2021

David Bate

# Characterising rectifiable metric spaces using tangent spaces

<EMAIL_ADDRESS>Zeeman Building, University of Warwick, Coventry, CV4
7AL, UK, ORCiD: 0000-0003-0808-2453

(8 October 2021)

###### Abstract

We characterise rectifiable subsets of a complete metric space $X$ in terms of
local approximation, with respect to the Gromov–Hausdorff distance, by an
$n$-dimensional Banach space. In fact, if $E\subset X$ with
$\mathcal{H}^{n}(E)<\infty$ and has positive lower density almost everywhere,
we prove that it is sufficient that, at almost every point and each
sufficiently small scale, $E$ is approximated by a bi-Lipschitz image of
Euclidean space.

We also introduce a generalisation of Preiss’s tangent measures that is
suitable for the setting of arbitrary metric spaces and formulate our
characterisation in terms of tangent measures. This definition is equivalent
to that of Preiss when the ambient space is Euclidean, and equivalent to the
measured Gromov–Hausdorff tangent space when the measure is doubling.

###### pacs:

[

MSC Classification]30L99 (Primary), 28A75 (secondary)

### Acknowledgements

This work was supported by the European Union’s Horizon 2020 research and
innovation programme (Grant agreement No. 948021).

I would like to thank the referee for a careful reading and many helpful
comments that improved the exposition.

##  1 Introduction

A set $E\subset\mathbb{R}^{m}$ is _$n$ -rectifiable_ if it can be covered, up
to a set of $\mathcal{H}^{n}$ measure zero, by countably many Lipschitz images
of $\mathbb{R}^{n}$ (throughout this paper, $\mathcal{H}^{n}$ denotes the
$n$-dimensional Hausdorff measure). Rademacher’s theorem implies that an
$n$-rectifiable $E\subset\mathbb{R}^{m}$ with $\mathcal{H}^{n}(E)<\infty$ has
a unique $n$-dimensional _approximate tangent plane_ at $\mathcal{H}^{n}$-a.e.
$x\in E$: an $n$-dimensional affine subspace $W\ni x$ such that, for each
$r>0$, there exists $E_{r}\subset E\cap B(x,r)$ with

$\frac{\mathcal{H}^{n}(E\cap B(x,r)\setminus E_{r})}{r^{n}}\to 0$

and

$\frac{d_{\mathrm{H}}(W\cap B(x,r),E_{r})}{r}\to 0$

as $r\to 0$. Here $d_{\mathrm{H}}$ denotes the Hausdorff distance between two
subsets of $\mathbb{R}^{m}$. A central concept of geometric measure theory is
to understand sufficient conditions for rectifiability in terms of local
approximations by tangent planes.

The simplest such result is the converse to the above statement: if
$E\subset\mathbb{R}^{m}$ with $\mathcal{H}^{n}(E)<\infty$, the existence of an
$n$-dimensional approximate tangent plane at $\mathcal{H}^{n}$-a.e. point
implies that $E$ is $n$-rectifiable (see (Mattila_1995, , Theorem 15.19)). A
result of Marstrand marstrand  (for $n=2$ and $m=3$) significantly strengthens
this by allowing the approximating affine subspace $W$ to depend on $r$ — the
approximating tangent plane may “rotate” as one zooms into $E$ — under the
additional assumption of the positive lower Hausdorff density of $E$ (see (1)
below). This was later generalised by Mattila mattila  to all higher
dimensions. The case of $n=1$ is addressed in the work of Besicovitch
besicovitchII . This much stronger result is crucial for many results in
geometric measure theory, such as the density theorems of Marstrand marstrand
, Mattila mattila  and Preiss MR890162 .

Analogously, the work of David and Semmes davidsemmes  characterises _uniform_
rectifiability in terms of good approximation by (rotating) tangents in the
form of Jones’s beta numbers jones . Tolsa and Azzam ta ; t  provide similar
results for measures.

Recently there has been significant interest in studying analysis on metric
spaces. Inevitably, such study leads to questions regarding the geometric
nature of rectifiable subsets of a metric space. For a metric space $X$,
$E\subset X$ is _$n$ -rectifiable_ if it can be covered, up to a set of
$\mathcal{H}^{n}$ measure zero, by countably many Lipschitz images of
_subsets_ of $\mathbb{R}^{n}$. Kirchheim MR1189747  gives a precise
description of the local structure of rectifiable subsets of a metric space
analogous to the description granted by Rademacher’s theorem. From this
description it is possible to deduce the existence of a tangent structure of a
rectifiable metric space akin to an approximate tangent plane after two
modifications.

Firstly, one must replace the Hausdorff distance $d_{\mathrm{H}}$ with the
_Gromov–Hausdorff_ distance $d_{\mathrm{GH}}$. Two metric spaces $M_{1}$ and
$M_{2}$ satisfy

$d_{\mathrm{GH}}(M_{1},M_{2})<\epsilon$

if there exist a metric space $Z$ and isometric embeddings $\iota_{i}\colon
M_{i}\to Z$, $i=1,2$, such that

$d_{\mathrm{H}}(\iota_{1}(M_{1}),\iota_{2}(M_{2}))<\epsilon$

(see Definition 2.15 for a modification that is more suitable for unbounded
metric spaces). Secondly, the affine subspace must be replaced by an
$n$-dimensional Banach space; the tangent space may not be equipped with the
Euclidean norm.

The question of whether an $E\subset X$ with approximate tangent planes
defined in this way is rectifiable has existed since the work of Kirchheim.
More recently, this question appeared in the work of Ambrosio, Brué and Semola
MR3990192  when studying sets of finite perimeter in RCD metric spaces. To
date the question is unknown for any ambient metric space that is not a
Euclidean space (in this case $d_{\mathrm{GH}}$ can be replaced by
$d_{\mathrm{H}}$ in (3) and so the statement is equivalent to the results of
Marstrand and Mattila), even if all of the tangent spaces are assumed to be a
fixed $n$-dimensional Banach space.

In this paper we answer this question affirmatively. In fact we prove the
following stronger result. For $K\geq 1$ let $\operatorname{biLip}(K)$ be the
set of metric spaces $Y=(\mathbb{R}^{n},\rho)$ such that $\rho$ is $K$-bi-
Lipschitz equivalent to the Euclidean norm, that is

$\frac{1}{K}\|x-y\|_{2}\leq\rho(x,y)\leq K\|x-y\|_{2}\quad\forall
x,y\in\mathbb{R}^{n}.$

Rather than requiring local approximation by a fixed $n$-dimensional Banach
space, we allow approximation by an element of $\operatorname{biLip}(K)$ that
is allowed to depend on the scale.

######  Theorem 1.1.

Let $(X,d)$ be a complete metric space, $n\in\mathbb{N}$ and let $E\subset X$
be $\mathcal{H}^{n}$-measurable with $\mathcal{H}^{n}(E)<\infty$ and

$\Theta_{*}^{n}(E,x):=\liminf_{r\to 0}\frac{\mathcal{H}^{n}(B(x,r)\cap
E)}{r^{n}}>0\quad\text{for }\mathcal{H}^{n}\text{-a.e. }x\in E.$ (1)

Suppose that, for $\mathcal{H}^{n}$-a.e. $x\in E$, there exist $K_{x}\geq 1$
and, for each $r>0$, a $Y_{r}\in\operatorname{biLip}(K_{x})$ and $E_{r}\subset
E\cap B(x,r)$ such that

$\frac{\mathcal{H}^{n}(E\cap B(x,r)\setminus E_{r})}{r^{n}}\to 0$ (2)

and

$\frac{d_{\mathrm{GH}}(Y_{r}\cap B(0,r),E_{r})}{r}\to 0$ (3)

as $r\to 0$. Then $E$ is $n$-rectifiable.

Note that even in the case that $X$ is a Euclidean space, Theorem 1.1 is new
since it allows for local approximation by bi-Lipschitz images of
$\mathbb{R}^{n}$.

There is a more subtle aspect in which Theorem 1.1 is stronger than the
results of Marstrand and Mattila. Since our notion of a tangent space is
defined via Gromov–Hausdorff convergence, we do not rely on any structure of
the ambient metric space $X$. On the other hand, the proof of Marstrand
crucially relies on tangents that are affine subspaces of $\mathbb{R}^{m}$. In
particular, Marstrand’s proof fails for tangents that are isometric copies of
$\mathbb{R}^{2}$ when $X=(\mathbb{R}^{3},\|\cdot\|_{\infty})$. However, we
note that for the special case $n=1$, the idea of Besicovitch can be modified
to work in any metric space; an account of this will appear in regular .

The use of Gromov–Hausdorff tangents may be considered as a further weakening
of the concept of the rotating tangent plane. Such a framework is necessary
when working in this generality; Even if we assume $X$ is a Banach space, it
is not possible to define an approximate tangent plane using ambient linear
structure. For example consider

$\\{\chi_{[0,t]}\in L^{1}([0,1]):t\in[0,1]\\},$

which is an isometric copy of $[0,1]$ but does not have an approximate tangent
plane at any point.

The results of Marstrand and Mattila are a starting point for the work of
Preiss MR890162 . It is natural to consider generalisations of Preiss’s
tangent measures when investigating results like Theorem 1.1. Recall that a
non-zero Radon measure $\nu$ on $\mathbb{R}^{m}$ is a _tangent measure_ to
another Radon measure $\mu$ at $x\in\operatorname{spt}\mu$ if there exist
$r_{i}\to 0$ such that the pushforwards of $\mu$ under the maps
$y\mapsto(y-x)/r$, scaled by $1/\mu(B(x,r_{i}))$, weak* converge to $\nu$.

To define a tangent measure of a measure $\mu$ on a metric space $X$, one
requires a definition of a limit of a sequence of pointed metric measure
spaces. In this paper, a _pointed metric measure space_ $(X,d,\mu,x)$ will
consist of a metric space $(X,d)$, a Borel measure
$\mu\in\mathcal{M}_{\mathrm{loc}}(X)$ (that is, $\mu$ is finite on bounded
sets), and a distinguished point $x\in\operatorname{spt}\mu$. Given some
notion of convergence of pointed metric measure spaces, we can define a
_tangent_ of a metric measure space $(X,d,\mu,x)$ as any metric measure space
$(Y,\rho,\nu,y)$ for which there exist $r_{i}\to 0$ such that

$\left(X,\frac{d}{r_{i}},\frac{\mu}{\mu(B(x,r_{i}))},x\right)\to(Y,\rho,\nu,y).$
(4)

One such notion of convergence is _measured Gromov–Hausdorff convergence_ and
the resulting tangent spaces are known as _measured Gromov–Hausdorff
tangents_. This convergence can be metrised similarly to the Gromov–Hausdorff
distance: In addition to taking the Hausdorff distance between the metric
spaces embedded in $Z$, one also considers the distance between the
pushforwards of the measures, using some canonical metric that metrises weak*
convergence in $Z$. We construct one such metrisation of measured
Gromov–Hausdorff convergence, $d_{\mathrm{pmGH}}$, in Definition 4.1.

However, the requirement that the underlying metric spaces must
Gromov–Hausdorff converge is too rigid to study rectifiable sets. Indeed, it
is easy to find examples of rectifiable metric spaces with no tangent measures
according to this definition (see Example 4.8). Instead, we define a metric,
$d_{*}$, that only considers the distance between the pushforwards of the
measures in $Z$ and disregards the Hausdorff distance between the embedded
metric spaces (this will be discussed further below). We write
$\operatorname{Tan}(X,d,\mu,x)$ for the set of all tangent measures to
$(X,d,\mu,x)$ defined using $d_{*}$ and (4).

Our main result on tangent measures is the following. For $K\geq 1$ let
$\operatorname{biLip}(K)^{*}$ be the set of all pointed metric measure spaces
that are supported on an element of $\operatorname{biLip}(K)$.

######  Theorem 1.2.

Let $(X,d)$ be a complete metric space, $n\in\mathbb{N}$ and let $E\subset X$
be $\mathcal{H}^{n}$-measurable with $\mathcal{H}^{n}(E)<\infty$. The
following are equivalent:

  1. 1.

$E$ is $n$-rectifiable;

  2. 2.

For $\mathcal{H}^{n}$-a.e. $x\in E$, $\Theta_{*}^{n}(E,x)>0$ and there exists
an $n$-dimensional Banach space $(\mathbb{R}^{n},\|.\|_{x})$ such that

$\operatorname{Tan}(X,d,\mathcal{H}^{n}|_{E},x)=\\{(\mathbb{R}^{n},\|.\|_{x},\mathcal{H}^{n}/2^{n},0)\\}.$

  3. 3.

For $\mathcal{H}^{n}$-a.e. $x\in E$, $\Theta^{n}_{*}(E,x)>0$ and there exists
a $K_{x}\geq 1$ such that
$\operatorname{Tan}(X,d,\mathcal{H}^{n}|_{E},x)\subset\operatorname{biLip}(K_{x})^{*}$.

Once Theorem 1.1 is established, the main steps to proving Theorem 1.2 are to
develop the properties of $d_{*}$ and the properties of tangent measures.

The proof of Theorem 1.1 follows from the combination of two results, Theorem
3.4 and Corollary 6.4. Roughly speaking, Theorem 3.4 shows that, under the
hypotheses of Theorem 1.1, for $\mathcal{H}^{n}$-a.e. $x\in E$, the following
is true: for any $\epsilon>0$ and any sufficiently small $r>0$, there exists a
metric space $\tilde{E}\supset E$ and a continuous (in fact Hölder) map
$\iota\colon[0,r]^{n}\to\tilde{E}$ such that $\iota|_{\partial[0,r]^{n}}$ is
close to having Lipschitz inverse and

$\mathcal{H}^{n}_{\infty}(\iota([0,r]^{n})\setminus E)<\epsilon r^{n}.$ (5)

Here $\mathcal{H}^{n}_{\infty}$ denotes the $n$-dimensional Hausdorff content.

