{\mathbf{W}_{0}(x)^{\perp}}(x-u)\right|$.

Abbreviate $g=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}$. If each $\mathbf{v}_{i}$ is differentiable at $x\in U$, and $h\in\mathbf{R}^{n}$, then

[TABLE]

where here and elsewhere $e_{1},\ldots,e_{n-m}$ denotes the canonical basis of $\mathbf{R}^{n-m}$. Thus if $\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$ constitute an orthonormal family in $\mathbf{R}^{n}$ then

[TABLE]

where

[TABLE]

according to (7), and in turn

[TABLE]

This allows for a lower bound of the coarea factor of $g$ at $x$ as follows.

[TABLE]

In view of (9) we obtain the next lemma.

2.12. —

Given $\Lambda>0$ and $0<\varepsilon<1$ there exists $\boldsymbol{\delta}_{2.12}(n,\Lambda,\varepsilon)>0$ with the following property. Assume that

1. (1)

$U\subseteq\mathbf{R}^{n}$* is open and $u\in\mathbf{R}^{n}$;* 2. (2)

$\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:U\to\mathbf{R}^{n}$* are Lipschitz;* 3. (3)

$\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$* is an orthonormal family for every $x\in U$.*

If

1. (4)

$\operatorname{\mathrm{Lip}}\mathbf{v}_{i}\leqslant\Lambda$* for each $i=1,\ldots,n-m$;* 2. (5)

$\operatorname{\mathrm{diam}}\left(U\cup\{u\}\right)\leqslant\boldsymbol{\delta}_{2.12}(n,\Lambda,\varepsilon)$**

then

[TABLE]

*at $\mathscr{L}^{n}$ almost every $x\in U$. *

2.13Definition of $\pi_{u}$ and its relation with $g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}$. —

With $u\in\mathbf{R}^{n}$ we associate

[TABLE]

When $(\xi_{1},\ldots,\xi_{n-m})\in\mathbf{V}(n,n-m)$ is fixed we also abbreviate as $\pi_{\xi_{1},\ldots,\xi_{n-m},u}$ the map $\mathbf{R}^{n}\to\mathbf{R}^{n-m}$ defined by $\pi_{\xi_{1},\ldots,\xi_{n-m},u}(x)=\pi_{u}(\xi_{1},\ldots,\xi_{n-m},x)$. It is then rather useful to observe that in the context described in 2.8 and 2.10 the following holds:

[TABLE]

Indeed,

[TABLE]

In the sequel we will sometimes abbreviate $\xi=(\xi_{1},\ldots,\xi_{n-m})\in\mathbf{V}(n,n-m)$. It also helps to notice that for given $\xi\in\mathbf{V}(n,n-m)$ and $y\in\mathbf{R}^{n-m}$ the set $\pi_{\xi,u}^{-1}\{y\}$ is an $m$ dimensional affine subspace of $\mathbf{R}^{n}$.

2.14. —

Assume $B\in\mathscr{B}(\mathbf{R}^{n})$ and $u\in\mathbf{R}^{n}$. It follows that

[TABLE]

*is Borel measurable. *

Proof.

We start by showing that when $B$ is compact, $h_{B}$ is upper semicontinuous. Thus if $(\xi_{k},y_{k})\in\mathbf{V}(n,n-m)\times\mathbf{R}^{n-m}$ converge to $(\xi,y)$, we ought to show that

[TABLE]

where $K=B\cap\pi_{\xi,u}^{-1}\{y\}$ and $K_{k}=B\cap\pi_{\xi_{k},u}^{-1}\{y_{k}\}$. This is indeed equivalent to the same inequality with $\mathscr{H}^{m}_{\infty}$ replaced by $\mathscr{H}^{m}$ according to 2.2(3) and the last sentence of 2.13. Considering if necessary a subsequence of $(K_{k})_{k}$ we may assume that none of the compact sets $K_{k}$ is empty, and that the $\limsup$ in (11) is a $\lim$. Since the set of nonempty compact subsets of the compact set $B$, equipped with the Hausdorff metric is compact, the sequence $(K_{k})_{k}$ admits a subsequence (denoted the same way) converging to a compact set $L\subseteq B$. Given $z\in L$ there are $z_{k}\in K_{k}$ converging to $z$. Thus $\pi_{u}(\xi,z)=\lim_{k}\pi_{u}(\xi_{k},z_{k})=\lim_{k}y_{k}=y$. In other words $z\in K$. Thus $\mathscr{H}^{m}_{\infty}(K)\geqslant\mathscr{H}^{m}_{\infty}(L)$ and (11) follows from 2.2(1).

Next we abbreviate $\mathscr{A}=\mathscr{B}(\mathbf{R}^{n})\cap\{B:h_{B}\text{ is Borel measurable}\}$. Thus we have just shown that $\mathscr{A}$ contains the collection $\mathscr{K}(\mathbf{R}^{n})$ of all compact subsets of $\mathbf{R}^{n}$. Observe that if $(B_{j})_{j}$ is an increasing sequence in $\mathscr{A}$ and $B=\cup_{j}B_{j}$ then $h_{B}=\lim_{j}h_{B_{j}}$ pointwise, thus $B\in\mathscr{A}$. In particular $\mathbf{R}^{n}\in\mathscr{A}$. Finally if $B,B^{\prime}\in\mathscr{A}$ and $B^{\prime}\subseteq B$ then $h_{B\thicksim B^{\prime}}=h_{B}-h_{B^{\prime}}$ because all measures involved are finite, indeed $h_{B}(\xi,y)\leqslant\boldsymbol{\alpha}(m)r^{m}$ for all $(\xi,y)$. Accordingly $B\thicksim B^{\prime}\in\mathscr{A}$. This means that $\mathscr{A}$ is a Dynkin class. Since $\mathscr{K}(\mathbf{R}^{n})$ is a $\pi$ system, $\mathscr{A}$ contains the $\sigma$ algebra generated by $\mathscr{K}(\mathbf{R}^{n})$, i.e. $\mathscr{B}(\mathbf{R}^{n})$, [1, Theorem 1.6.2]. ∎

2.15. —

Assume $B\in\mathscr{B}(\mathbf{R}^{n})$, $r>0$ and $\mathbf{W}_{0}:\mathbf{R}^{n}\to\mathbf{G}(n,m)$ is Borel measurable. The following function is Borel measurable.

[TABLE]

Proof.

Let $h_{\mathbf{W},B}$ denote this function. Let $\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:\mathbf{R}^{n}\to\mathbf{R}^{n}$ be Borel measurable maps associated with $\mathbf{W}_{0}$ as in 2.9. Fix $u\in\mathbf{R}^{n}$ arbitrarily. Define

[TABLE]

so that

[TABLE]

(where $h_{B}$ is the function associated with $B$ and $u$ in 2.14), according to (10). One notes that $\Upsilon$ is Borel measurable, and the conclusion ensues from 2.14. ∎

2.16Definition of $\phi_{E,\mathbf{W}}$. —

Let $\mathbf{W}_{0}:\mathbf{R}^{n}\to\mathbf{G}(n,m)$ be Borel measurable and let $E\in\mathscr{B}(\mathbf{R}^{n})$ be such that $\mathscr{L}^{n}(E)<\infty$. For each $B\in\mathscr{B}(\mathbf{R}^{n})$ we define

[TABLE]

This is well defined according to 2.15(2). It is easy to check that $\phi_{E,\mathbf{W}}$ is a locally finite (hence $\sigma$ finite) Borel measure on $\mathbf{R}^{n}$; indeed $\phi_{E,\mathbf{W}}(B)\leqslant\boldsymbol{\alpha}(m)(\operatorname{\mathrm{diam}}B)^{m}\mathscr{L}^{n}(E)$.

To close this section we discuss the relevance of $\phi_{E,\mathbf{W}}$ to the problem of existence of <<nearly Nikodým sets>>.