Corollary 6.4 is a consequence of a theorem of perturbations  (see Theorem
6.3). This theorem can be viewed as a replacement for the Besicovitch–Federer
projection theorem that is valid in any complete metric space. It considers
1-Lipschitz “non-linear” projections in place of orthogonal projections.
Corollary 6.4 states that any $E\subset X$ for which each $E^{\prime}\subset
E$ satisfies the conclusion of Theorem 3.4 is $n$-rectifiable. Note that the
hypotheses of Theorem 1.1 are inherited by any subset (by Lemma 2.3).

The construction of Theorem 3.4 is the central result of the paper. Naturally,
the idea is to take a set $G\subset E$ for which the approximations given by
(3) are in some sense uniform and “glue” together the approximating tangent
planes to construct the Hölder surface. Of course, following this approach,
one will eventually encounter points in $E\setminus G$ where we halt the
gluing process and we begin to define $\tilde{E}\setminus G$. Therefore, at
every step of the construction, it is crucial to control the size of the set
of points we encounter in $E\setminus G$ to establish (5). The details of this
construction are discussed further in Section 3.

We will see that the basic theory of tangent measures in our setting follows
analogously to Preiss’s tangent measures, once the properties of $d_{*}$ are
established.

There are several equivalent ways to define convergence of metric measure
spaces that considers only the convergence of measures and not the
Gromov–Hausdorff convergence of the metric spaces: one simply has to decide
upon a way to metrise weak* convergence in the metric space $Z$. Sturm sturm
first considered the $L_{2}$ transportation distance between probability
measures with finite variance. Greven, Pfaffelhuber, and Winter MR2520129
consider the Prokhorov metric between probability measures on compact metric
spaces. Gigli, Mondino and Savaré MR2401600  simply define
$(X_{i},d_{i},\mu_{i},x_{i})\to(X,d,\mu,x)$, for $\mu_{i},\mu\neq 0$, if there
exist isometric embeddings into $Z$ such that $x_{i}\to x$ and $\mu_{i}$ weak*
converges to $\mu$.

To define $d_{*}$, we first define a metric $F$ between
$\mu,\nu\in\mathcal{M}_{\mathrm{loc}}(Z)$ by duality with Lipschitz functions.
A common approach for _finite_ measures is to consider the metric $F^{1}$
defined as the dual norm to the set of 1-Lipschitz functions on $Z$. It is
well known that $F^{1}$ contains geometric information on the support of
$\mu,\nu$. One can define a metric between $\mu$ and $\nu$ by taking the
infimal $\epsilon>0$ for which

$F^{1}(\mu|_{B(z,1/\epsilon)},\nu|_{B(z,1/\epsilon)})<\epsilon,$ (6)

for some fixed $z\in Z$. Taking this idea further, we let $F^{\epsilon}$ be
the dual norm of the set of $1/\epsilon$-Lipschitz functions, and define $F$
by replacing $F^{1}$ with $F^{\epsilon}$ in (6) (see Definition 2.12). Then
from $F$ one precisely obtains the Hausdorff distance between two large
subsets of the supports of $\mu$ and $\nu$ (see Lemma 4.10). Consequently, by
defining $d_{*}$ using $F$ to metrise weak* convergence in
$\mathcal{M}_{\mathrm{loc}}(Z)$, we obtain an explicit relationship between
$d_{*}$ and the $d_{\mathrm{pmGH}}$ distance between two large subsets (see
Theorem 4.11). This allows us to use standard Gromov–Hausdorff techniques and
Prokhorov’s theorem to develop the theory of $d_{*}$.

However, we do not introduce $d_{*}$ as a new notion of convergence. Indeed,
it is equivalent to any of the previously mentioned notions whenever the
latter is defined: they all correspond to weak* convergence in $Z$ (see Remark
4.14). Rather, we use it as another, convenient description of the established
notions.

Note that $\operatorname{Tan}(\mathbb{R}^{n},\|\cdot\|_{2},\mu,x)$ agrees with
(isometry classes of) Preiss’s tangent measures. Also, for a sequence of
uniformly doubling spaces, convergence in $d_{*}$ is equivalent to measured
Gromov–Hausdorff convergence (by (MR2401600, , Theorem 3.33) or Corollary
4.18). Consequently, tangent measures of doubling metric measure spaces are
precisely measured Gromov–Hausdorff tangents.

To conclude the introduction we discuss results related to this work.

Assuming a set $E\subset\mathbb{R}^{m}$ has _uniform_ approximation by
tangents that are isomorphic copies of $\mathbb{R}^{n}$, a classical
construction of Reifenberg reifenberg  constructs a locally bi-Hölder
parametrisation of $E$ by an $n$-dimensional ball. This has been generalised
by David and Toro toro_metric , using Gromov–Hausdorff tangents, to any metric
space. Such a parametrisation can be used to obtain the map $\iota$ in Theorem
3.4 and in fact one may take $\tilde{E}=E$. However, these assumptions are too
strong for our situation: they do not allow the exclusion of sets of small
measure as in (2) and the convergence in (3) is not uniform in $x$. Moreover,
one cannot deduce that such a map exists for subsets of $E$ and so cannot use
Corollary 6.4 to deduce rectifiability.

Relationships between tangent spaces and parametrising maps have been explored
in specific non-Euclidean metric spaces such as the Heisenberg group (or other
Carnot groups) merlo ; donne . In this case, the additional structure of a
particular ambient metric space such as the Heisenberg group enables one to
define a much stronger notion of a tangent than a Gromov–Hausdorff tangent.

The outline of the paper is as follows.

In Section 2 we recall preliminary notions of geometric measure theory and
properties of doubling measures. In Section 2.3 we recall facts about weak*
convergence of measures in metric spaces and define $F$. In Section 2.4 we
recall several constructions related to the Gromov–Hausdorff distance,
incorporating both $d_{\mathrm{H}}$ and $F$, in a way that the definitions and
properties of $d_{\mathrm{pGH}}$, $d_{\mathrm{pmGH}}$ and $d_{*}$ all follow
from the same construction.

In Section 3 we prove Theorem 3.4: the construction of a Hölder surface under
the hypotheses of sets $X\supset C\supset G$ such that, for each $x\in G$ and
each sufficiently small $r>0$, $B(x,r)\cap C$ is approximated (up to sets of
small measure) by bi-Lipschitz images of $\mathbb{R}^{n}$ (see Definition
3.2).

In Section 4 we define $d_{\mathrm{pmGH}}$ and $d_{*}$ and prove various
properties, in particular the relationship between the two in Theorem 4.11.
This section may be of independent interest.

Section 5 contains the definition of a tangent measure using $d_{*}$ and
develops the theory of tangent measures in parallel to the basic theory of
Preiss MR890162 .

In Section 6 we conclude the proof of Theorems 1.1 and 1.2.

##  2 Preliminaries

Throughout this article $n$ will denote a fixed positive integer. For
$m\in\mathbb{N}$, $\mathbb{R}^{m}$ will denote the $m$-dimensional real vector
space, $\ell_{2}^{m}=(\mathbb{R}^{m},\|\cdot\|_{2})$ and
$\ell_{\infty}^{m}=(\mathbb{R}^{m},\|\cdot\|_{\infty})$. We also write
$\ell_{\infty}$ for the Banach space of all bounded real sequences equipped
with the supremum norm.

Let $(X,d)$ be a metric space. If $S\subset X$ and $x\in X$ we will write

$d(x,S)=\inf\\{d(x,s):s\in S\\}$

and

$B(S,r)=\\{x\in X:d(x,S)\leq r\\},$

the _closed_ neighbourhood of $S$ of radius $r$. We write $B(x,r)$ for
$B(\\{x\\},r)$, the closed ball of radius $r$ centred at $x$. Similarly, we
define

$U(S,r)=\\{x\in X:d(x,S)<r\\}$

and $U(x,r)=U(\\{x\\},r)$, the open neighbourhood of $S$ and open ball centred
on $x$, respectively. We will often omit notation for the metric $d$ when it
is not necessary.

By a _measure_ on $X$ we mean a non-negative countably sub-additive function
$\mu$ defined on the power set of $X$ with $\mu(\emptyset)=0$. A set $A\subset
X$ is _$\mu$ -measurable_ if

$\mu(F)=\mu(F\cap E)+\mu(F\setminus E)\quad\forall F\subset X.$

The set of all $\mu$-measurable subsets of $X$ is a $\sigma$-algebra and $\mu$
is countably additive when restricted to the set of $\mu$-measurable sets. A
_Borel measure_ on $X$ is a measure for which all Borel subsets of $X$ are
$\mu$-measurable. A Borel measure is _Borel regular_ if, for any $S\subset X$,
there exists a Borel $B\supset S$ with $\mu(B)=\mu(S)$. Note that, for any
$Y\subset X$, the monotonicity of $\mu$ implies $\mu(Y\cap S)=\mu(Y\cap B)$.
The _support_ of $\mu$, denoted $\operatorname{spt}\mu$, is the smallest
closed set $C\subset X$ with $\mu(X\setminus C)=0$.

We write $\mathcal{M}(X)$ for the set of all Borel regular measures on $X$ and
define $\mathcal{M}_{\mathrm{loc}}(X)$ to be those $\mu\in\mathcal{M}(X)$ that
are finite on all balls in $X$. By (federer, , Theorem 2.2.2), for any
$\mu\in\mathcal{M}_{\mathrm{loc}}(X)$, and any Borel $B\subset X$,

$\mu(B)=\sup\\{\mu(C):C\subset B\text{ closed}\\}=\inf\\{\mu(U):U\supset
B\text{ open}\\}.$

Also note that, for any $\mu\in\mathcal{M}_{\mathrm{loc}}(X)$,
$\operatorname{spt}\mu$ is separable. Consequently, if $X$ is complete, any
$\mu\in\mathcal{M}_{\mathrm{loc}}(X)$ is a Radon measure. That is, for every
Borel $B\subset X$,

$\mu(B)=\sup\\{\mu(K):K\subset B\text{ compact}\\}.$

###  2.1 Rectifiable subsets of a metric space

For $s\geq 0$, $0\leq\delta\leq\infty$ and $A\subset X$, define

$\mathcal{H}_{\delta}^{s}(A)=\inf\left\\{\sum_{i\in\mathbb{N}}\operatorname{diam}(A_{i})^{s}:A\subset\bigcup_{i\in\mathbb{N}}A_{i},\
\operatorname{diam}(A_{i})\leq\delta\right\\}$

and

$\mathcal{H}^{s}(A)=\lim_{\delta\to 0}\mathcal{H}_{\delta}^{s}(A),$

the $s$-dimensional _Hausdorff measure_ of $A$. For any $s\geq 0$ and any
metric space $X$, $\mathcal{H}^{s}$ is a Borel regular measure on $X$, see
(Mattila_1995, , Section 4). We note that $\mathcal{H}^{s}_{\infty}(A)$, the
_$s$ -dimensional Hausdorff content_ of $A$, is bounded above by
$\operatorname{diam}(A)^{s}$.

For $L\geq 0$, a function $f\colon(X,d)\to(Y,\rho)$ between two metric spaces
is $L$-Lipschitz if

$\rho(f(x),f(y))\leq Ld(x,y)\quad\forall x,y\in X.$

Note that, for any $s\geq 0$, $0\leq\delta\leq\infty$ and $A\subset Y$,

$\mathcal{H}_{L\delta}^{s}(f(A))\leq L^{s}\mathcal{H}_{\delta}^{s}(A).$

In particular, Lipschitz functions increase $\mathcal{H}^{s}$ and
$\mathcal{H}^{s}_{\infty}$ by at most a multiplicative factor of $L^{s}$. A
function $f\colon(X,d)\to(Y,\rho)$ is _$L$ -bi-Lipschitz_ if it is
$L$-Lipschitz and invertible onto its image with $L$-Lipschitz inverse. That
is, for any $x,y\in Y$,

$\frac{1}{L}\rho(x,y)\leq\rho^{\prime}(f(x),f(y))\leq L\rho(x,y)\quad\forall
x,y\in X.$

For a $\mathcal{H}^{s}$-measurable $S\subset X$ and $x\in X$ define the
_upper_ and _lower Hausdorff densities_ of $S$ at $x$ by

$\Theta^{*,s}(S,x):=\limsup_{r\to 0}\frac{\mathcal{H}^{n}(B(x,r))}{(2r)^{s}}$

and

$\Theta_{*}^{s}(S,x):=\liminf_{r\to
0}\frac{\mathcal{H}^{n}(B(x,r))}{(2r)^{s}}$

respectively.

######  Definition 2.1.

A $\mathcal{H}^{n}$-measurable $E\subset X$ is _$n$ -rectifiable_ if there
exist a countable number of Lipschitz $f_{i}\colon
A_{i}\subset\mathbb{R}^{n}\to X$ such that

$\mathcal{H}^{n}\left(E\setminus\bigcup_{i\in\mathbb{N}}f_{i}(A_{i})\right)=0.$

A $\mathcal{H}^{n}$-measurable $S\subset X$ is _purely $n$-unrectifiable_ if
every $n$-rectifiable $E\subset X$ satisfies $\mathcal{H}^{n}(S\cap E)=0$.

By the same proof as for rectifiable subsets of Euclidean space (see
(Mattila_1995, , Theorem 15.6)), we have the following decomposition result.

######  Lemma 2.2.

Let $Y\subset X$ be $\mathcal{H}^{n}$-measurable with
$\mathcal{H}^{n}(Y)<\infty$. There exists a decomposition $Y=E\cup S$ where
$E$ is $n$-rectifiable and $S$ is purely $n$-unrectifiable.

Recall the following classical result regarding Hausdorff densities.

######  Lemma 2.3 (federer  2.10.18).

Let $s>0$ and $S\subset X$ be $\mathcal{H}^{s}$-measurable with
$\mathcal{H}^{s}(S)<\infty$. Then

  1. 1.

For $\mathcal{H}^{s}$-a.e. $x\in S$

$\Theta^{*,s}(S,x)\leq 1\mathchar 24635\relax\;$

  2. 2.

For $\mathcal{H}^{s}$-a.e. $x\in X\setminus S$

$\Theta^{*,s}(S,x)=0.$

The following theorem of Kirchheim precisely describes the local structure of
rectifiable subsets of a metric space. Note the statements of theorems 2.4 and
2.5 are slightly different to those in MR1189747  due to the choice of
normalisation of Hausdorff measure.

######  Theorem 2.4 (Theorem 9 MR1189747 ).

Let $E\subset X$ be $n$-rectifiable. For $\mathcal{H}^{n}$-a.e. $x\in E$ there
exists a norm $\|\cdot\|_{x}$ on $\mathbb{R}^{n}$, a map $f_{x}\colon
X\to\mathbb{R}^{n}$ and a closed set $C_{x}\subset E$ such that $f_{x}(x)=0$,

$\lim_{r\to 0}\frac{\mathcal{H}^{n}(B(x,r)\cap C_{x})}{(2r)^{n}}=1$ (7)

and

$\limsup_{r\to
0}\left\\{\left|1-\frac{\|f_{x}(y)-f_{x}(z)\|_{x}}{d(y,z)}\right|:y\neq z\in
C_{x}\cap B(x,r)\right\\}=0.$

Note that, if $\mathcal{H}^{n}(E)<\infty$, (7) and Lemma 2.3 item 1 imply

$\frac{\mathcal{H}^{n}(B(x,r))}{(2r)^{n}}\to 1$ (8)

and

$\frac{\mathcal{H}^{n}(B(x,r)\setminus C_{x})}{(2r)^{n}}\to 0.$ (9)

We also note the following.

######  Lemma 2.5 (Lemma 6 MR1189747 ).

If $\|\cdot\|$ is a norm on $\mathbb{R}^{n}$ then
$\mathcal{H}^{n}(B(0,r))=(2r)^{n}$. Here $\mathcal{H}^{n}$ and $B(0,1)$ are
defined with respect to $\|\cdot\|$.

We also require the McShane extension theorem.

######  Theorem 2.6.

Let $S\subset X$ and $0<\alpha\leq 1$. Suppose that $f\colon
S\to\ell_{\infty}^{m}$ (respectively into $\ell_{\infty}$) is an
$\alpha$-Hölder map with Hölder constant $H$. Then there exists an
$\alpha$-Hölder extension $F\colon X\to\ell_{\infty}^{m}$ (respectively into
$\ell_{\infty})$ of $f$ with Hölder constant $H$. In particular, $F$ is
$\alpha$-Hölder with Hölder constant $\sqrt{m}H$ as a map into $\ell_{2}^{m}$.

Moreover, for each $1\leq i\leq m$ (respectively, for each $i\in\mathbb{N}$),
$F$ may be chosen such that

$\inf\\{f_{i}(x):x\in S\\}\leq F_{i}\leq\sup\\{f_{i}(x):x\in S\\}.$

Finally, if $S\subset B(x,r)$ for some ball $B(x,r)\subset X$ then $F$ may be
chosen with $\operatorname{spt}F\subset B(x,r^{\prime})$, for
$r^{\prime}=r+(H^{-1}\|f\|_{\infty})^{1/\alpha}$.

Here and throughout, $\operatorname{spt}F$ denotes the closure of the set

$\\{x\in X:F(x)\neq 0\\}.$

Proof: The proof of the first part is standard, see (MR1800917, , Section 6).
For the second part, one simply replaces the $i$th coordinate of $F$ by

$\max\\{\inf\\{f_{i}(x):x\in S\\},\min\\{F_{i},\sup\\{f_{i}(x):x\in
S\\}\\}\\}$

for each $1\leq i\leq m$ (respectively for each $i\in\mathbb{N}$).

Finally, if $S\subset B(x,r)$ then $f$ is bounded. We first extend $f$ to a
map $f^{\prime}$ defined on

$S^{\prime}:=S\cup(X\setminus B(x,r^{\prime}))$

by defining $f^{\prime}$ to equal zero on $X\setminus B(x,r^{\prime})$. Then
$f^{\prime}$ is also $\alpha$-Hölder with Hölder constant $H$. Applying the
second part of the theorem to $f^{\prime}$ gives the final statement.

###  2.2 Doubling measures

Much of the theory of tangent measures requires the original measure to be
asymptotically doubling.

######  Definition 2.7.

For $M\geq 1$, $\mu\in\mathcal{M}(X)$ is _$M$ -doubling_ if

$0<\mu(B(x,2r))\leq M\mu(B(x,r))<\infty$

for each $x\in X$ and each $r>0$. A $\mu\in\mathcal{M}(X)$ is _asymptotically
doubling_ if

$\limsup_{r\to 0}\frac{\mu(B(x,2r))}{\mu(B(x,r))}<\infty\quad\text{for
}\mu\text{-a.e. }x\in X.$

To proceed we need the following standard facts.

######  Lemma 2.8.

Let $\mu\in\mathcal{M}_{\mathrm{loc}}(X)$. For any $r>0$,

$x\mapsto\mu(U(x,r))\text{ is lower semicontinuous}$ (10)

and

$x\mapsto\mu(B(x,r))\text{ is upper semicontinuous}.$ (11)

Consequently, for any $R,M,s>0$ the set

$A:=\\{x\in X:\mu(B(x,2r))\leq M\mu(B(x,r))\ \forall 0<r<R\\}$

is a Borel set, as are

$B:=\\{x\in X:\mu(B(x,r))<Mr^{s}\ \forall 0<r<R\\}$

and

$C:=\\{x\in X:\mu(B(x,r))>Mr^{s}\ \forall 0<r<R\\}.$

Proof: If $x_{i}\to x$ with $d(x_{i},x)$ monotonically decreasing then

$U(x,r-d(x,x_{i}))$

monotonically increases to $U(x,r)$ and is contained in $U(x_{i},r)$, proving
(10). Similarly,

$B(x,r+d(x,x_{i}))$

monotonically decreases to $B(x,r)$ and contains in $B(x_{i},r)$, proving
(11). Consequently, for any $R,M>0$ and $\delta\geq 0$, the set

$A_{R,M,\delta}:=\\{x\in X:\mu(U(x,(2+\delta)r))\leq M\mu(B(x,r))\ \forall
0<r<R\\}$

is closed, as is

$A=\bigcap_{\delta\in\mathbb{Q}^{+}}A_{R,M,\delta}.$

The proof that $B$ and $C$ are Borel is similar.

######  Lemma 2.9.

Let $\mu\in\mathcal{M}(X)$. Suppose that for some $M,R>0$ a set $Y\subset X$
satisfies

$0<\mu(B(x,2r))\leq M\mu(B(x,r))<\infty\quad\forall 0<r<R,\ \forall x\in Y.$
(12)

Then for any $x\in Y$ and $r<R/2$, $B(x,r)\cap Y$ is contained in $M^{4}$
balls of radius $r/2$ centred at points of $B(x,r)\cap Y$. In particular, for
any $\epsilon>0$, $B(x,r)\cap Y$ is contained in $M^{-4\log_{2}\epsilon}$
balls of radius $\epsilon r$.

Proof: Let $\mathcal{N}$ be a maximal disjoint $r/4$-net of $B(x,r)\cap Y$.
Then for any $y\in\mathcal{N}$, (12) gives

$\mu(B(y,r/4))\geq\frac{\mu(B(y,r/2))}{M}\geq\frac{\mu(B(y,r))}{M^{2}}\geq\frac{\mu(B(y,2r))}{M^{3}}\geq\frac{\mu(B(x,r))}{M^{3}}.$
(13)

For $y\in\mathcal{N}$ the $B(y,r/4)$ are disjoint subsets of $B(x,2r)$.
Therefore, if $y_{1},\ldots,y_{N}\in\mathcal{N}$,

$\mu(B(x,2r))\geq\mu\left(\bigcup_{i=1}^{N}B(y_{i},r/4)\right)=\sum_{i=1}^{N}\mu(B(y_{i},r/4))\\\
\geq\sum_{i=1}^{N}\frac{\mu(B(x,r))}{M^{3}}\geq\sum_{i=1}^{N}\frac{\mu(B(x,2r))}{M^{4}}$

using (13) for the penultimate inequality and (12) for the final inequality.
Thus $N\leq M^{4}$.

The statement about covering by balls of radius $\epsilon r$ follows by
induction.

######  Theorem 2.10.

Any asymptotically doubling $\mu\in\mathcal{M}(X)$ satisfies the Lebesgue
density theorem. That is, for any $S\subset X$ and for $\mu$-a.e. $x\in S$,

$\lim_{r\to 0}\frac{\mu(S\cap B(x,r))}{\mu(B(x,r))}=1.$

Such an $x\in S$ is a _Lebesgue density point_ of $S$.

Proof: This is a standard result for doubling measures. First one proves the
Vitali covering theorem (MR1800917, , Theorem 1.6) and uses it to deduce the
Lebesgue differentiation theorem (MR1800917, , Theorem 1.8) and hence the
Lebesgue density theorem for Borel sets $S\subset X$.

The proof for asymptotically doubling measures is the same since the proof of
the Vitali covering theorem works, with minor modifications, only assuming an
almost everywhere countable decomposition into Borel sets, each satisfying
(12) for some $M,R>0$. Assuming $\mu$ is asymptotically doubling, Lemma 2.8
gives such a decomposition of $\operatorname{spt}\mu$.

The statement for arbitrary $S\subset X$ follows by considering a Borel
$S^{\prime}\supset S$ with $\mu(S^{\prime})=\mu(S)$.

###  2.3 Prokhorov’s theorem

A sequence $\mu_{i}\in\mathcal{M}_{\mathrm{loc}}(X)$ _weak*_ converges to
$\mu\in\mathcal{M}_{\mathrm{loc}}(X)$ if, for every bounded and continuous
$g\colon X\to\mathbb{R}$ with bounded support,

$\int g\,\mathrm{d}\mu_{i}\to\int g\,\mathrm{d}\mu.$

We write $\mu_{i}\to\mu$ to denote this convergence.

Recall that, if $\mu_{i}\to\mu\in\mathcal{M}_{\mathrm{loc}}(X)$ and $x_{i}\to
x\in X$, then for every $r>0$,

$\mu(U(x,r))\leq\liminf_{i\to\infty}\mu_{i}(U(x_{i},r))$ (14)

and

$\mu(B(x,r))\geq\limsup_{i\to\infty}\mu_{i}(B(x_{i},r)).$ (15)

Indeed, these follow by combining the standard proof of lower/upper
semicontinuity for a fixed open/closed set and the proof of eqs. 10 and 11.

Recall Prokhorov’s theorem, reformulated to describe convergence in
$\mathcal{M}_{\mathrm{loc}}(X)$.

######  Theorem 2.11 (Theorem 2.3.4 bogachev ).

Let $X$ be a complete and separable metric space and $x\in X$. A set
$\mathcal{S}\subset\mathcal{M}_{\mathrm{loc}}(X)$ is pre-compact (with respect
to weak* convergence) if and only if, for every $r,\epsilon>0$:

  * •

$\\{\mu(B(x,r)):\mu\in\mathcal{S}\\}$ is bounded and

  * •

There exists a compact $K\subset X$ such that

$\mu(B(x,r)\setminus K)\leq\epsilon\quad\forall\mu\in\mathcal{S}.$ (16)

######  Definition 2.12.

Let $X$ be a metric space and $x\in X$. For
$\mu,\nu\in\mathcal{M}_{\mathrm{loc}}(X)$, and $L,r>0$, define

$F^{L,r}_{x}(\mu,\nu)=\sup\left\\{\int g\,\mathrm{d}(\mu-\nu):g\colon
X\to[-1,1],\ L\text{-Lipschitz},\ \operatorname{spt}g\subset B(x,r)\right\\}.$

Define $F_{x}^{L}(\mu,\nu)$ to be the infimum, over all $0<\epsilon<1/2$, for
which

$F_{x}^{L,1/\epsilon}(\mu,\nu)<\epsilon.$

If no such $0<\epsilon<1/2$ exists, set $F_{x}^{L}(\mu,\nu)=1/2$. Also define
$F_{x}(\mu,\nu)$ to be the infimum, over all $0<\epsilon<1/2$, for which

$F_{x}^{1/\epsilon,1/\epsilon}(\mu,\nu)<\epsilon.$

If no such $0<\epsilon<1/2$ exists, set $F_{x}(\mu,\nu)=1/2$.

Note that $F_{x}^{L,r}$ is increasing in $L,r$ and

$F_{x}^{L^{\prime},r}\leq\frac{L^{\prime}}{L}F_{x}^{L,r}.$ (17)

Note that, for any Borel $S\subset X$,

$F_{x}^{L,r}(\mu,\mu|_{S})\leq\mu(B(x,r)\setminus S).$ (18)

The reason for allowing the Lipschitz constants of the functions to increase
when defining $F_{x}$ will become clear in Lemma 4.10.