2.17Definition of Nearly Nikodým set. —

Let $E\in\mathscr{B}(\mathbf{R}^{n})$. We say that $B\in\mathscr{B}(E)$ is nearly $m$ Nikodým in $E$ if

1. (1)

$\mathscr{L}^{n}(B)>0$; 2. (2)

For $\mathscr{L}^{n}$ almost each $x\in E$ there is $W\in\mathbf{G}(n,m)$ such that $\mathscr{H}^{m}\left(B\cap(x+W)\right)=0$.

In case $n=2$, $m=1$, $E=[0,1]\times[0,1]$, the existence of such $B$ (with $\mathscr{L}^{2}(B)=1$) was established by O. Nikodým [9], see also [2, Chapter 8]. For arbitrary $n\geqslant 2$ and $m=n-1$ the existence of such $B$ was established by K. Falconer [5]. In fact in both cases these authors proved the stronger condition that for every $x\in B$, $\mathscr{H}^{m}(B\cap(x+W))=0$ can be replaced by $B\cap(x+W)=\{x\}$. Thus in case $1\leqslant m<n-1$, if $B$ is a set exhibited by K. Falconer, $x\in B$ and $W\subseteq\in\mathbf{G}(n,n-1)$ is such that $B\cap(x+W)=\{x\}$, picking arbitrarily $V\in\mathbf{G}(n,m)$ such that $V\subseteq W$ we see that $B\cap(x+V)=\{x\}$. Whence $B$ is also nearly $m$ Nikodým in $B$.

Assuming also that $\mathbf{W}_{0}:E\to\mathbf{G}(n,m)$ is Borel measurable we say that $B\in\mathscr{B}(E)$ is nearly $m$ Nikodým in $E$ relative to $\mathbf{W}$ if

1. (1)

$\mathscr{L}^{n}(B)>0$; 2. (2)

For $\mathscr{L}^{n}$ almost each $x\in E$ one has $\mathscr{H}^{m}\left(B\cap\mathbf{W}(x)\right)=0$.

2.18. —

Let $E\in\mathscr{B}(\mathbf{R}^{n})$ and let $\mathbf{W}_{0}:\mathbf{R}^{n}\to\mathbf{G}(n,m)$ be Borel measurable. The following are equivalent.

1. (1)

$\mathscr{L}^{n}|_{\mathscr{B}(E)}$* is absolutely continuous with respect to $\phi_{E,\mathbf{W}}|_{\mathscr{B}(E)}$.* 2. (2)

There does not exist a nearly $m$ Nikodým set relative to $\mathbf{W}$.

Proof.

A set $B\in\mathscr{B}(E)$ such that $\phi_{E,\mathbf{W}}(B)=0$ and $\mathscr{L}^{n}(B)>0$ is, by definition a nearly $m$ Nikodým set relative to $\mathbf{W}$. Condition (1) is equivalent to their nonexistence. ∎

2.19. —

Assume that $E\in\mathscr{B}(\mathbf{R}^{n})$ and that $B\in\mathscr{B}(E)$ is nearly $m$ Nikodým. It follows that:

1. (1)

There exists $\mathbf{W}_{0}:\mathbf{R}^{n}\to\mathbf{G}(n,m)$ Borel measurable such that $B$ is nearly $m$ Nikodým in $E$ relative to $\mathbf{W}$. 2. (2)

There exists $C\subseteq B$ compact and $\mathbf{W}_{0}:\mathbf{R}^{n}\to\mathbf{G}(n,m)$ continuous such that $C$ is nearly $m$ Nikodým in $C$ relative to $\mathbf{W}$.

Proof.

Define a Borel measurable map $\boldsymbol{\xi}:\mathbf{G}(n,m)\to\mathbf{V}(n-m)$ by $\boldsymbol{\xi}(W)=\Xi\left(W^{\perp}\right)$ where $\Xi:\mathbf{G}(n,n-m)\to\mathbf{V}(n,n-m)$ is as in 2.7. Choose arbitrarily $u\in\mathbf{R}^{n}$ and define a Borel measurable map

[TABLE]

Similarly to (10) observe that

[TABLE]

for every $W\in\mathbf{G}(n,m)$. We infer from 2.14 that

[TABLE]

is Borel measurable. Thus the set

[TABLE]

is Borel as well. The set $N=E\cap\{x:\mathscr{E}_{x}=\emptyset\}$ is coanalytic and $\mathscr{L}^{n}(N)=0$ by assumption. According to von Neumann’s selection Theorem [10, 5.5.3] there exists a universally measurable map $\tilde{\mathbf{W}}_{0}:E\thicksim N\to\mathbf{G}(n,m)$ such that $\tilde{\mathbf{W}}_{0}(x)\in\mathscr{E}_{x}$ for every $x\in E\thicksim N$, i.e. $\mathscr{H}^{m}\left(B\cap\left(x+\tilde{\mathbf{W}}_{0}(x)\right)\right)=0$. We extend $\tilde{\mathbf{W}}_{0}$ to be an arbitrary constant on $N$. This makes $\tilde{\mathbf{W}}_{0}$ an $\mathscr{L}^{n}$ measurable map defined on $E$. Therefore it is equal $\mathscr{L}^{n}$ almost everywhere to a Borel map $\mathbf{W}_{0}:E\to\mathbf{G}(n,m)$. This proves (1).

In order to prove (2) we recall of 2.4, specifically the retraction $\rho:V\to M_{n,m}$ and the homeomorphic identification $\varphi:\mathbf{G}(n,m)\to M_{n,m}$. Owing to the compactness of $M_{n,m}$ there are finitely many open balls $U_{j}$, $j=1,\ldots,J$, whose closure are contained in $V$ and covering $M_{n,m}$. Since $\mathscr{L}^{n}(B)>0$ there exists $j=1,\ldots,J$ such that $\mathscr{L}^{n}\left(B\cap E_{j}\right)>0$ where $E_{j}=\left(\varphi\circ\mathbf{W}_{0}\right)^{-1}(U_{j})$. It follows from Lusin’s Theorem [7, 2.5.3] that there exists a compact set $C\subseteq B\cap E_{j}$ such that $\mathscr{L}^{n}(C)>0$ and the restriction $\mathbf{W}_{0}|_{C}$ is continuous. The map $\varphi\circ\mathbf{W}_{0}|_{C}$ takes its values in the closed ball $\operatorname{\mathrm{Clos}}U_{j}$, therefore admits a continuous extension $\mathbf{Y}:\mathbf{R}^{n}\to\operatorname{\mathrm{Clos}}U_{j}\subseteq V$. Letting $\mathbf{W}=\varphi^{-1}\circ\rho\circ\mathbf{Y}$ completes the proof. ∎

3. Common setting

3.1Setting for the next three sections. —

In the next three sections we shall assume the following.

1. (1)

$E\subseteq\mathbf{R}^{n}$ is Borel and $\mathscr{L}^{n}(E)<\infty$. 2. (2)

$U\subseteq\mathbf{R}^{n}$ is open and $E\subseteq U$. 3. (3)

$B\subseteq\mathbf{R}^{n}$ is Borel. 4. (4)

$\mathbf{W}_{0}:E\to\mathbf{G}(n,m)$ is Lipschitz. 5. (5)

$\mathbf{W}(x)=x+\mathbf{W}_{0}(x)$ for each $x\in E$. 6. (6)

$\Lambda>0$. 7. (7)

$\mathbf{w}_{1},\ldots,\mathbf{w}_{m}:U\to\mathbf{R}^{n}$ and $\operatorname{\mathrm{Lip}}\mathbf{w}_{i}\leqslant\Lambda$, $i=1,\ldots,m$. 8. (8)

$\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}:U\to\mathbf{R}^{n}$ and $\operatorname{\mathrm{Lip}}\mathbf{v}_{i}\leqslant\Lambda$, $i=1,\ldots,n-m$. 9. (9)

$\mathbf{W}_{0}(x)=\operatorname{\mathrm{span}}\{\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x)\}$ for every $x\in E$. 10. (10)

$\mathbf{W}_{0}(x)^{\perp}=\operatorname{\mathrm{span}}\{\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)\}$ for every $x\in E$. 11. (11)

$\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x),\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$ constitute an orthonormal basis of $\mathbf{R}^{n}$, for every $x\in E$.