######  Proposition 2.13.

Let $X$ be a metric space, $x\in X$ and $L>0$. Convergence in $F_{x}$,
equivalently $F_{x}^{L}$, implies weak* convergence in
$\mathcal{M}_{\mathrm{loc}}(X)$ and, if $X$ is complete and separable, the
converse is true.

Proof: Equivalence of convergence in $F_{x}$ and $F_{x}^{L}$ follows from
(17).

Suppose $F_{x}(\mu_{i},\mu)\to 0$ and that $g\colon X\to\mathbb{R}$ is bounded
and continuous with $\operatorname{spt}g\subset B(x,r)$. The fact that

$\int g\,\mathrm{d}\mu_{i}\to\int g\,\mathrm{d}\mu$

follows from a standard approximation argument (MR2401600, , Section 5.1).
Indeed, observe that the sequence

$g_{k}(y):=\inf\\{g(w)+kd(w,y):w\in X\\}$

monotonically increases to $g$ and that each $g_{k}$ is $k$-Lipschitz with

$\operatorname{spt}g_{k}\subset B(x,r+\|g\|_{\infty}).$

Thus,

$\displaystyle\liminf_{i\to\infty}\int g\,\mathrm{d}\mu_{i}$
$\displaystyle\geq\limsup_{k\to\infty}\liminf_{i\to\infty}\int
g_{k}\,\mathrm{d}\mu_{i}$ $\displaystyle=\limsup_{k\to\infty}\int
g_{k}\,\mathrm{d}\mu$ $\displaystyle=\int g\,\mathrm{d}\mu,$

using monotonicity of the integral for the inequality, the fact that
$F_{x}(\mu_{i},\mu)\to 0$ for the first equality and the monotone convergence
theorem for the final equality. The reverse “limsup” equality follows from a
similar argument that involves approximating $g$ from above.

Now suppose that $X$ is complete and separable, that $\mu_{i}\to\mu$ in
$\mathcal{M}_{\mathrm{loc}}(X)$ but $F_{x}(\mu_{i},\mu)\not\to 0$. By taking a
(non-relabelled) subsequence, there exist an $r>0$ such that
$F_{x}(\mu_{i},\mu)>1/r$ for all $i\in\mathbb{N}$. That is, for each
$i\in\mathbb{N}$ there exists an $r$-Lipschitz $g_{i}\colon X\to[-1,1]$ with
$\operatorname{spt}g_{i}\subset B(x,r)$ such that

$\left|\int g_{i}\,\mathrm{d}(\mu_{i}-\mu)\right|\geq 1/r.$

Set $r^{\prime}=r+1/r$. By Theorem 2.11, there exist an $M>0$ and a compact
$K\subset B(x,r^{\prime})$ such that, for all $i\in\mathbb{N}$,
$\mu_{i}(B(x,r^{\prime}))\leq M$ and

$\mu_{i}(B(x,r^{\prime})\setminus K)<1/8r.$ (19)

By increasing $K$ if necessary, we may also suppose
$\mu(B(x,r^{\prime})\setminus K)<1/8r$.

The Arzelà–Ascoli theorem implies that, after taking a further subsequence,
the $g_{i}$ converge uniformly on $K$ to an $r$-Lipschitz $g\colon K\to[-1,1]$
with $\operatorname{spt}g\subset B(x,r)$. By Theorem 2.6, we may extend $g$ to
an $r$-Lipschitz function defined on the whole of $X$ with
$\operatorname{spt}g\subset B(x,r^{\prime})$. Then, for any sufficiently large
$i$ such that

$\sup\\{|g(y)-g_{i}(y)|:y\in K\\}(\mu_{i}+\mu)(B(x,r^{\prime}))\leq 1/4r,$

which exists by (15),

$\displaystyle\left|\int g\,\mathrm{d}(\mu_{i}-\mu)\right|$
$\displaystyle\geq\left|\int_{K}g\,\mathrm{d}(\mu_{i}-\mu)\right|-1/4r$
$\displaystyle\geq\left|\int_{K}g_{i}\,\mathrm{d}(\mu_{i}-\mu)\right|-1/2r$
$\displaystyle\geq\left|\int g_{i}\,\mathrm{d}(\mu_{i}-\mu)\right|-3/4r\geq
1/4r.$

This contradicts $\mu_{i}\to\mu$.

######  Lemma 2.14.

For any metric space $X$, $x\in X$ and $L>0$, $F_{x}$ and $F_{x}^{L}$ are
metrics on $\mathcal{M}_{\mathrm{loc}}(X)$. If $X$ is complete and separable
then so are $(\mathcal{M}_{\mathrm{loc}}(X),F_{x})$ and
$(\mathcal{M}_{\mathrm{loc}}(X),F_{x}^{L})$.

Proof: Certainly $F_{x}$ are positive, finite on
$\mathcal{M}_{\mathrm{loc}}(X)$ and symmetric. If $F_{x}(\mu,\nu)=0$, then the
monotonicity of $F^{L,r}_{x}$ implies $F^{L,r}_{x}(\mu,\nu)=0$ for every
$L,r>0$. Consequently, the monotone convergence theorem implies
$\mu(C)=\nu(C)$ for all closed and bounded $C\subset X$ and hence
$\mu(B)=\nu(B)$ for all Borel $B\subset X$. Since $\mu$ and $\nu$ are both
Borel regular, this implies $\mu=\nu$.

To see that $F_{x}$ satisfies the triangle inequality, let
$\mu,\nu,\lambda\in\mathcal{M}_{\mathrm{loc}}(X)$. If either of
$F_{x}(\mu,\lambda)$ or $F_{x}(\lambda,\nu)$ equal 1/2 then the triangle
inequality holds since $F_{x}(\mu,\nu)\leq 1/2$. Otherwise, for $\delta>0$,
let $0<\epsilon_{1}<F_{x}(\mu,\lambda)+\delta$ and
$0<\epsilon_{2}<F_{x}(\lambda,\nu)+\delta$ satisfy

$F_{x}^{\frac{1}{\epsilon_{1}},\frac{1}{\epsilon_{1}}}(\mu,\lambda)<\epsilon_{1}\quad\text{and}\quad
F_{x}^{\frac{1}{\epsilon_{2}},\frac{1}{\epsilon_{2}}}(\lambda,\nu)<\epsilon_{2}.$

Then the triangle inequality and monotonicity of $F_{x}^{L,r}$ imply

$\displaystyle
F_{x}^{\frac{1}{\epsilon_{1}+\epsilon_{2}},\frac{1}{\epsilon_{1}+\epsilon_{2}}}(\mu,\nu)$
$\displaystyle\leq
F_{x}^{\frac{1}{\epsilon_{1}+\epsilon_{2}},\frac{1}{\epsilon_{1}+\epsilon_{2}}}(\mu,\lambda)+F_{x}^{\frac{1}{\epsilon_{1}+\epsilon_{2}},\frac{1}{\epsilon_{1}+\epsilon_{2}}}(\lambda,\nu)$
$\displaystyle\leq
F_{x}^{\frac{1}{\epsilon_{1}},\frac{1}{\epsilon_{1}}}(\mu,\lambda)+F_{x}^{\frac{1}{\epsilon_{2}},\frac{1}{\epsilon_{2}}}(\lambda,\nu)$
$\displaystyle\leq\epsilon_{1}+\epsilon_{2}$ $\displaystyle\leq
F_{x}(\mu,\lambda)+F_{x}(\lambda,\nu)+2\delta.$

Since $\delta>0$ is arbitrary, this shows that $F_{x}$ is a metric.

If $x_{i}$, $i\in\mathbb{N}$ is a dense subset of $X$, one easily checks that
the set

$\\{q_{1}\delta_{x_{1}}+\dots q_{m}\delta_{x_{m}}:m\in\mathbb{N},\
q_{i}\in\mathbb{Q}\\}$

is dense in $\mathcal{M}_{\mathrm{loc}}(X)$.

Finally, the completeness of $F_{x}$ immediately follows from (bogachev, ,
Corollary 2.3.5) (that convergence of $\int g\,\mathrm{d}\mu_{i}$ for every
bounded Lipschitz $g$ implies (16)), Theorem 2.11 and Proposition 2.13. The
proof for $F_{x}^{L}$ is similar.

###  2.4 Gromov distances, embeddings and convergence

Pointed Gromov–Hausdorff _convergence_ of metric spaces is prevalent in the
literature. However, the metric defining this convergence is less commonly
used. In this subsection we recall Gromov’s metric for pointed
Gromov–Hausdorff convergence and prove some basic facts about it that are
missing from the literature (namely that it is a complete metric on the
isometry classes of pointed metric spaces). Our presentation of this is
sufficiently general so that the corresponding statements for the pointed
measured Gromov–Hausdorff distance, and our metric $d_{*}$, easily follow (see
Section 4).

We use the following modification of the Hausdorff distance.

######  Definition 2.15 (Section 6 gromov2 ).

Let $Z$ be a metric space, $z\in Z$ and $X,Y\subset Z$. Define $H_{z}(X,Y)$ to
be the infimum over all $0<\epsilon<1/2$ for which

$X\cap B(z,1/\epsilon)\subset B(Y,\epsilon)\quad\text{and}\quad Y\cap
B(z,1/\epsilon)\subset B(X,\epsilon).$ (20)

If no such $0<\epsilon<1/2$ exists, set $H_{z}(X,Y)=1/2$. Note that, if (20)
is satisfied for some $\epsilon>0$ then it is satisfied for all
$\epsilon^{\prime}\geq\epsilon$.

######  Lemma 2.16.

For any metric space $Z$ and $z\in Z$, $H_{z}$ is a pseudometric on the power
set of $Z$ and is a metric on the closed subsets of $Z$. If $Z$ is complete
then so is the set of closed subsets of $Z$ equipped with $H_{z}$.

Proof: Let $\zeta$ be the metric on $Z$, let $X,Y,W\subset Z$ and let

$d_{XW}=H_{z}(X,W)\quad\text{and}\quad d_{WY}=H_{z}(W,Y).$

If one of $d_{XW}$ or $d_{WY}$ equals $1/2$ there is nothing to prove.
Otherwise, let $0<\delta<1/2-\min\\{d_{XZ},d_{ZY}\\}$. If

$x\in X\cap B(z,1/(d_{XZ}+d_{ZY}+2\delta))$ (21)

then (20) for $X$ and $W$ gives a $w\in W$ with

$\zeta(w,x)\leq d_{XW}+\delta.$ (22)

In particular, since $d_{XW},d_{WZ}<1/2-\delta$,

$\zeta(w,z)\leq\zeta(w,x)+\zeta(x,z)\leq
d_{XW}+\delta+1/(d_{XW}+d_{WY}+2\delta)\leq 1/(d_{WY}+\delta).$

Therefore (20) for $W$ and $Y$ gives a $y\in Y$ with

$\zeta(w,y)\leq d_{WY}+\delta.$ (23)

In particular,

$\zeta(x,w)\leq d_{XW}+d_{WY}+2\delta.$ (24)

Thus eqs. 21 and 24 show that (20) holds between $X$ and $W$ for
$\epsilon=d_{XW}+d_{WY}+2\delta$. Since $\delta>0$ is arbitrary, this
completes the proof.

If $C\subset Z$ is closed and $H_{z}(C,S)=0$, then for any $x\in S$ there
exist $x_{i}\in C$ with $x_{i}\to x$. Since $C$ is closed, $x\in C$ and so
$C\subset S$. Consequently, $H_{z}$ is a metric on the closed subsets of $Z$.

Completeness follows from the same argument as the regular Hausdorff metric
(burago, , Proposition 7.3.7).

In this section we consider isometric embeddings $X\to Z$ of one metric space
into another. We identify $X$, its elements and any measure on $X$, with their
isometric images in $Z$.

We require the following standard construction.

######  Lemma 2.17.

Let $X_{i}$ be a sequence of metric spaces and, for each $i\leq
j\in\mathbb{N}$, suppose that there exist isometric embeddings
$X_{i},X_{j}\to(Z_{i,j},\zeta_{i,j})$ into some metric space. There exists a
metric space $(Z,\zeta)$ and, for each $i\in\mathbb{N}$, isometric embeddings
$X_{i}\to Z$ such that, for each $i\leq j\in\mathbb{N}$, $w_{i}\in X_{i}$ and
$w_{j}\in X_{j}$, $\zeta(w_{i},w_{j})\leq\zeta_{i,j}(w_{i},w_{j})$, with
equality if $j=i+1$.

Proof: Define a function $\zeta$ on

$\tilde{Z}=\bigsqcup_{i\in\mathbb{N}}X_{i}$

as follows. For $i\leq j\in\mathbb{N}$, $w_{i}\in X_{i}$ and $w_{j}\in X_{j}$,
define $\zeta(w_{i},w_{j})$ to be the infimum of

$\sum_{p=1}^{m}\zeta_{k_{p-1},k_{p}}(w_{p-1},w_{p})$

over all $m\in\mathbb{N}$, $i=k_{0}\leq k_{1}\leq\ldots\leq k_{m}=j$ and
$w_{k}\in X_{k}$ for each $i<k<j$. Then $\zeta$ is a pseudometric on $Z$ such
that $\zeta(w_{i},w_{j})\leq\zeta_{i,j}(w_{i},w_{j})$ for all $i\leq
j\in\mathbb{N}$, $w_{i}\in X_{i}$ and $w_{j}\in X_{j}$. If $j=i+1$, the
triangle inequality for $\zeta_{i,i+1}$ gives the reverse inequality. Finally,
let $(Z,\zeta)$ be the metric space obtained from $(\tilde{Z},\zeta)$. Since
each $X_{i}\to\tilde{Z}$, the projection to $Z$ gives an isometric embedding
$X_{i}\to Z$.

######  Definition 2.18.

A _pointed_ metric space $(X,x)$ consists of a metric space $X$ and a
distinguished point $x\in X$. An _isometric embedding_ $(X,x)\to(Y,y)$ of
pointed metric spaces is an isometric embedding of metric spaces with $x=z$.
An _isometry_ $(X,x)\to(Y,y)$ is a surjective isometric embedding.

A _pointed metric measure space_ $(X,\mu,x)$ consists of a complete and
separable pointed metric space $(X,x)$ and a
$\mu\in\mathcal{M}_{\mathrm{loc}}(X)$.

######  Definition 2.19.

Let $(X,\mu,x)$ and $(Y,\nu,y)$ be pointed metric measure spaces and $a,b\geq
0$. Define

$G_{a,b}((X,\mu,x),(Y,\nu,y))$

to be the infimum, over all pointed metric spaces $(Z,z)$ and all isometric
embeddings $(X,x),(Y,y)\to(Z,z)$ of

$aH_{z}(X,Y)+bF_{z}(\mu,\nu).$

######  Proposition 2.20.

For any $a,b\geq 0$, $G_{a,b}$ is a complete pseudometric on the class of
pointed metric measure spaces. Moreover,

$G_{a,b}((X_{i},\mu_{i},x_{i}),(X,\mu,x))\to 0$

if and only if there exists a complete and separable pointed metric space
$(Z,z)$ and isometric embeddings $(X_{i},x_{i})\to(Z,z)$ and $(X,x)\to(Z,z)$
such that

$aH_{z}(X_{i},X)+bF_{z}(\mu_{i},\mu)\to 0.$

Proof:

First observe the following. Let $\zeta,\zeta^{\prime}$ be metrics on a set
$Z$ with $\zeta\leq\zeta^{\prime}$. Let $z\in Z$ and suppose that $X,Y\subset
Z$. Then

$H^{\zeta}_{z}(X,Y)\leq H^{\zeta^{\prime}}_{z}(X,Y),$ (25)

where the superscript denotes the metric used to define $H_{z}$. Also, since
any Lipschitz function on $X\cup Y\subset Z$ with respect to $\zeta$ is
Lipschitz with respect to $\zeta^{\prime}$, if
$\mu\in\mathcal{M}_{\mathrm{loc}}(X)$ and
$\nu\in\mathcal{M}_{\mathrm{loc}}(Y)$,

$F^{\zeta}_{z}(\mu,\nu)\leq F^{\zeta^{\prime}}_{z}(\mu,\nu).$ (26)

To prove the triangle inequality for $G_{a,b}$, for $i=1,2,3$ let
$(X_{i},d_{1},\mu_{i},x_{i})$ be a pointed metric measure space and suppose
there exist isometric embeddings

$(X_{1},d_{1},x_{1}),(X_{2},d_{2},x_{2})\to(Z_{1,2},\zeta_{1,2},z_{1,2})$

and

$(X_{2},d_{2},x_{2}),(X_{3},d_{3},x_{3})\to(Z_{2,3},\zeta_{2,3},z_{2,3}).$

Let $\zeta_{1,3}$ be the metric on $X_{1}\sqcup X_{3}$ that equals $d_{1}$ and
$d_{3}$ on $X_{1}$ and $X_{3}$ respectively and
$\zeta(w_{1},w_{3})=1+d_{1}(w_{1},x_{1})+d_{3}(w_{3},x_{3})$ whenever
$w_{1}\in X_{1}$ and $w_{3}\in X_{3}$. Let $(Z,\zeta)$ be the metric space
obtained from Lemma 2.17 and note that necessarily $x_{1}=x_{2}=x_{3}=:z$ in
$Z$. The triangle inequality for $H_{z}$ and $F_{z}$ in $Z$ imply

$aH_{z}(X_{1},X_{3})+bF_{z}(\mu_{1},\mu_{3})\leq
aH_{z}(X_{1},X_{2})+bF_{z}(\mu_{1},\mu_{2})\\\
+aH_{z}(X_{2},X_{3})+bF_{z}(\mu_{2},\mu_{3}).$ (27)

Combining eqs. 27, 25 and 26 gives the triangle inequality for $G_{a,b}$.

Now let $(X_{i},\mu_{i},x_{i})$ be a Cauchy sequence with respect to $G_{a,b}$
and, for each $i\leq j\in\mathbb{N}$, let $(Z_{i,j},\zeta_{i,j},z_{i,j})$ be a
pointed metric space for which there exist isometric embeddings

$(X_{i},x_{i}),(X_{j},x_{j})\to(Z_{i,j},\zeta_{i,j},z_{i,j})$

with

$aH_{z_{i,j}}(X_{i},X_{j})+bF_{z_{i,j}}(\mu_{i},\mu_{j})\leq
2G_{a,b}((X_{i},\mu_{i},x_{i}),(X_{j},\mu_{j},x_{j})).$ (28)

Let $(Z,\zeta)$ be the completion of the metric space given by Lemma 2.17 and
note that necessarily $x_{1}=x_{2}=\ldots=:z$ in $Z$. Since each $X_{i}$ is
separable, so is $Z$. Since each $X_{i}$ is complete, they are closed subsets
of $Z$ and so the $\mu_{i}\in\mathcal{M}_{\mathrm{loc}}(Z)$. Equations 25, 26
and 28 imply that the $X_{i}$ and $\mu_{i}$ are Cauchy sequences with respect
to $H_{z}$ and $F_{z}$ in $Z$. Since $Z$ is complete, $X_{i}$ and $\mu_{i}$
converge to some $X$ and $\mu$. Consequently

$G_{a,b}((X_{i},\mu_{i},x_{i}),(X,\mu,x))\to 0.$

Since any convergent sequence is Cauchy, this also proves the moreover
statement.

######  Definition 2.21 (Section 6 gromov2 ).

The _Gromov–Hausdorff distance_ between pointed metric spaces $(X,x)$ and
$(Y,y)$ is defined to be

$d_{\mathrm{pGH}}((X,x),(Y,y))=G_{1,0}((X,0,x),(Y,0,y)).$

The definition of $d_{\mathrm{pGH}}$ in gromov2  only requires isometric
embeddings into $(Z,\zeta)$ satisfying $\zeta(x,y)\leq\epsilon$, not
necessarily $x=y$. However, the next lemma shows that two definitions are bi-
Lipschitz equivalent. We will see that fixing the distinguished point
simplifies several results.

######  Lemma 2.22.

Let $(Z,\zeta)$ be a metric space, $X,Y\subset Z$, $x\in X$ and $y\in Y$ with
$x\neq y$.

  1. 1.

The metric $\tilde{\zeta}$ on $X\sqcup Y$ that equals $\zeta$ on each of $X$
and $Y$ and $\tilde{\zeta}(w,w^{\prime})=\zeta(w,w^{\prime})+\zeta(x,y)$ for
any $w\in X$ and $w^{\prime}\in Y$ satisfies

$\tilde{\zeta}(w,x)\leq\tilde{\zeta}(w,y)\ \forall w\in
X\quad\text{and}\quad\tilde{\zeta}(w,y)\leq\tilde{\zeta}(w,x)\ \forall w\in
Y.$ (29)

  2. 2.

Let $\tilde{\zeta}$ be _any_ metric on $X\sqcup Y$ that equals $\zeta$ on each
of $X$ and $Y$ that satisfies (29). If $(Z^{\prime},\zeta^{\prime})$ is the
quotient metric space of $(X\sqcup Y,\tilde{\zeta})$ with respect to the
relation $x\sim y$, there exist isometric embeddings $X,Y\to Z^{\prime}$ with
$x=y$.

  3. 3.

There exists a metric space $(Z^{\prime},\zeta^{\prime})$ and isometric
embeddings $X,Y\to Z^{\prime}$ with $x=y$ such that
$\zeta^{\prime}\leq\zeta+\zeta(x,y)$ on $X\cup Y$.

Proof: The triangle inequality for $\zeta$ implies the triangle inequality for
$\tilde{\zeta}$. Equation 29 is equivalent to the triangle inequality for
$\zeta$. This proves item 1.

The quotient metric of any metric $\tilde{\zeta}$ with respect to the relation
$x\sim y$ is given by

$\zeta^{\prime}(w,w^{\prime})=\min\\{\tilde{\zeta}(w,w^{\prime}),\tilde{\zeta}(w,x)+\tilde{\zeta}(y,w^{\prime}),\tilde{\zeta}(w^{\prime},x)+\tilde{\zeta}(y,z)\\}.$

If $\tilde{\zeta}$ satisfies (29) then $\zeta^{\prime}$ equals $\zeta$ on each
of $X$ and $Y$. Indeed, if $w,w^{\prime}\in X$,

$\tilde{\zeta}(w,w^{\prime})\leq\tilde{\zeta}(w,x)+\tilde{\zeta}(x,w^{\prime})\leq\tilde{\zeta}(w,x)+\tilde{\zeta}(y,w^{\prime}),$

by the triangle inequality and (29). Thus
$\zeta^{\prime}(w,w^{\prime})=\tilde{\zeta}(w,w^{\prime})$. The corresponding
statement for elements of $Y$ implies item 2.

Combining items 1 and 2 gives item 3.

The following Lemma is an adaptation of the corresponding concepts of an
$\epsilon$-isometry of bounded metric spaces, see (burago, , Corollary
7.3.28).

######  Definition 2.23.

Let $(X,d,x)$ and $(Y,\rho,y)$ be pointed metric spaces and $\epsilon>0$. An
_$\epsilon$ -isometry_ $f\colon(X,d,x)\to(Y,\rho,y)$ is a Borel map $f\colon
B(x,1/\epsilon)\to Y$ with $f(x)=y$ such that

$|\rho(f(z),f(z^{\prime}))-d(z,z^{\prime})|\leq\epsilon\quad\forall
z,z^{\prime}\in B(x,1/\epsilon)$ (30)

and

$B(y,1/\epsilon-\epsilon)\subset B(f(B(x,1/\epsilon)),\epsilon).$ (31)

######  Lemma 2.24.

Let $(Z,\zeta,z)$ be a pointed metric space, $X,Y$ separable subsets of $Z$
and $0<\epsilon<1/2$. If $H_{z}(X,Y)<\epsilon$ then there exists a Borel map

$f\colon X\cap B(z,1/\epsilon)\to Y$

with $\zeta(f(x),x)\leq\epsilon$ for all $x\in X\cap B(z,1/\epsilon)$ and

$Y\cap B\left(z,1/\epsilon-\epsilon\right)\subset B\left(f\left(X\cap
B\left(z,1/\epsilon\right)\right),2\epsilon\right).$ (32)

In particular, if $d_{\mathrm{pGH}}((X,x),(Y,y))<\epsilon$, there exists a
$2\epsilon$-isometry from $(X,x)$ to $(Y,y)$.

Conversely, if there exists an $\epsilon$-isometry from $(X,x)$ to $(Y,y)$
then there exists a pointed metric space $(Z,\zeta,z)$ and isometric
embeddings

$(X,x),(Y,y)\to(Z,z)$

such that, for all $x^{\prime}\in B(x,1/\epsilon)$,

$\zeta(x^{\prime},f(x^{\prime}))\leq 2\epsilon.$

In particular,

$d_{\mathrm{pGH}}((X,x),(Y,y))<2\epsilon.$

Proof: Let $\delta>0$ be such that $H_{z}(X,Y)<\epsilon-\delta$. Since $X$ is
separable, there exist by countably many disjoint and non-empty Borel sets
$X_{j}\subset X$, $j\in\mathbb{N}$, of diameter at most $\delta$ with
$B(z,1/\epsilon)\cap X=\cup_{j}X_{j}$. For each $j\in\mathbb{N}$, pick
$x_{j}\in X_{j}$. Since $H_{z}(X,Y)<\epsilon-\delta$, there exists $y_{j}\in
Y$ with $\zeta(x_{j},y_{j})\leq\epsilon-\delta$. Define $f(w)=y_{j}$ for each
$w\in X_{j}$. Then $f$ is a Borel map and, by the triangle inequality, for $x$
in any $X_{j}$,

$\zeta(f(x),x)\leq\zeta(y_{j},x_{j})+\zeta(x_{j},x)\leq\epsilon-\delta+\delta\leq\epsilon.$
(33)

To see that (32) holds, let

$y\in B\left(z,\frac{1}{\epsilon}-\epsilon\right)$

By assumption there exists $x\in X$ with $\zeta(x,y)\leq\epsilon$. Then by the
triangle inequality,

$\zeta(x,z)\leq\zeta(x,y)+\zeta(y,z)\leq\epsilon+(1/\epsilon-\epsilon)\leq
1/\epsilon.$

Consequently $f(x)$ is defined and, by (33) and the triangle inequality,

$\zeta(f(x),y)\leq\zeta(f(x),x)+\zeta(x,y)\leq 2\epsilon,$

proving (32).