4. Two fibrations

4.1A fibered space associated with $E,B,\mathbf{w}_{1},\ldots,\mathbf{w}_{m}$. —

We define

[TABLE]

as well as

[TABLE]

It is obvious that $F$ is Lipschitz and therefore $\Sigma$ is countably $n+m$ rectifiable and $\mathscr{H}^{n+m}$ measurable. We also consider the two canonical projections

[TABLE]

as well as

[TABLE]

which is clearly also countably $n+m$ rectifiable and $\mathscr{H}^{n+m}$ measurable. In view of applying the coarea formula to $\Sigma_{B}$ and $\pi_{1}$ first, to $\Sigma_{B}$ and $\pi_{2}$ next, we observe that

[TABLE]

so that

[TABLE]

whenever $x\in E$, and that

[TABLE]

according to (8), so that

[TABLE]

whenever $u\in B$. It now follows from the coarea formula that

[TABLE]

and

[TABLE]

For these formulæ to be useful we need to establish bounds for the coarea Jacobian factors $J_{\Sigma}\pi_{1}$ and $J_{\Sigma}\pi_{2}$. In order to do so we notice that if $\Sigma\ni(x,u)=F(x,t_{1},\ldots,t_{m})$ and if $F$ is differentiable at $(x,t_{1},\ldots,t_{m})$ then the approximate tangent space $T_{(x,u)}\Sigma$ exists and is generated by the following $n+m$ vectors of $\mathbf{R}^{n}\times\mathbf{R}^{n}$:

[TABLE]

As usual $e_{1},\ldots,e_{n}$ denotes the canonical basis of $\mathbf{R}^{n}$.

4.2Coarea Jacobian factor of $\pi_{1}$. —

For $\mathscr{H}^{n+m}$ almost every $(x,u)\in\Sigma$ one has

[TABLE]

Proof.

We recall 2.3. The right hand inequality follows from $\operatorname{\mathrm{Lip}}\pi_{1}=1$. Regarding the left hand inequality fix $(x,u)=F(x,t)$ such that $F$ is differentiable at $(x,t)$ and let $L:T_{(x,u)}\Sigma\to\mathbf{R}^{n}$ denote the restriction of $\pi_{1}$ to $T_{(x,u)}\Sigma$. Put $v_{j}=\frac{\partial F}{\partial x_{j}}(x,t)$, $j=1,\ldots,n$, and recall (6) that

[TABLE]

since $L(v_{j})=e_{j}$, $j=1\ldots,n$. Now notice that

[TABLE]

Since $u=x+\sum_{i=1}^{m}t_{i}\mathbf{w}_{i}(x)$ one also has

[TABLE]

Finally,

[TABLE]

and the conclusion follows. ∎

4.3. —

Let $1\leqslant q\leqslant n-1$ be an integer and let $v_{1},\ldots,v_{q}$ be an orthonormal family in $\mathbf{R}^{n}$. There then exists $\lambda\in\Lambda(n,q)$ such that

[TABLE]

*Here $\Lambda(n,q)$ denotes the set of increasing maps $\{1,\ldots,q\}\to\{1,\ldots,n\}$. *

Proof.

We define a linear map $L:\mathbf{R}^{q}\to\mathbf{R}^{n}:(s_{1},\ldots,s_{q})\mapsto\sum_{k=1}^{q}s_{k}v_{k}$ and we observe that $L$ is an isometry. Therefore its area Jacobian factor $JL=1$, by definition. Now also

[TABLE]

according to the Binet-Cauchy formula [4, Chapter 3 §2 Theorem 4]. The conclusion easily follows. ∎

4.4Coarea Jacobian factor of $\pi_{2}$. —

The following hold.

1. (1)

For $\mathscr{H}^{n+m}$ almost every $(x,u)\in\Sigma$ one has

[TABLE] 2. (2)

For $\mathscr{H}^{n+m}$ almost every $(x,u)\in\Sigma$ one has $J_{\Sigma}\pi_{2}(x,u)>0$.

Proof.

Clearly $J_{\Sigma}\pi_{2}(x,u)\leqslant(\operatorname{\mathrm{Lip}}\pi_{2})^{n}\leqslant 1$. Regarding the left hand inequality fix $(x,u)=F(x,t)$ such that $F$ is approximately differentiable at $(x,t)$ and this time let $L:T_{(x,u)}\Sigma\to\mathbf{R}^{n}$ denote the restriction of $\pi_{2}$ to $T_{(x,u)}\Sigma$. We will now define a family of $n$ vectors $v_{1},\ldots,v_{n}$ belonging to $T_{(x,u)}\Sigma$. We choose $v_{k}=\frac{\partial F}{\partial t_{k}}(x,t)=(0,\mathbf{w}_{k}(x))$ for $k=1,\ldots,m$. For choosing the $n-m$ remaining vectors we proceed as follows. We select $\lambda\in\Lambda(n,n-m)$ as in 4.3 applied with $q=n-m$ to $\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$, and we let $v_{m+j}=\frac{\partial F}{\partial x_{\lambda(j)}}(x,t)$, $j=1,\ldots,n-m$. Recalling (6) we have

[TABLE]

As in the proof of 4.2 we find that

[TABLE]

and it remains only to find a lower bound for $|L(v_{1})\wedge\ldots\wedge L(v_{n})|$. This equals the absolute value of the determinant of the matrix of coefficients of $L(v_{i})$, $i=1,\ldots,n$, with respect to any orthonormal basis of $\mathbf{R}^{n}$. We choose the basis $\mathbf{w}_{1}(x),\ldots,\mathbf{w}_{m}(x),\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$. Thus

[TABLE]

Abbreviate

[TABLE]

and observe that $\left|h_{\lambda(j)}\right|\leqslant m\Lambda|t|=m\Lambda|x-u|$, $j=1,\ldots,n-m$ (recall the proof of 4.2). It remains only to remember that $\lambda$ has been selected in order that

[TABLE]

and to infer from the multilinearity of the determinant that

[TABLE]

This completes the proof of conclusion (1).

Let $E_{0}$ denote the subset of $E$ consisting of those $x$ such that each $\mathbf{w}_{i}$, $i=1,\ldots,m$, is differentiable at $x$. Thus $E_{0}$ is Borel and so is

[TABLE]

If $(x,u)\in\Sigma\thicksim F(A)$ then the restriction of $\pi_{2}$ to $T_{(x,u)}\Sigma$ is surjective and therefore $J_{\Sigma}\pi_{2}(x,u)>0$. Thus we ought to show that $\mathscr{H}^{n+m}(F(A))=0$. Since $F$ is Lipschitz it suffices to establish that $\mathscr{L}^{n+m}(A)=0$. As $A$ is Borel it is enough to prove that $\mathscr{L}^{m}(A_{x})=0$ for every $x\in E_{0}$, according to Fubini’s Theorem. Fix $x\in E_{0}$. As in the proof of conclusion (1), choose $\lambda\in\Lambda(n,n-m)$ associated with $\mathbf{v}_{1}(x),\ldots,\mathbf{v}_{n-m}(x)$ according to 4.3. Based on (16) we see that

[TABLE]

The set on the right is of the form $S_{x}=\mathbf{R}^{m}\cap\{(t_{1},\ldots,t_{m}):P_{x}(t_{1},\ldots,t_{m})=0\}$ for some polynomial $P_{x}\in\mathbf{R}[T_{1},\ldots,T_{m}]$, and $P_{x}(0,\ldots,0)=\det\left(\langle e_{\lambda(j)},\mathbf{v}_{k}(x)\rangle\right)_{j,k=1,\ldots,n-m}\neq 0$. It follows that $\mathscr{L}^{m}(S_{x})=0$, see e.g. [7, 2.6.5] and the proof of (2) is complete. ∎

4.5 Proposition. —

The measure $\phi_{E,\mathbf{W}}$ is absolutely continuous with respect to $\mathscr{L}^{n}$.