For the converse statement, let $f$ be the assumed $\epsilon$-isometry and
define $\zeta$ on $X\sqcup Y$ by $\zeta=d$ on $X$, $\zeta=\rho$ on $Y$ and

$\zeta(z,t)=\inf\\{d(z,w)+\rho(f(w),t)+\epsilon:w\in B(x,1/\epsilon)\\}$

whenever $z\in X$ and $t\in Y$. Then $\zeta$ is a metric: if
$z,z^{\prime\prime}\in X$ and $w,w^{\prime\prime}\in B(x,1/\epsilon)$, the
triangle inequality for $d,\rho$ and the fact that $f$ is an
$\epsilon$-isometry imply

$\displaystyle d(z,z^{\prime})$ $\displaystyle\leq
d(z,w)+d(w,w^{\prime})+d(w^{\prime},z^{\prime})$ $\displaystyle\leq
d(z,w)+\rho(f(w),f(w^{\prime}))+\epsilon+d(w^{\prime},z^{\prime})$
$\displaystyle\leq
d(z,w)+\rho(f(w),t)+\epsilon+d(w^{\prime},z^{\prime})+\rho(t,f(w^{\prime}))+\epsilon$

for any $t\in Y$. Taking the infimum over all $w,w^{\prime}\in
B(x,1/\epsilon)$ shows
$\zeta(z,z^{\prime})\leq\zeta(z,t)+\zeta(t,z^{\prime})$. A similar argument
for $t,t^{\prime}\in Y$ and $z\in X$ shows that $\zeta$ satisfies the triangle
inequality. Note that $\zeta$ satisfies $\zeta(x,y)=\epsilon$ and (29).
Indeed, for any $w\in B(x,1/\epsilon)$, since $f$ is an $\epsilon$-isometry,

$d(z,x)\leq d(z,w)+d(w,x)\leq d(w,z)+\rho(f(w),y)+\epsilon$

Taking the infimum over all $w\in B(x,1/\epsilon)$ gives (29).

Let $(Z^{\prime},\zeta^{\prime})$ be the metric space obtained from Lemma 2.22
item 2. Observe that for any $x^{\prime}\in B(x,1/\epsilon)$,
$\zeta^{\prime}(x^{\prime},f(x^{\prime}))\leq\epsilon$, proving and the first
containment in (20). If $y^{\prime}\in Y\cap B(y,1/\epsilon-\epsilon)$ then
there exists $x^{\prime}\in X\cap B(x,1/\epsilon)$ with
$\rho(y^{\prime},f(x^{\prime}))\leq\epsilon$, so that
$\zeta^{\prime}(x^{\prime},y^{\prime})\leq 2\epsilon$, proving the second
containment in (20), for $\epsilon$ replaced by $2\epsilon$.

Recall that a metric space $X$ is _proper_ if all closed balls in $X$ are
compact.

######  Corollary 2.25.

The Gromov–Hausdorff distance is a complete and separable metric on the set
$\mathbb{M}_{p}$ of isometry classes of proper pointed metric spaces. That is,
if $(X,x)$ and $(Y,y)$ are proper with

$d_{\mathrm{pGH}}((X,x),(Y,y))=0$ (34)

then there exists an isometry $(X,x)\to(Y,y)$.

######  Remark 2.26.

Here and below, the issue of considering the set of “all” separable metric
spaces can be easily avoided by using the Kuratowski (isometric) embedding of
any separable metric space into $\ell_{\infty}$.

We also make no distinction in notation between spaces and their equivalence
classes. It can be easily verified that the concepts we consider to not depend
on particular representatives of an equivalence class.

Proof: By Proposition 2.20 we know that $d_{\mathrm{pGH}}$ is a complete
pseudometric. Moreover, if

$d_{\mathrm{pGH}}((X,x),(Y,y))<\epsilon$

and $B(x,1/\epsilon)$ is covered by $N$ balls of radius $\epsilon$, then
$B(y,1/2\epsilon)$ is covered by $N$ balls of radius $2\epsilon$. Thus, if
$(Y,y)$ is a complete pointed metric space and is the $d_{\mathrm{pGH}}$ limit
of a sequence of proper pointed metric spaces, then $Y$ is proper too. It is
easily verified that the set of finite pointed metric spaces with rational
distances is dense in $(\mathbb{M}_{p},d_{\mathrm{pGH}})$.

Now let $(X,x)$ and $(Y,y)$ be proper satisfying (34) and fix
$k\in\mathbb{N}$. For each $j\in\mathbb{N}$ let $\mathcal{N}_{j}$ be a finite
$1/j$-net of $B(x,k)$. For each fixed $j\in\mathbb{N}$, by taking a convergent
subsequence, we may suppose that the $f_{i}$ converge to an isometry when
restricted to $\mathcal{N}_{j}$ whose image is a $2/j$-net of $B(y,k)$. By
taking a diagonal subsequence, we may suppose the $f_{i}$ converge to an
isometry when restricted to each $\mathcal{N}_{j}$. Let
$\iota\colon\cup_{j}\mathcal{N}_{j}\to B(y,k)$ be the limiting isometry. Since
$\cup_{j}\mathcal{N}_{j}$ is dense in $B(x,k)$,
$\iota(\cup_{j}\mathcal{N}_{j})$ is dense in $B(y,k)$ and since $B(y,k)$ is
complete, $\iota$ extends to an isometry $\iota\colon B(x,k)\to B(y,k)$.
Taking a convergent diagonal subsequence over $k\to\infty$ completes the
proof.

The Gromov compactness theorem is an immediate corollary of the completeness
of $(\mathbb{M}_{p},d_{\mathrm{pGH}})$.

Recall that a set of metric spaces $\mathcal{S}$ is _uniformly totally
bounded_ if, for every $\epsilon>0$ there exists $N_{\epsilon}\in\mathbb{N}$
such that each $X\in\mathcal{S}$ is contained in $N_{\epsilon}$ balls of
radius $\epsilon$.

######  Theorem 2.27 (Section 6 gromov2 ).

A set $\mathcal{S}\subset(\mathbb{M}_{p},d_{\mathrm{pGH}})$ is pre-compact if
and only if, for every $r>0$,

$\\{B(x,r)\subset X:(X,x)\in\mathcal{S}\\}$

is uniformly totally bounded.

Proof: The given condition is equivalent to $\mathcal{S}$ being totally
bounded in $(\mathbb{M}_{p},d_{\mathrm{pGH}})$. Indeed, for $\epsilon>0$ and
$(X,x)\in\mathcal{S}$, suppose that $B(x,1/\epsilon)$ is covered by
$N_{\epsilon}$ balls of radius $\epsilon$. Let $\mathcal{N}$ be the set of
metric spaces of the form

$(\\{1,2,\ldots,N_{\epsilon}\\},\rho)$

with

$\rho(x,y)\in\\{\epsilon j:j\in\mathbb{N},\ 0<j\leq 1/\epsilon^{2}\\}$

for each $x\neq y\in Y$ (and such that $\rho$ satisfies the triangle
inequality). Then $\mathcal{S}$ is contained in the $2\epsilon$ neighbourhood
of $\mathcal{N}$.

Conversely, suppose that $\mathcal{S}$ is totally bounded and
$\mathcal{N}\subset\mathcal{S}$ is a finite set such that $\mathcal{S}\subset
B(\mathcal{N},\epsilon)$. Since each $(Y,y)\in\mathcal{N}$ is proper and
$\mathcal{N}$ is finite, there exists $N_{\epsilon}\in\mathbb{N}$ such that,
for each $(Y,y)\in\mathcal{N}$, $B(y,1/\epsilon)$ is covered by $N_{\epsilon}$
balls of radius $\epsilon$. Then for any $(X,x)\in\mathcal{S}$,
$B(x,1/2\epsilon)$ is contained in $N_{\epsilon}$ balls of radius $2\epsilon$.

##  3 Construction of a Hölder surface

Recall that throughout the paper we work with a fixed $n\in\mathbb{N}$. For
notational convenience, we define $\operatorname{biLip}(K)$ as the set of
$K$-bi-Lipschitz images of $\ell_{\infty}^{n}$, rather than images of
$\ell_{2}^{n}$.

######  Definition 3.1.

For $K\geq 1$ let $\operatorname{biLip}(K)$ be the set of isometry classes of
all pointed proper metric spaces $(X,x)$ for which there exists a surjective
and $K$-bi-Lipschitz $\psi\colon\ell_{\infty}^{n}\to X$.

######  Definition 3.2.

Let $(X,d)$ be a complete metric space, let $G\subset C\subset X$ be closed
subsets and $K\geq 1$, $\eta,R_{0}>0$ and $0<\delta<1/2$.

The triple $(X,C,G)$ has _good tangential approximation_ , written
$GTA(\eta,K,\delta,R_{0})$, if both of the following conditions hold:

  1. 1.

For each $x\in C$ and each $0<r\leq R_{0}$,

$\mathcal{H}^{n}(B(x,r))\geq\eta r^{n}\mathchar 24635\relax\;$ (35)

  2. 2.

For each $x\in G$ and each $0<r\leq R_{0}$, there exists a closed

$C^{\prime}\subset B(x,r)\cap C$

containing $x$ with

$\mathcal{H}^{n}(B(x,r)\setminus C^{\prime})<\eta(\delta r)^{n}$ (36)

and

$d_{\mathrm{pGH}}((C^{\prime},d/r,x),\operatorname{biLip}(K))<\min\\{\delta,1/K(1+2\delta)\\}.$
(37)

######  Observation 3.3.

For $K\geq 1$ and $0<\delta<1/2$ let

$\Phi=[-K(1+2\delta),K(1+2\delta)]^{n}\subset\ell_{\infty}^{n}.$

If $(X,C,G)$ satisfies $GTA(\eta,K,\delta,R_{0})$ then for any $0<r\leq R_{0}$
there exists a Borel $\xi\colon\Phi\to C$ with $\xi(0)=x$ such that

$\mathcal{H}^{n}(B(x,r)\setminus B(\xi(\Phi),\delta r))<\eta(\delta r)^{n}$
(38)

and

$\frac{\|q-q^{\prime}\|_{\infty}}{K}-\delta\leq\frac{d(\xi(q),\xi(q^{\prime}))}{r}\leq
K\|q-q^{\prime}\|_{\infty}+\delta$ (39)

for all $q,q^{\prime}\in\Phi$. Indeed, let
$(Y,\rho,y)\in\operatorname{biLip}(K)$ with

$d_{\mathrm{pGH}}((C^{\prime},d/r,x),(Y,\rho,y))<\min\\{\delta,1/K(1+2\delta)\\}=:\delta^{\prime}$

and let $p\colon\ell_{\infty}^{n}\to Y$ be $K$-bi-Lipschitz and surjective
with $p(0)=x$. Define $\xi$ by composing $p$ and the
$\delta^{\prime}$-isometry from $(Y,\rho,y)$ to $(C^{\prime},d/r,x)$ granted
by Lemma 2.24.

For $m\in\mathbb{N}$ and $R>0$ let

$\mathcal{D}(R,m)=\\{R2^{-m}(j_{1},\ldots,j_{n})\in[0,R]^{n}:j_{1},\ldots,j_{n}\in\mathbb{Z}\\}.$

The main result of this section is the following theorem.

######  Theorem 3.4.

For any $K\geq 1$, $0<\gamma<1$ and $\eta>0$ there exists $m_{0}\in\mathbb{N}$
such that the following is true. Suppose that $m\geq m_{0}$ and $G\subset
C\subset X\subset\ell_{\infty}$ are closed sets such that $(X,C,G)$ satisfies

$GTA(\eta,K,4^{-m},R_{0}).$ (40)

Then for any $x\in G$ and $0<r\leq R_{0}$, there exists a $\gamma$-Hölder map

$\iota\colon[0,r]^{n}\subset\ell_{\infty}^{n}\to
B(x,20Kr)\subset\ell_{\infty}$

with $\iota(0)=x$ such that:

  1. 1.

$\iota|_{\mathcal{D}(r,m)}\colon\mathcal{D}(r,m)\to C$ is $(K+2^{-m})$-bi-
Lipschitz onto its image;

  2. 2.

For any $y,z\in[0,r]^{n}$ with $\|y-z\|_{\infty}\leq 2^{-m}r$,

$\|\iota(y)-\iota(z)\|_{\infty}\leq 10K2^{-m\gamma/2}r\mathchar 24635\relax\;$

  3. 3.

$\mathcal{H}^{n}_{\infty}(\iota([0,r]^{n})\setminus
C)\leq\Lambda\mathcal{H}^{n}(B(\iota(0),20Kr)\cap X\setminus G),$

for some $\Lambda>0$ depending only upon $K,\gamma,m,n,\eta$.

Before discussing the proof of Theorem 3.4, we mention how the properties of
$\iota$ are used in the proof of Theorem 1.2 (the complete details are given
in Theorem 6.5).

Item 1 implies that the map $f:=\iota|_{\mathcal{D}(r,m)}^{-1}$ exists and is
$L$-Lipschitz for some $L$ independent of $m$ and $r$. After extending $f$ to
all of $\ell_{\infty}$ using Theorem 2.6, Theorem 3.4 items 1 and 2 imply that

$\|f(\iota(y))-y\|_{\infty}\leq A2^{-m}r$ (41)

for some constant $A$ independent of $m$ and $r$. Provided $m$ is sufficiently
large, this implies that $f(\iota([0,r]^{n}))$ contains $[r/4,3r/4]^{n}$ and
hence

$\mathcal{H}^{n}(f(\iota([0,r]^{n})))\geq cr^{n}$ (42)

for an absolute constant $c>0$ (see Corollary 6.4). Provided

$\frac{\mathcal{H}^{n}(B(\iota(0),20Kr)\cap X\setminus G)}{r^{n}}$

is sufficiently small (which can be guaranteed if $\iota(0)$ is a density
point of $G$ and $r$ is sufficiently small), item 3 and (42) imply that

$\mathcal{H}^{n}(f(\iota([0,r]^{n})\cap C))>0.$ (43)

Moreover, if $f$ is replaced by any $L$-Lipschitz $g$ with
$\|f-g\|_{\infty}<A2^{m}r/4$, then (41) and hence (42) and (43) also hold for
$g$. The main result of perturbations  (see Theorem 6.3) then implies that $C$
cannot be purely $n$-unrectifiable. In Proposition 6.2 it is shown that if $X$
satisfies the hypotheses of Theorem 1.2, any subset of $X$ can be covered (up
to a $\mathcal{H}^{n}$ null set) by a countable union of sets which satisfy
the hypotheses of $C$. Consequently, $X$ must be $n$-rectifiable.

We now discuss the proof of Theorem 3.4. First note that, by scaling the norm
in $\ell_{\infty}$, it suffices to consider the case $R_{0}=r=1$. The
construction of the map $\iota$ involves a multi-scale iteration over sub-
cubes of $[0,1]^{n}$. To begin we define $\iota_{1}$ on $\mathcal{D}(1,m)$ to
equal the map $\xi$ given by 3.3. By choosing $m_{0}$ sufficiently large, (39)
ensures that item 1 of Theorem 3.4 holds.

We now wish to extend $\iota_{1}$ to a function $\iota_{2}$ defined on (a
large subset of) $\mathcal{D}(1,2m)$. The basic idea is the following:

  * •

Suppose that $p\in\mathcal{D}(1,m)$ is such that $\iota_{1}(p)$ is
sufficiently close to a point of $G$. Then the map $\xi$ from 3.3 provides a
rich structure to the neighbourhood of $\iota_{1}(p)$. We wish to use $\xi$ to
locally define $\iota_{2}$ on points in $\mathcal{D}(1,2m)$ that are close to
$p$, whilst maintaining some control on the local Lipschitz constant of
$\iota_{2}$. (See Lemma 3.8 for the fundamental building block of this
extension.)

  * •

On the other hand, since $\xi$ takes values in $C$, there may well exist
$p\in\mathcal{D}(1,m)$ for which $\iota_{1}(p)$ lies relatively far away from
$G$. At this stage we do not extend $\iota_{2}$ near to $p$ and this will
produce a hole in the domain of $\iota$. Later we will patch over such holes,
but we must control the total size of the holes we need to patch. The lower
density bound (35) allows us to bound the measure of those points that lie far
away from $G$ and consequently on the amount of patching required. (This is
addressed much later in Proposition 3.15.)

We then iterate this extension process for each $i\geq 2$, defining
$\iota_{i}$ on a large subset of $\mathcal{D}(1,im)$ as an extension of
$\iota_{i-1}$ and defining $\iota$ as the limit of the $\iota_{i}$. The
control on the local Lipschitz constant of each $\iota_{i}$ leads to a Hölder
bound on $\iota$ as in Theorem 3.4 item 2; The control on the total amount of
patching over all scales leads to item 3.

Before proceeding with more details, we fix notation for this section.

######  Notation 3.5.

We fix $K\geq 1$, $0<\gamma<1$ and $\eta>0$.

Let $N\in\mathbb{N}$ be such that

$(5K^{2})^{n}\leq 2^{N(1-\gamma)}$

and choose $\gamma\leq\alpha<1$ such that

$(5K^{2})^{n}=2^{N(1-\alpha)}.$ (44)

Write $l=2^{-N}$ and $\sigma=(5K^{2})^{n}$, so that

$\sigma l=(5K^{2})^{n}2^{-N}=2^{-N\alpha}=l^{\alpha}.$ (45)

Fix $(X,d)$ a complete metric space and $C,G\subset X$ such that $(X,C,G)$
satisfies $GTA(\eta,K,l/20,1)$.

We note the following. If $i\in\mathbb{N}$ with $i>2/\alpha$, eqs. 45 and 44
give

$\sigma(\sigma l)^{i}=l^{\alpha-1}l^{i\alpha}<l^{1}\leq 1/2.$

That is,

$\sigma^{i}<\frac{1}{2\sigma l^{i}}.$ (46)

######  Definition 3.6.

For $i\in\mathbb{N}$ let

$\mathcal{D}(i)=\\{l^{i}(j_{1},\ldots,j_{n})\in[0,1]^{n}:j_{1},\ldots,j_{n}\in\mathbb{Z}\\}.$

For an integer $0\leq m\leq n$, an $m$-dimensional _face_ of side length
$l^{i}$ is any set of the form

$\\{x\in[0,1]^{n}:p_{j}\leq x_{j}\leq p_{j}+b_{j}l^{i}\ \forall 1\leq j\leq
n\\},$

with $p\in\mathcal{D}(i)$, $b\in\\{0,1\\}^{n}$ and exactly $m$ of the
$b_{i}=1$. We denote by $\mathcal{F}(m,i)$ the set of all such faces. If
$1\leq m\leq n$ and $F\in\mathcal{F}(m,i)$, define the _boundary_ of $F$ by

$\partial F:=\bigcup\\{F^{\prime}\in\mathcal{F}(m-1,i):F^{\prime}\subset
F\\}.$

We also write $\mathcal{Q}(i)=\mathcal{F}(n,i)$, the set of _cubes_ of side
length $l^{i}$. For $Q\in\mathcal{Q}(i)$ the $m$-dimensional _skeleton_ of $Q$
is

$\operatorname{skel}(Q,m):=\bigcup\\{F\in\mathcal{F}(m,i):F\subset Q\\}.$

For $\mathcal{Q}\subset\mathcal{Q}(i)$, we define the _corners_ of
$\mathcal{Q}$ by

$\operatorname{cor}(\mathcal{Q})=\bigcup_{Q\in\mathcal{Q}}\operatorname{skel}(Q,0).$

We also define the _children_ of $\mathcal{Q}$ to be

$\operatorname{child}(\mathcal{Q}):=\\{Q^{\prime}\in\mathcal{Q}(i+1):Q^{\prime}\subset
Q,\text{ some }Q\in\mathcal{Q}\\}.$

If $Q\in\mathcal{Q}_{i}$ we write $\operatorname{child}(Q)$ for
$\operatorname{child}(\\{Q\\})$.

To begin the construction, we define a decomposition of a given collection of
cubes in order to implement the idea sketched above.

######  Lemma 3.7.

Let $i\in\mathbb{N}$, $\mathcal{Q}\subset\mathcal{Q}(i)$ and, for some
$\beta>0$, suppose that

$\iota\colon\operatorname{cor}(\mathcal{Q})\to C$

satisfies

$d(\iota(p),\iota(p^{\prime}))\leq\beta\|p-p^{\prime}\|_{\infty}$ (47)

for each $Q\in\mathcal{Q}$ and each $p,p^{\prime}\in\operatorname{cor}(Q)$.
Then there exists a decomposition $\mathcal{Q}=\mathcal{G}\cup\mathcal{B}$
such that:

  1. 1.

For each $Q\in\mathcal{G}$ there exists $x\in G$ with

$\iota(\operatorname{cor}(Q))\subset B(x,2\beta l^{i}).$

  2. 2.

For every $p\in\operatorname{cor}(\mathcal{B})$,

$B(\iota(p),\beta l^{i})\cap G=\emptyset.$

Proof: Let $Q\in\mathcal{Q}$. The triangle inequality and (47) imply that
either item 1 or item 2 holds. Decompose $\mathcal{Q}$ accordingly.

The proof of Theorem 3.4 now consists of two main parts. First, for
$i\in\mathbb{N}$ and $\iota$ and $\mathcal{Q}$ satisfying the hypotheses of
Lemma 3.7, we define an extension of $\iota$ to

$D^{\prime}:=\bigcup\mathcal{G}\cap\mathcal{D}(i+1).$

This extension satisfies the hypotheses of Lemma 3.7 for $i+1$ with a larger
value of $\beta$. Note that we do not extend $\iota$ into

$\bigcup\mathcal{B}\cap\mathcal{D}(i+1).$

In the second part we iteratively construct such extensions for all
$i\in\mathbb{N}$, using the resulting function from one iteration as the input
to the next. We show that the limiting function,

$\iota\colon S\subset[0,1]^{n}\to C,$

is $\alpha$-Hölder continuous. A Hölder extension extends $\iota$ to

$\iota\colon[0,1]^{n}\to\ell_{\infty}.$

This extension simultaneously patches all holes that were created when we did
not extend the function into the cubes in $\mathcal{B}$ after each application
of Lemma 3.7. Finally we show how the properties of $\mathcal{B}$ imply
Theorem 3.4 item 3.

###  3.1 Constructing the extension to the next scale

The hypothesis that $(X,C,G)$ satisfies $GTA(\eta,K,l/20,1)$ is used to prove
the following extension result. This serves as the fundamental building block
that is used construct the extension from $\mathcal{D}(i)$ to a large subset
of $\mathcal{D}(i+1)$.

######  Lemma 3.8.

For $0\leq m\leq n$ and $i\in\mathbb{N}$ let $F\in\mathcal{F}(m,i)$. For
$0<\beta<1/(2l^{i})$ suppose that

$\iota\colon\partial F\cap\mathcal{D}(i+1)\to C$

satisfies

$d(\iota(p),\iota(p^{\prime}))\leq\beta\|p-p^{\prime}\|_{\infty}$ (48)

for each $p,p^{\prime}\in\partial F\cap\mathcal{D}(i+1)$. Suppose that there
exists $x\in G$ with

$\iota(\partial F\cap\mathcal{D}(i+1))\subset B(x,2\beta l^{i}).$ (49)

Then there exists an extension of $\iota$ to

$\iota\colon F\cap\mathcal{D}(i+1)\to C$

such that

$d(\iota(p),\iota(p^{\prime}))\leq 2K^{2}\beta\|p-p^{\prime}\|_{\infty}$ (50)

for each $p,p^{\prime}\in F\cap\mathcal{D}(i+1)$.

Proof: By hypothesis, we have $r:=2\beta l^{i}<1$. Let $\xi\colon\Phi\to C$ be
given by 3.3 for $x$ and this value of $r$. Let $\delta=l/20$.

For a moment fix a point $p\in\partial F\cap\mathcal{D}(i+1)$. Since
$\iota(p)\in C$, Definition 3.2 item 1 implies

$\mathcal{H}^{n}(B(\iota(p),\delta r))\geq\eta(\delta r)^{n}.$

Since $\iota(p)\in B(x,2\beta l^{i})$, (38) implies that there exists $z\in
B(\xi(\Phi),\delta r)$ with

$d(z,\iota(p))\leq\delta r.$

That is, there exists a point $\psi(p)\in\Phi$ with

$d(\iota(p),\xi(\psi(p)))\leq 2\delta r.$ (51)

The function $\psi\colon\partial F\cap\mathcal{D}(i+1)\to\Phi$ is
$\frac{3K\beta}{2r}$-Lipschitz. Indeed, for any $p,p^{\prime}\in\partial
F\cap\mathcal{D}(i+1)$, eq. 39 implies

$\|\psi(p)-\psi(p^{\prime})\|_{\infty}\leq\frac{K}{r}d(\xi(\psi(p)),\xi(\psi(p^{\prime})))+\delta
K.$

Combining this with eq. 51 and using the triangle inequality gives

$\|\psi(p)-\psi(p^{\prime})\|_{\infty}\leq\frac{K}{r}d(\iota(p),\iota(p^{\prime}))+5\delta
K.$

Finally, eq. 48 and the choice of $r$ gives

$\displaystyle\|\psi(p)-\psi(p^{\prime})\|_{\infty}$
$\displaystyle\leq\frac{K\beta}{r}\|p-p^{\prime}\|_{\infty}+5\delta K$
$\displaystyle=\frac{K\beta}{r}(\|p-p^{\prime}\|_{\infty}+10\delta l^{i})$
$\displaystyle=\frac{K\beta}{r}(\|p-p^{\prime}\|_{\infty}+l^{i+1}/2)$
$\displaystyle\leq\frac{3K\beta}{2r}\|p-p^{\prime}\|_{\infty}.$

This implies that $\psi$ is $\frac{3K\beta}{2r}$-Lipschitz as claimed. Since
$\psi$ takes values in $\Phi$, the second conclusion of Theorem 2.6 gives an
extension of $\psi$ to a $\frac{3K\beta}{2r}$-Lipschitz function (with respect
to $\|\cdot\|_{\infty}$ in both the domain and image)

$\psi\colon F\cap\mathcal{D}(i+1)\to\Phi.$

For each $p\in F\cap\mathcal{D}(i+1)\setminus\partial F$, define
$\iota(p)=\xi(\psi(p))$. To check (50), let $p,p^{\prime}\in
F\cap\mathcal{D}(i+1)$. Note that, regardless of whether $p$ or
$p^{\prime}\in\partial F$, eq. 51 and the triangle inequality give

$d(\iota(p),\iota(p^{\prime}))\leq
d(\xi(\psi(p)),\xi(\psi(p^{\prime})))+4\delta r.$

Combining this with (39) gives

$d(\iota(p),\iota(p^{\prime}))\leq
Kr\|\psi(p)-\psi(p^{\prime})\|_{\infty}+5\delta r$

Using the fact that $\psi$ is $\frac{3K\beta}{2r}$-Lipschitz gives

$d(\iota(p),\iota(p^{\prime}))\leq\frac{3K^{2}\beta}{2}\|p-p^{\prime}\|_{\infty}+5\delta
r.$

Finally, substituting in for $r$ and $\delta$ gives

$d(\iota(p),\iota(p^{\prime}))\leq\frac{3K^{2}\beta}{2}\|p-p^{\prime}\|_{\infty}+\frac{\beta
l^{i+1}}{2}\leq 2K^{2}\beta\|p-p^{\prime}\|_{\infty},$

giving (50).