Proof.

Let $B\in\mathscr{B}(\mathbf{R}^{n})$ be such that $\mathscr{L}^{n}(B)=0$. It follows from (15) that

[TABLE]

It next follows from 4.4(2) that $\mathscr{H}^{n+m}(\Sigma_{B})=0$. In turn (14) implies that

[TABLE]

∎

4.6Definition of $\mathscr{Z}_{E}\mathbf{W}$. —

Note that $\phi_{E,\mathbf{W}}$ is a $\sigma$ finite Borel measure on $\mathbf{R}^{n}$ (see 2.16) and that it is absolutely continuous with respect to $\mathscr{L}^{n}$ (see 4.5). It then ensues from the Radon-Nikodým Theorem that there exists a Borel measurable function

[TABLE]

such that for every $B\in\mathscr{B}(\mathbf{R}^{n})$ one has

[TABLE]

Furthermore $\mathscr{Z}_{E}\mathbf{W}$ is univoquely defined only up to a $\mathscr{L}^{n}$ null set. This will not affect the reasonings in this paper. Each time we will write $\mathscr{Z}_{E}\mathbf{W}$ we will mean one particular Borel measurable function verifying the above equality for every $B\in\mathscr{B}(\mathbf{R}^{n})$.

4.7Definition of $\mathscr{Y}_{E}^{0}\mathbf{W}$. —

We define $\mathscr{Y}^{0}_{E}\mathbf{W}:\mathbf{R}^{n}\to[0,\infty]$ by the formula

[TABLE]

$u\in\mathbf{R}^{n}$. Letting $B=\mathbf{R}^{n}$ in (13) one infers from 2.3 that $\mathscr{Y}^{0}_{E}\mathbf{W}$ is $\mathscr{L}^{n}$ measurable. Using the estimates we have established so far regarding coarea Jacobian factors we now show that $\mathscr{Z}_{E}\mathbf{W}$ and $\mathscr{Y}_{E}^{0}\mathbf{W}$ are comparable when the diameter of $E$ is not too large.

4.8 Proposition. —

Given $0<\varepsilon<1$ there exists $\boldsymbol{\delta}_{4.8}(n,\Lambda,\varepsilon)>0$ with the following property. If $\operatorname{\mathrm{diam}}E\leqslant\boldsymbol{\delta}_{4.8}(n,\Lambda,\varepsilon)$ then

[TABLE]

for $\mathscr{L}^{n}$ almost every $u\in E$.

Proof.

We readily infer from 4.2 and 4.4(1) that there exists $\boldsymbol{\delta}(n,\Lambda,\varepsilon)>0$ such that for $\mathscr{H}^{n+m}$ almost all $(x,u)\in\Sigma$ if $|x-u|\leqslant\boldsymbol{\delta}(n,\Lambda,\varepsilon)$ then

[TABLE]

and

[TABLE]

where the above define $\alpha$ and $\beta$.

Assume now that $\operatorname{\mathrm{diam}}E\leqslant\boldsymbol{\delta}(n,\Lambda,\varepsilon)$. Given $B\in\mathscr{B}(E)$ we infer from (14), 4.2, 4.4(1), (15) and the above lower bounds that

[TABLE]

and

[TABLE]

Thus

[TABLE]

for every $B\in\mathscr{B}(\mathbf{R}^{n})$. The conclusion follows from the $\mathscr{L}^{n}$ measurability of both $\mathscr{Z}_{E}\mathbf{W}$ and $\mathscr{Y}^{0}_{E}\mathbf{W}$. ∎

4.9Rest stop. —

The above upper bound for $\mathscr{Z}_{E}\mathbf{W}$ is already enough to bound it in turn, by a constant times $(\operatorname{\mathrm{diam}}E)^{m}$, see 5.4. However I would not know how to use the above lower bound to establish that $\mathscr{Z}_{E}\mathbf{W}>0$ almost everywhere in $E$, which is what we are after. Indeed in the definition (17) of $\mathscr{Y}^{0}_{E}\mathbf{W}(u)$, $u$ does not appear as the covariable of the function whose level set we are measuring, thereby preventing the use of the coarea formula in an attempt to estimate $\mathscr{Y}^{0}_{E}\mathbf{W}(u)$. This naturally leads to adding a variable $y\in\mathbf{R}^{n-m}$ to the fibered space $\Sigma$, a covariable for $g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}$.

4.10A fibered space associated with $E,B,\mathbf{w}_{1},\ldots,\mathbf{w}_{m},\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m}$. —

Let $r>0$, and abbreviate $C_{r}=\mathbf{R}^{n-m}\cap\left\{y:|y|\leqslant r\right\}$ the Euclidean ball centered at the origin, of radius $r$ in $\mathbf{R}^{n-m}$. We define

[TABLE]

and

[TABLE]

so that $\hat{F}_{r}$ is Lipschitz and $\hat{\Sigma}_{r}$ is countably $2n$ rectifiable and $\mathscr{H}^{2n}$ measurable. Similarly to 4.1 we define

[TABLE]

which clearly is also countably $2n$ rectifiable and $\mathscr{H}^{2n}$ measurable. We aim to apply the coarea formula to $\hat{\Sigma}_{r,B}$ and to the two projections

[TABLE]

and

[TABLE]

To this end we notice that

[TABLE]

and thus

[TABLE]

for every $(x,y)\in E\times C_{r}$. We further notice that

[TABLE]

because

[TABLE]

and therefore

[TABLE]

whenever $u\in B$ and $y\in C_{r}$.

It now follows from the coarea formula and Fubini’s Theorem that

[TABLE]

and that

[TABLE]

4.11Coarea Jacobian factors of $\pi_{1}\times\pi_{3}$ and $\pi_{2}\times\pi_{3}$. —

The following inequalities hold for $\mathscr{H}^{2n}$ almost every $(x,u,y)\in\hat{\Sigma}_{r}$.

[TABLE]

and

[TABLE]

Proof.

The second conclusion is obvious since $\operatorname{\mathrm{Lip}}\pi_{2}\times\pi_{3}=1$. Regarding the first conclusion we reason similarly as in the proof of 4.2. Fix $(x,u,y)=\hat{F}_{r}(x,t,y)$ such that $\hat{F}_{r}$ is differentiable at $(x,t,y)$ and denote by $L$ the restriction of $\pi_{1}\times\pi_{3}$ to $T_{(x,u,y)}\hat{\Sigma}_{r}$. This tangent space is generated by the following $2n$ vectors of $\mathbf{R}^{n}\times\mathbf{R}^{n}\times\mathbf{R}^{n-m}$

[TABLE]

The range of $\pi_{1}\times\pi_{3}$ being $2n-m$ dimensional we need to select $2n-m$ vectors $v_{1},\ldots,v_{2n-m}$ in $T_{(x,u,y)}\hat{\Sigma}_{r}$ to obtain a lower bound

[TABLE]

The obvious choice consists of $v_{j}=\frac{\partial\hat{F}_{r}}{\partial x_{j}}(x,t,y)$, $j=1,\ldots,n$, and $v_{n+\ell}=\frac{\partial\hat{F}_{r}}{\partial y_{\ell}}(x,t,y)$, $\ell=1,\ldots,n-m$, so that $L(v_{1}),\ldots,L(v_{n-m})$ is the canonical basis of $\mathbf{R}^{n}\times\mathbf{R}^{n-m}$ and therefore the numerator in (22) equals 1. In order to determine an upper bound for its denominator we start by fixing $j=1,\ldots,n$, we abbreviate $a_{j}(x,t,y)=\sum_{i=1}^{m}t_{i}\frac{\partial\mathbf{w}_{i}(x)}{\partial x_{j}}(x)$ and $b_{j}(x,t,y)=\sum_{i=1}^{n-m}y_{i}\frac{\partial\mathbf{v}_{i}}{\partial x_{j}}(x)$ and we notice that $|a_{j}(x,t,y)|\leqslant m\Lambda|t|\leqslant n\Lambda|t|$, $|b_{j}(x,t,y)|\leqslant(n-m)\Lambda|y|\leqslant n\Lambda|y|$. Furthermore since $u-x=\sum_{i=1}^{m}t_{i}\mathbf{w}_{i}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)$ one has $|u-x|^{2}=|t|^{2}+|y|^{2}\leqslant\max\{|t|^{2},|y|^{2}\}$. Therefore

[TABLE]

Moreover

[TABLE]

for each $\ell=1,\ldots,n-m$. We conclude that

[TABLE]

and the proof is complete. ∎

4.12Definition of $\mathscr{Y}_{E}\mathbf{W}$. —

It follows from the Coarea Theorem that the function

[TABLE]

is $\mathscr{L}^{n}\otimes\mathscr{L}^{n-m}$ measurable (recall 4.10 applied with $B=\mathbf{R}^{n}$). It now follows from Fubini’s Theorem that for each $r>0$ the function

[TABLE]

is $\mathscr{L}^{n}$ measurable. In turn the function

[TABLE]

is $\mathscr{L}^{n}$ measurable. It is a replacement for $\mathscr{Y}^{0}_{E}\mathbf{W}$ defined in 4.7. We shall establish for $\mathscr{Z}_{E}\mathbf{W}$ a similar lower bound to that in 4.8, this time involving $\mathscr{Y}_{E}\mathbf{W}$. Before doing so, we notice the rather trivial fact that if $F\subseteq E$ then

[TABLE]

for all $u\in\mathbf{R}^{n}$.

4.13Preparatory remark for the proof of 4.15. —

It follows from the Coarea Theorem that the function

[TABLE]

is $\mathscr{L}^{n}\otimes\mathscr{L}^{n-m}$ measurable (recall 4.10 applied with $B=\mathbf{R}^{n}$). It therefore follows from Fubini’s Theorem as in 4.12 that

[TABLE]

is $\mathscr{L}^{n}$ measurable. Furthermore if $B$ is bounded then $|f_{j}(x)|\leqslant\boldsymbol{\alpha}(m)(\operatorname{\mathrm{diam}}B)^{m}$ for every $x\in\mathbf{R}^{n}$.

4.14. —

If $B$ is compact then for every $x\in E$ the function

[TABLE]

*is upper semicontinuous. *

Proof.

The proof is analogous to that of 2.14. For each $y\in\mathbf{R}^{n-m}$ define the compact set $K_{y}=B\cap\left(\mathbf{W}(x)+\sum_{i=1}^{n-m}y_{i}\mathbf{v}_{i}(x)\right)$. If $(y_{k})_{k}$ is a sequence converging to $y$ we ought to show that

[TABLE]

Since each $K_{y}$ is a subset of an $m$ dimensional affine subspace of $\mathbf{R}^{n}$ this is indeed equivalent to the same inequality with $\mathscr{H}^{m}_{\infty}$ replaced by $\mathscr{H}^{m}$ according to 2.2(3). Considering if necessary a subsequence of $(y_{k})_{k}$ we may assume that none of the compact sets $K_{y_{k}}$ is empty and the the above $\limsup$ is a $\lim$. Considering yet a further subsequence we may now assume that $(K_{y_{k}})_{k}$ converges in Hausdorff distance to some compact set $L\subseteq B$. One checks that $L\subseteq K_{y}$. It then follows from 2.2(1) that $\mathscr{H}^{m}_{\infty}\left(K_{y}\right)\geqslant\mathscr{H}^{m}_{\infty}(L)\geqslant\limsup_{k}\mathscr{H}^{m}_{\infty}\left(K_{y_{k}}\right)$. ∎

4.15 Proposition. —

Given $0<\varepsilon<1$ there exists $\boldsymbol{\delta}_{4.15}(n,\Lambda,\varepsilon)>0$ with the following property. If $\operatorname{\mathrm{diam}}(E\cup B)\leqslant\boldsymbol{\delta}_{4.15}(n,\Lambda,\varepsilon)$ and if $B$ is compact then

[TABLE]

Proof.

We first observe that we can choose $\boldsymbol{\delta}_{\ref{lb.1}}(n,\Lambda,\varepsilon)>0$ small enough so that

[TABLE]

for $\mathscr{H}^{2n}$ almost every $(x,u,y)\in\hat{\Sigma}_{r}$ provided $|u-x|\leqslant\boldsymbol{\delta}_{\ref{lb.1}}(n,\Lambda,\varepsilon)$, according to 4.11. Thus (23) holds for $\mathscr{H}^{2n}$ almost every $(x,u,y)\in\hat{\Sigma}_{r,B}$ under the assumption that $\operatorname{\mathrm{diam}}(E\cup B)\leqslant\boldsymbol{\delta}_{\ref{lb.1}}(n,\Lambda,\varepsilon)$. When (20), (21) and 4.11 imply that

[TABLE]

Fix $x\in E$ and $\beta>0$. According to 4.14 there exists a positive integer $j(x,\beta)$ such that if $j\geqslant j(x,\beta)$ then

[TABLE]

Taking the $\limsup$ as $j\to\infty$ on the right hand side, and letting $\beta\to 0$ we obtain

[TABLE]

As this holds for all $x\in E$ we may integrate over $E$ with respect to $\mathscr{L}^{n}$. Noticing that for every $j=1,2,\ldots$ (with the notation of 4.13) $|f_{j}|\leqslant\boldsymbol{\alpha}(m)(\operatorname{\mathrm{diam}}B)^{m}\mathbbm{1}_{E}$, the latter being $\mathscr{L}^{n}$ summable, justifies the application of the reverse Fatou lemma below. Thus the following ensues from (25), the reverse Fatou lemma, (24), and the Fatou lemma:

[TABLE]

∎

4.16 Corollary. —

If $0<\varepsilon<1$ and $\operatorname{\mathrm{diam}}E\leqslant\boldsymbol{\delta}_{\ref{lb.1}}(n,\Lambda,\varepsilon)$ then

[TABLE]

for $\mathscr{L}^{n}$ almost every $u\in E$.

5. Upper bound for $\mathscr{Y}_{E}\mathbf{W}$ and $\mathscr{Z}_{E}\mathbf{W}$

5.1Bow Tie Lemma. —

Let $S\subseteq\mathbf{R}^{n}$, $W\in\mathbf{G}(n,m)$ and $0<\tau<1$. Assume that

[TABLE]

There then exists $F:P_{W}(S)\to\mathbf{R}^{n}$ such that $S=\operatorname{\mathrm{im}}F$ and $\operatorname{\mathrm{Lip}}F\leqslant\frac{1}{\sqrt{1-\tau^{2}}}$. In particular

[TABLE]

Proof.

Let $x,x^{\prime}\in S$ and define $\rho=|x-x^{\prime}|\leqslant\operatorname{\mathrm{diam}}S$. Thus $x^{\prime}\in S\cap\mathbf{B}(x,\rho)$ and therefore $\left|P_{W^{\perp}}(x-x^{\prime})\right|\leqslant\tau\rho=\tau|x-x^{\prime}|$. Since $|x-x^{\prime}|^{2}=\left|P_{W}(x-x^{\prime})\right|^{2}+\left|P_{W^{\perp}}(x-x^{\prime})\right|^{2}$ we infer that

[TABLE]

Therefore $P_{W}|_{S}$ is injective, and the Lipschitz bound on $F=\left(P_{W}|_{S}\right)^{-1}$ clearly follows from the above inequality. Regarding the second conclusion,

[TABLE]

and $P_{W}(S)$ is contained in a ball of radius $\operatorname{\mathrm{diam}}P_{W}(S)\leqslant\operatorname{\mathrm{diam}}S$. ∎

5.2. —

Given $0<\tau<1$ there exists $\boldsymbol{\delta}_{5.2}(n,\Lambda,\tau)>0$ with the following property. If

1. (1)

$x_{0}\in U$* and $u\in\mathbf{R}^{n}$;* 2. (2)

$\operatorname{\mathrm{diam}}\left(E\cup\{x_{0}\}\cup\{u\}\right)\leqslant\boldsymbol{\delta}_{5.2}(n,\Lambda,\tau)$;

Then: For every $y\in\mathbf{R}^{n-m}$, for every $x\in E\cap g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\{y\}$ and for every $0<\rho<\infty$ one has

[TABLE]

Proof.

We shall show that $\boldsymbol{\delta}_{\ref{ub.1}}(n,\Lambda,\tau)=\frac{\tau}{2\Lambda\sqrt{n}}$ will do. Let $x,x^{\prime}\in E\cap g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\{y\}$ for some $y\in\mathbf{R}^{n-m}$. Thus $g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x)=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x^{\prime})$ and hence

[TABLE]

thus

[TABLE]

In turn,

[TABLE]

∎

5.3 Proposition. —

There are $\boldsymbol{\delta}_{5.3}(n,\Lambda)>0$ and $\mathbf{c}_{5.3}(m)\geqslant 1$ with the following property. If $u\in U$ and $\operatorname{\mathrm{diam}}(E\cup\{u\})\leqslant\boldsymbol{\delta}_{5.3}(n,\Lambda)$ then

[TABLE]

Proof.

Let $\boldsymbol{\delta}_{\ref{upper.bound}}(n,\Lambda)=\boldsymbol{\delta}_{\ref{ub.1}}(n,\Lambda,1/2)$. Recall the definitions of $\mathscr{Y}^{0}_{E}\mathbf{W}$ and $\mathscr{Y}_{E}\mathbf{W}$ from 4.7 and 4.12 respectively. If $E=\emptyset$ the conclusion is obvious. If not pick $x_{0}\in E$ arbitrarily. Given any $y\in\mathbf{R}^{n-m}$ we see that 5.2 applies with $\tau=1/2$ and in turn the bow-tie lemma 5.1 applies to $S=E\cap g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}\{y\}$ and $W=\mathbf{W}_{0}(x_{0})$. Thus

[TABLE]

The proposition is proved. ∎

5.4 Corollary. —

There are $\boldsymbol{\delta}_{5.4}(n,\Lambda)>0$ and $\mathbf{c}_{5.4}(n)\geqslant 1$ with the following property. If $\operatorname{\mathrm{diam}}E\leqslant\boldsymbol{\delta}_{5.4}(n,\Lambda)$ then

[TABLE]

for $\mathscr{L}^{n}$ almost every $u\in E$.

Proof.

Let $\boldsymbol{\delta}_{\ref{cor.ub}}(n,\Lambda)=\min\{\boldsymbol{\delta}_{\ref{upper.bound}}(n,\Lambda),\boldsymbol{\delta}_{\ref{Z.1}}(n,\Lambda,1/2)\}$. ∎

6. Lower bound for $\mathscr{Y}_{E}\mathbf{W}$ and $\mathscr{Z}_{E}\mathbf{W}$

6.1Setting for this section. —

We enforce again the exact same assumptions as in 3.1, and as in 4.10 we let $C_{r}=\mathbf{R}^{n-m}\cap\{y:|y|\leqslant r\}$.

6.2Polyballs. —

Given $x_{0}\in\mathbf{R}^{n}$ and $r>0$ we define

[TABLE]

We notice that if $x\in\mathbf{C}_{\mathbf{W}}(x_{0},r)$ then $|x-x_{0}|\leqslant r\sqrt{2}$, in particular $\operatorname{\mathrm{diam}}\mathbf{C}_{\mathbf{W}}(x_{0},r)\leqslant 2\sqrt{2}$. We also notice that $\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)=\boldsymbol{\alpha}(m)\boldsymbol{\alpha}(n-m)r^{n}$.

6.3. —

Given $0<\varepsilon<1$ there exists $\boldsymbol{\delta}_{6.3}(n,\Lambda,\varepsilon)>0$ with the following property. If

1. (1)

$0<r<\boldsymbol{\delta}_{6.3}(n,\Lambda,\varepsilon)$; 2. (2)

$u\in\mathbf{C}_{\mathbf{W}}(x_{0},r)\subseteq U$; 3. (3)

$|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x_{0})|\leqslant(1-3\varepsilon)r$; 4. (4)

$C\subseteq C_{\varepsilon r}$* is closed;*

then

[TABLE]

6.4 Remark. —

With hopes that the following will help the reader form a geometrical imagery: Under the circumstances 6.3, $\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}(C)$ may be seen as a <<nonlinear stripe>>, <<horizontal>> with respect to $\mathbf{W}_{0}(x_{0})$, <<at height>> $g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x_{0})$ with respect to $x_{0}$, and of <<width>> $C$.

Proof of 6.3.

Given $z\in\mathbf{W}_{0}(x_{0})\cap\mathbf{B}(0,r)$ we define

[TABLE]

and we consider the isometric parametrization $\gamma_{z}:C_{r}\to V_{z}$ defined by the formula

[TABLE]

We also abbeviate $f_{z,u}=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}\circ\gamma_{z}$.

Claim #1. $\operatorname{\mathrm{Lip}}f_{z,u}\leqslant(1+\varepsilon)^{\frac{1}{n-m}}$.

Since $\gamma_{z}$ is an isometry it suffices to obtain an upper bound for $\operatorname{\mathrm{Lip}}g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}|_{\mathbf{C}_{\mathbf{W}}(x_{0},r)}$. Let $x,x^{\prime}\in\mathbf{C}_{\mathbf{W}}(x_{0},r)$,

[TABLE]

Recalling hypothesis (1) it is now apparent that $\boldsymbol{\delta}_{\ref{53}}$ can be chosen small enough according to $n$, $\Lambda$ and $\varepsilon$ so that Claim #1 holds.

Claim #2. For $\mathscr{L}^{n-m}$ almost every $y\in C_{r}$ one has $\|Df_{z,u}(y)-\operatorname{\mathrm{id}}_{\mathbf{R}^{n-m}}\|\leqslant\varepsilon$.

Let $y\in C_{r}$ be such that $f_{z,u}$ is differentiable at $y$. We shall estimate the coefficients of the matrix representing $Df_{z,u}(y)$ with respect to the canonical basis. Fix $i,j=1,\ldots,n-m$ and recall (7):

[TABLE]

Next notice that

[TABLE]

where the last inequality follows from hypothesis (1) upon choosing $\boldsymbol{\delta}_{\ref{53}}$ small enough according to $n$, $\Lambda$ and $\varepsilon$. Moreover,

[TABLE]

Therefore if $(a_{ij})_{i,j=1,\ldots,n-m}$ is the matrix representing $Df_{z,u}(y)$ with respect to the canonical basis we have shown that $|a_{ij}-\delta_{ij}|\leqslant\frac{\varepsilon}{n-m}$ for all $i,j=1,\ldots,n-m$. This completes the proof of Claim #2.

Claim #3. $C_{\varepsilon r}\subseteq f_{z,u}(C_{r})$.

We shall show that $|y-f_{z,u}(y)|\leqslant(1-\varepsilon)r$ for every $y\in\operatorname{\mathrm{Bdry}}C_{r}$ and the conclusion will become a consequence of the Intermediate Value Theorem in case $m=n-1$, and a standard application of homology theory, see e.g. [3, 4.6.1] in case $m<n-1$. If $m<n-1$ it is clearly enough to establish this inequality for $\mathscr{H}^{n-m-1}$ almost every $y\in\operatorname{\mathrm{Bdry}}C_{r}$: In that case, owing to the Coarea Theorem [7, 3.2.22] we choose such $y$ in order that $f_{z,u}$ is differentiable $\mathscr{H}^{1}$ almost everywhere on the line segment $\mathbf{R}^{n-m}\cap\{sy:0\leqslant s\leqslant 1\}$. Whether $m<n-1$ or $m=n-1$ it then follows from Claim #2 that

[TABLE]

Accordingly,

[TABLE]

and the claim will be established upon showing that $\left|f_{z,u}(0)\right|\leqslant(1-2\varepsilon)r$. Note that $f_{z,u}(0)=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}(x_{0}+z)$, and we shall use hypothesis (3) to bound its norm from above. Given $j=1,\ldots,n-m$ recall that $\langle\mathbf{v}_{j}(x_{0}),z\rangle=0$ thus

[TABLE]

where the last inequality holds according to hypothesis (1) provided $\boldsymbol{\delta}_{\ref{53}}$ is chosen sufficiently small. In turn

[TABLE]

according to hypothesis (3).

Claim #4. For every $z\in\mathbf{W}_{0}(x_{0})\cap\mathbf{B}(0,r)$ and every closed $C\subseteq C_{\varepsilon r}$ one has $\mathscr{H}^{n-m}(C)\leqslant(1+\varepsilon)\mathscr{H}^{n-m}\left(g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u}^{-1}(C)\cap V_{z}\right)$.

First notice that

[TABLE]

and therefore

[TABLE]

since $\gamma_{z}$ is an isometry. Now since $C\subseteq C_{\varepsilon r}\subseteq f_{z,u}(C_{r})$ according to Claim #3 we have

[TABLE]

It therefore follows from Claim #1 that

[TABLE]

We are now ready to finish the proof by an application of Fubini’s Theorem :

[TABLE]

∎

6.5Lower bound for $\mathscr{Y}_{E}\mathbf{W}$. —

Given $0<\varepsilon<1/3$ there exists $\boldsymbol{\delta}_{6.5}(n,\Lambda,\varepsilon)>0$ with the following property. If

1. (1)

$0<r<\boldsymbol{\delta}_{6.5}(n,\Lambda,\varepsilon)$; 2. (2)

$\mathbf{C}_{\mathbf{W}}(x_{0},r)\subseteq U$; 3. (3)

$A\subseteq U$* is closed;* 4. (4)

$\mathscr{L}^{n}\left(A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)\geqslant(1-\varepsilon)\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)$;

then

[TABLE]

*where $\mathbf{c}_{6.5}(n)=5+6n$. *

Proof.

Similarly to the proof of 6.3 we will first establish a lower bound for $\mathscr{Y}_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}$ on <<vertical slices>> $V_{z}$ of the given polyball and then apply Fubini. Given $z\in\mathbf{W}_{0}(x_{0})\cap\mathbf{B}(0,r)$ we let $V_{z}$ and $\gamma_{z}$ be as in 6.3 and we also define

[TABLE]

(notice it is slightly smaller than $V_{z}$ used in the proof of 6.3) and we consider the isometric parametrization $\check{\gamma}_{z}:C_{(1-3\varepsilon)r}\to\check{V}_{z}$ defined by

[TABLE]

For part of the proof we find it convenient to abbreviate $E=A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)$. We also let $\check{\mathscr{Y}}_{E}\mathbf{W}=\left(\mathscr{Y}_{E}\mathbf{W}\right)\circ\check{\gamma}_{z}$.

By definition of $\mathscr{Y}_{E}\mathbf{W}$ for each $\check{\gamma}_{z}(y)\in\check{V}_{z}$ there exists a collection $\mathscr{C}_{y}$ of closed balls in $\mathbf{R}^{n-m}$ with the following properties: For every $C\in\mathscr{C}_{y}$, $C$ is a ball centered at 0, $C\subseteq C_{\varepsilon r}$,

[TABLE]

and $\inf\{\operatorname{\mathrm{diam}}C:C\in\mathscr{B}_{y}\}=0$. Furthermore $\check{\mathscr{Y}}_{E}\mathbf{W}$ being $\mathscr{L}^{n-m}$ summable according to 5.3 there exists $N\subseteq C_{(1-3\varepsilon)r}$ such that $\mathscr{L}^{n-m}(N)=0$ and every $y\not\in N$ is a Lebesgue point of $\check{\mathscr{Y}}_{E}\mathbf{W}$. For such $y$ we may reduce $\mathscr{C}_{y}$ if necessary, keeping all the previously stated properties and enforcing that

[TABLE]

whenever $C\in\mathscr{C}_{y}$. We infer that for each $y\in C_{(1-3\varepsilon)r}\thicksim N$ and each $C\in\mathscr{C}_{y}$,

[TABLE]

It follows from the Vitali Covering Theorem that there is a sequence $(y_{k})_{k}$ in $C_{(1-3\varepsilon)r}\thicksim N$, and $C_{k}\in\mathscr{C}_{y_{k}}$, such that the balls $y_{k}+C_{k}$, $k=1,2,\ldots,$ are pairwise disjoint, and $\mathscr{L}^{n-m}\left(C_{(1-3\varepsilon)r}\thicksim\cup_{k=1}^{\infty}(y_{k}+C_{k})\right)=0$. It therefore follows from (26) and the fact that $\gamma_{z}$ is an isometry that

[TABLE]

where we have abbreviated $u_{k}=\check{\gamma}_{z}(y_{k})$. We also abbreviate $S_{k}=g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}^{-1}(C_{k})$ and we infer from the coarea formula that for each $k=1,2,\ldots$,

[TABLE]

where the last inequality follows from 2.12 applied with $U=\operatorname{\mathrm{Int}}\mathbf{C}_{\mathbf{W}}(x_{0},r)$ provided that $\boldsymbol{\delta}_{\ref{54}}(n,\Lambda,\varepsilon)$ is chosen smaller than $(2\sqrt{2})^{-1}\boldsymbol{\delta}_{\ref{jac.g}}(n,\Lambda,\varepsilon)$. Letting $S=\cup_{k=1}^{\infty}S_{k}$, and recalling that $E=A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)$, we infer from (27) and (28) that

[TABLE]

Applying 6.3 to each $S_{k}$ does not immediately yield a lower bound for $\mathscr{L}^{n}(\mathbf{C}_{\mathbf{W}}(x_{0},r)\cap S)$ because the $S_{k}$ are not necessarily pairwise disjoint. This is why we now introduce slightly smaller versions of these:

[TABLE]

Claim. The sets $\check{S}_{k}\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)$, $k=1,2,\ldots$, are pairwise disjoint.

Assume if possible that there are $j\neq k$ and $x\in\check{S}_{j}\cap\check{S}_{k}\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)$. Letting $\rho_{j}$ and $\rho_{k}$ denote respectively the radius of $C_{j}$ and $C_{k}$ we notice that $\rho_{j}+\rho_{k}<|y_{j}-y_{k}|$ because $(y_{j}+C_{j})\cap(y_{k}+C_{k})=\emptyset$. Since $\check{\gamma}_{z}$ is an isometry we have $|u_{j}-u_{k}|=\left|\check{\gamma}_{z}(y_{j})-\check{\gamma}_{z}(y_{k})\right|=|y_{j}-y_{k}|$ and therefore also

[TABLE]

We now introduce the following vectors of $\mathbf{R}^{n-m}$,

[TABLE]

and we notice that

[TABLE]

where the second equality holds because $u_{j}-u_{k}\in\mathbf{W}_{0}(x_{0})^{\perp}$ as clearly follows from the definition of $\check{\gamma}_{z}$. Furthermore

[TABLE]

since we may choose $\boldsymbol{\delta}_{\ref{54}}(n,\Lambda,\varepsilon)$ to be so small that the last inequality holds according to hypothesis (1). Whence

[TABLE]

in contradiction with (30). The Claim is established.

Thus

[TABLE]

where the last ineqality follows from 6.3. We notice that indeed 6.3 applies since $\check{C}_{k}\subseteq C_{k}\subseteq C_{\varepsilon r}$ and $\left|g_{\mathbf{v}_{1},\ldots,\mathbf{v}_{n-m},u_{k}}(x_{0})\right|=\left|P_{\mathbf{W}_{0}(x_{0})^{\perp}}(u_{k}-x_{0})\right|=\left|y_{k}\right|\leqslant(1-3\varepsilon)r$.

Now,

[TABLE]

We infer from (31) and (32) that

[TABLE]

It therefore ensues from (29) and hypothesis (4) that

[TABLE]

Integrating over $z$ we infer from Fubini’s Theorem

[TABLE]

∎

6.6 Proposition. —

Given $0<\varepsilon<1/3$ there exist $\boldsymbol{\delta}_{6.6}(n,\Lambda,\varepsilon)>0$ and $\mathbf{c}_{6.6}(n)\geqslant 1$ with the following property. If

1. (1)

$0<r<\boldsymbol{\delta}_{6.6}(n,\Lambda,\varepsilon)$; 2. (2)

$\mathbf{C}_{\mathbf{W}}(x_{0},r)\subseteq U$; 3. (3)

$A\subseteq U$* is closed;* 4. (4)

$\mathscr{L}^{n}\left(A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)\geqslant(1-\varepsilon)\mathscr{L}^{n}\left(\mathbf{C}_{\mathbf{W}}(x_{0},r)\right)$;

then

[TABLE]

6.7 Remark. —

The difference with 6.5 is the domain of integration (being smaller) in the integral on the left hand side in the conclusion.

Proof of 6.6.

The reader will happily check that

[TABLE]

suits their needs. ∎

6.8 Proposition. —

There exists $\boldsymbol{\delta}_{6.8}(n,\Lambda)>0$ with the following property. If $\operatorname{\mathrm{diam}}E\leqslant\boldsymbol{\delta}_{6.8}(n,\Lambda)$ then

[TABLE]

for $\mathscr{L}^{n}$ almost every $u\in E$.

Proof.

We let

[TABLE]

According to 4.16 it suffices to show that $\mathscr{Y}_{E}\mathbf{W}(u)>0$ for $\mathscr{L}^{n}$ almost every $u\in E$. Define $Z=E\cap\{u:\mathscr{Y}_{E}\mathbf{W}(u)=0\}$ and assume if possible that $\mathscr{L}^{n}(Z)>0$. Since $Z$ is $\mathscr{L}^{n}$ measurable (recall 4.12) there exists a compact set $A\subseteq Z$ such that $\mathscr{L}^{n}(A)>0$. Observe that the sets $\mathbf{C}_{\mathbf{W}}(x,r)$, $x\in U$ and $r>0$, form a derivation basis for $\mathscr{L}^{n}$ measurable subsets of $U$ (because their excentricity is bounded away from zero) thus there exists $x_{0}\in A$ and $r_{0}>0$ such that

[TABLE]

whenever $0<r<r_{0}$. There is no restriction to assume that $r_{0}$ is small enough for $\mathbf{C}_{\mathbf{W}}(x_{0},r_{0})\subseteq U$. Thus if we let $r=\min\{r_{0},\boldsymbol{\delta}_{\ref{lower.bound}}(n,\Lambda,1/(4\mathbf{c}_{\ref{lower.bound}(n)}))\}$ it follows from 6.6 that

[TABLE]

On the other hand recalling 4.12 and the fact that $A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\subseteq E$ we infer that $\mathscr{Y}_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}(u)\leqslant\mathscr{Y}_{E}(u)$ for all $u\in\mathbf{R}^{n}$. In particular $\mathscr{Y}_{A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)}\mathbf{W}(u)=0$ for all $u\in A\cap\mathbf{C}_{\mathbf{W}}(x_{0},r)\subseteq Z$, contradicting (33). ∎

7. Proof of the theorem

7.1 Theorem. —

Assume that $S\subseteq\mathbf{R}^{n}$, $\mathbf{W}_{0}:S\to\mathbf{G}(n,m)$ is Lipschitz and $A\subseteq S$ is Borel. The following are equivalent.

1. (1)

$\mathscr{L}^{n}(A)=0$. 2. (2)

For $\mathscr{L}^{n}$ almost every $x\in A$, $\mathscr{H}^{m}(A\cap\mathbf{W}(x))=0$. 3. (3)

For $\mathscr{L}^{n}$ almost every $x\in S$, $\mathscr{H}^{m}(A\cap\mathbf{W}(x))=0$.

Recall our convention that $\mathbf{W}(x)=x+\mathbf{W}_{0}(x)$.

Proof.

Since $\mathbf{G}(n,m)$ is complete we can extend $\mathbf{W}_{0}$ to the closure of $S$. Furthermore if the Theorem holds for $\operatorname{\mathrm{Clos}}S$ then it also holds for $S$. Thus there is no restriction to assume that $S$ is closed.

$(1)\Rightarrow(3)$. It follows from 2.8 that each $x\in S$ admits an open neighborhood $U_{x}$ in $\mathbf{R}^{n}$ such that $\mathbf{W}(x)$ can be associated with a Lipschitz orthonormal frame verifying all the conditions of 3.1 for some $\Lambda_{x}>0$. Since $S$ is Lindelöf there are countably many $x_{1},x_{2},\ldots$ such that $S\subseteq\cup_{j}U_{x_{j}}$. Letting $E_{j}=S\cap U_{x_{j}}$ we infer from 4.5 that $\phi_{E_{j},\mathbf{W}}$ is absolutely continuous with respect to $\mathscr{L}^{n}$. Thus if $\mathscr{L}^{n}(A)=0$ then $\mathscr{H}^{m}\left(A\cap\mathbf{W}(x)\right)=0$ for $\mathscr{L}^{n}$ almost every $x\in E_{j}$ by definition of $\phi_{E_{j},\mathbf{W}}$. Since $j$ is arbitrary the proof is complete.

$(3)\Rightarrow(2)$ is trivial.

$(2)\Rightarrow(1)$ Let $A$ verify condition (3). It is enough to show that $\mathscr{L}^{n}(A\cap\mathbf{B}(0,r))=0$ for each $r>0$. Fix $r>0$ and define $S_{r}=S\cap\mathbf{B}(0,r)$. Consider the $U_{x_{j}}$ defined in the second paragraph of the present proof; since $S_{r}$ is compact only finitely many of those, say $U_{x_{1}},\ldots,U_{x_{N}}$, cover $S_{r}$. Let $\Lambda=\max_{j=1,\ldots,N}\Lambda_{x_{j}}$. Partition each $U_{x_{j}}$, $j=1,\ldots,N$, into Borel sets $E_{j,k}$, $k=1,\ldots,K_{j}$, such that $\operatorname{\mathrm{diam}}E_{j,k}\leqslant\boldsymbol{\delta}_{\ref{Z.positive}}(n,\Lambda)$. It then follows from 6.8 that

[TABLE]

for $\mathscr{L}^{n}$ almost every $u\in A\cap E_{j,k}$. Now fix $j$ and $k$. Observe that $\mathscr{H}^{m}\left(A\cap E_{j,k}\cap\mathbf{W}(x)\right)=0$ for $\mathscr{L}^{n}$ almost every $x\in A\cap E_{j,k}$. Thus $\phi_{A\cap E_{j,k},\mathbf{W}}(A\cap E_{j,k})=0$. Moreover,

[TABLE]

It follows from (34) that $\mathscr{L}^{n}(A\cap E_{j,k})=0$. Since $j$ and $k$ are arbitrary, $\mathscr{L}^{n}(A)=0$. ∎