Suppose we are given $\iota\colon\mathcal{D}(i)\to C$ that we wish to extend
to $\mathcal{D}(i+1)$. One way to do this is to consider each
$Q\in\mathcal{Q}(i)$ one at a time and apply Lemma 3.8 several times to extend
$\iota$ to $Q\cap\mathcal{D}(i+1)$ (for the purposes of this discussion,
ignore the requirement given in (49)). Such an extension must agree with the
value of $\iota$ at those points in $\partial Q\cap\mathcal{D}(i+1)$ that
belong to the boundary of adjacent cubes to which we have previously extended
$\iota$. However, each time Lemma 3.8 is applied in this way, the Lipschitz
constant of the extension, given by (50), is a multiplicative factor larger
than the input Lipschitz constant given in (48). Consequently, if this is
repeated for each $Q\in\mathcal{Q}(i)$, the Lipschitz constant of the
extension to the whole of $\mathcal{D}(i+1)$ is far too large, and such an
extension is useless.

The correct approach is to first extend $\iota$ to all points of

$S:=\mathcal{D}(i+1)\cap\bigcup\mathcal{F}(m,1)$

by applying Lemma 3.8 one face at a time. For any $F\in\mathcal{F}(m,1)$,

$\partial F\cap\mathcal{D}(i+1)\subset\mathcal{D}(i).$

Since $\iota$ is defined on $\mathcal{D}(i)$, it can be shown that the
Lipschitz constant increases by a fixed amount, independent of the number of
$F\in\mathcal{F}(m,1)$ to which the extension is applied. We then extend to

$\mathcal{D}(i+1)\cap\bigcup\mathcal{F}(m,2)$

one face at a time. Similarly to the previous case, for any
$F\in\mathcal{F}(m,2)$,

$\partial F\cap\mathcal{D}(i+1)\subset S.$

Since $\iota$ is now defined on $S$, as in the previous case, the Lipschitz
constant only increases once, regardless of the number of
$F\in\mathcal{F}(m,2)$ to which we apply Lemma 3.8. This is repeated $n$ times
in total, so that we extended $\iota$ to all of $\mathcal{D}(i+1)$.

To facilitate this iteration we make the following simple observation.

######  Observation 3.9.

For $0\leq m\leq n$ and $i\in\mathbb{N}$, let $Q,Q^{\prime}\in\mathcal{Q}(i)$,
$p\in\operatorname{skel}(Q,m)\cap\mathcal{D}(i+1)$ and
$p^{\prime}\in\operatorname{skel}(Q^{\prime},m)\cap\mathcal{D}(i+1)$. Suppose
that, for some $F\in\mathcal{F}(m,i)$, $p\in F$ and $p^{\prime}\not\in F$.
Then there exists $q\in\partial F\cap\mathcal{D}(i+1)$ such that

$\max\\{\|p-q\|_{\infty},\|q-p^{\prime}\|_{\infty}\\}\leq\|p-p^{\prime}\|_{\infty}.$
(52)

Proof: Without loss of generality, we may suppose that

$F=\\{x\in[0,1]^{n}:0\leq x_{j}\leq l^{i}\ \forall 1\leq j\leq m\\}.$

If $p\in\partial F$ then we pick $q=p$. Otherwise, we know that

$0<p_{1},\ldots,p_{m}<l^{i}.$

There are two cases:

  1. 1.

For some $1\leq j\leq m$, $p^{\prime}_{j}=0$ or $p^{\prime}_{j}\geq l^{i}$;

  2. 2.

$0<p^{\prime}_{1},\ldots,p^{\prime}_{m}<l^{i}$. In this case, since
$p^{\prime}\in\operatorname{skel}(Q^{\prime},m)$ and hence belongs to some
$m$-dimensional face, the other components of $p^{\prime}$ must be integer
multiples of $l^{i}$. Since $p^{\prime}\not\in F$, one such component, say
$p^{\prime}_{j}$ differs from $p_{j}$. Consequently,
$|p^{\prime}_{j}-p_{j}|\geq l^{i}$.

In the first case, define $q_{k}=p_{k}$ for each $k\neq j$ and let $q_{j}$
equal either $0$ or $l^{i}$, such that $q_{j}$ lies between $p_{j}$ and
$p^{\prime}_{j}$. Then $q\in\partial F$, $q_{k}=p_{k}$ for all $k\neq j$ and

$|p_{j}-p^{\prime}_{j}|=|p_{j}-q_{j}|+|q_{j}-p^{\prime}_{j}|,$

which implies (52). In the second case, arbitrarily pick $q\in\partial
F\cap\mathcal{D}(i+1)$. Then $\|p-q\|_{\infty}\leq
l^{i}\leq\|p-p^{\prime}\|_{\infty}$, implying one inequality in (52). Also,
$|p^{\prime}_{k}-q_{k}|\leq l^{i}\leq|p^{\prime}_{j}-p_{j}|$ for all $1\leq
k\leq m$ and $|p^{\prime}_{k}-q_{k}|=|p^{\prime}_{k}-p_{k}|$ for all $k>m$.
Therefore, $\|p^{\prime}-q\|_{\infty}\leq\|p^{\prime}-p\|_{\infty}$, proving
the other inequality in (52).

With this observation we demonstrate how to construct the extension from the
$m$-dimensional faces to the $m+1$-dimensional faces.

######  Lemma 3.10.

Fix $i\in\mathbb{N}$, let $D\subset\mathcal{D}(i+1)$ and let $0\leq m<n$. Let
$0<\beta<1/(2l^{i})$ and suppose that $\iota\colon D\to C$ satisfies

$d(\iota(p),\iota(p^{\prime}))\leq\beta\|p-p^{\prime}\|_{\infty}$ (53)

for each $p,p^{\prime}\in D$ with $\|p-p^{\prime}\|_{\infty}\leq l^{i}$.
Suppose that $\mathcal{G}\subset\mathcal{Q}(i)$ is such that

$\forall Q\in\mathcal{G},\ D\cap
Q=\operatorname{skel}(Q,m)\cap\mathcal{D}(i+1)$ (54)

and

$\forall Q\in\mathcal{G},\ \exists x\in G\text{ with }\iota(D\cap Q)\subset
B(x,2\beta l^{i}).$ (55)

Then there exists an extension of $\iota$ to

$D^{\prime}:=D\cup\bigcup_{Q\in\mathcal{G}}\operatorname{skel}(Q,m+1)\cap\mathcal{D}(i+1)$

such that

$d(\iota(p),\iota(p^{\prime}))\leq 5K^{2}\beta\|p-p^{\prime}\|_{\infty}$ (56)

for each $p,p^{\prime}\in D^{\prime}$ with $\|p-p^{\prime}\|_{\infty}\leq
l^{i}$.

Proof: Let $Q\in\mathcal{G}$ and $F\in\mathcal{F}(m+1,i)$ with $F\subset Q$.
By (54), $\iota$ is defined on $\partial F\cap\mathcal{D}(i+1)$. By (55) and
the fact that $\beta<1/(2l^{i})$, the hypotheses of Lemma 3.8 are satisfied.
An application of Lemma 3.8 extends $\iota|_{\partial F\cap\mathcal{D}(i+1)}$
to a map

$\iota\colon F\cap\mathcal{D}(i+1)\to C$

such that

$d(\iota(p),\iota(p^{\prime}))\leq 2K^{2}\beta\|p-p^{\prime}\|_{\infty}$ (57)

for each $p,p^{\prime}\in F\cap\mathcal{D}(i+1)$. We combine each of these
extensions to form an extension of $\iota$ to a map

$\iota\colon D^{\prime}\to C,$

ignoring any repetitions of faces that appear in multiple cubes.

Now let $p,p^{\prime}\in D^{\prime}$ with $\|p-p^{\prime}\|_{\infty}\leq
l^{i}$. If $p,p^{\prime}\in D$ then (56) holds for $p,p^{\prime}$ by
hypothesis. Therefore, we may suppose that $p\not\in D$. Let $Q\in\mathcal{G}$
with $p\in Q$ and $F\in\mathcal{F}(m+1,i)$ with $p\in
F\subset\operatorname{skel}(Q,m+1)$. If $p^{\prime}\in F$ then (57) implies
(56) for $p,p^{\prime}$. Otherwise, let $q\in\partial F\cap\mathcal{D}(i+1)$
be given by 3.9 such that

$\max\\{\|p-q\|_{\infty},\|q-p^{\prime}\|_{\infty}\\}\leq\|p-p^{\prime}\|_{\infty}$

If $p^{\prime}\not\in D$, let $Q^{\prime}\in\mathcal{G}$ and
$F^{\prime}\in\mathcal{F}(m+1,i)$ with $p^{\prime}\in
F^{\prime}\subset\operatorname{skel}(Q^{\prime},m+1)$ and let
$q^{\prime}\in\partial F^{\prime}\cap\mathcal{D}(i+1)$ be given by 3.9 such
that

$\max\\{\|p^{\prime}-q^{\prime}\|_{\infty},\|q-q^{\prime}\|_{\infty}\\}\leq\|p^{\prime}-q\|_{\infty}.$

If $p^{\prime}\in D$ let $q^{\prime}=p^{\prime}$. In either case, we have
$\|q-q^{\prime}\|_{\infty},\|p-p^{\prime}\|_{\infty}\leq l^{i}$ and so (53)
implies

$d(\iota(q),\iota(q^{\prime}))\leq\beta\|q-q^{\prime}\|_{\infty}.$

Further, (57) implies

$d(\iota(p),\iota(q))\leq 2K^{2}\beta\|p-q\|_{\infty}$

and

$d(\iota(q^{\prime}),\iota(p^{\prime}))\leq
2K^{2}\beta\|q^{\prime}-p^{\prime}\|_{\infty}.$

Therefore, the triangle inequality gives

$\displaystyle d(\iota(p),\iota(p^{\prime}))$ $\displaystyle\leq
2K^{2}\beta\|p-q\|_{\infty}+\beta\|q-q^{\prime}\|_{\infty}$
$\displaystyle\quad+2K^{2}\beta\|q^{\prime}-p^{\prime}\|_{\infty}$
$\displaystyle\leq 5K^{2}\beta\|p-p^{\prime}\|_{\infty},$

as required.

Recall $\sigma=(5K^{2})^{n}$ is defined in 3.5. We now iterate the
construction in Lemma 3.10 to obtain an extension to $n$-dimensional cubes.

######  Lemma 3.11.

Fix $i\in\mathbb{N}$, let $D\subset\mathcal{D}(i)$ and let

$0<\beta<\frac{1}{2\sigma l^{i}}.$ (58)

Suppose that $\iota\colon D\to C$ satisfies

$d(\iota(p),\iota(p^{\prime}))\leq\beta\|p-p^{\prime}\|_{\infty}$ (59)

for each $p,p^{\prime}\in D$ with $\|p-p^{\prime}\|_{\infty}\leq l^{i}$.
Suppose also that $\mathcal{G}\subset\mathcal{Q}(i)$ is such that
$\operatorname{cor}(\mathcal{G})\subset D$ and

$\forall Q\in\mathcal{G},\ \exists x\in G\text{ with }\iota(D\cap Q)\subset
B(x,2\beta l^{i}).$ (60)

Then there exists an extension of $\iota$ to

$D^{\prime}:=\left(\bigcup\mathcal{G}\cap\mathcal{D}(i+1)\right)\cup
D=\operatorname{cor}(\operatorname{child}(\mathcal{G}))\cup D$

such that

$d(\iota(p),\iota(p^{\prime}))\leq\sigma\beta\|p-p^{\prime}\|_{\infty}$ (61)

for each $p,p^{\prime}\in D^{\prime}$ with $\|p-p^{\prime}\|_{\infty}\leq
l^{i}$.

Proof: First note that, since $\operatorname{cor}(\mathcal{G})\subset D$, (54)
holds for $m=0$. Since (60) implies (55), we may apply Lemma 3.10 to obtain an
extension of $\iota$ to

$D_{1}:=D\cup\bigcup_{Q\in\mathcal{G}}\operatorname{skel}(Q,1)\cap\mathcal{D}(i+1)$

such that

$d(\iota(p),\iota(p^{\prime}))\leq 5K^{2}\beta\|p-p^{\prime}\|_{\infty}$

for each $p,p^{\prime}\in D_{1}$ with $\|p-p^{\prime}\|_{\infty}\leq l^{i}$.

Now notice that,

$\forall Q\in\mathcal{G},\ D_{1}\cap Q=\operatorname{skel}(Q,1).$

Since

$(5K^{2})\beta<\frac{(5K^{2})}{2\sigma l^{i}}\leq\frac{1}{2l^{i}},$

we may apply Lemma 3.10 again, but with $\beta$ replaced by $(5K^{2})\beta$
and with $m=1$. This gives an extension of $\iota$ to

$D_{2}:=D\cup\bigcup_{Q\in\mathcal{G}}\operatorname{skel}(Q,2)\cap\mathcal{D}(i+1)$

such that

$d(\iota(p),\iota(p^{\prime}))\leq(5K^{2})^{2}\beta\|p-p^{\prime}\|_{\infty}$

for each $p,p^{\prime}\in D_{2}$ with $\|p-p^{\prime}\|_{\infty}\leq l^{i}$.
Thus, the hypotheses of Lemma 3.10 are satisfied again, with $\beta$ replaced
by $(5K^{2})^{2}\beta$ and with $m=2$.

Since $\beta<1/(2(5K^{2})^{n}l^{i})$, we repeat this for a total of $n$ times,
extending $\iota$ to

$D_{n}:=D\cup\bigcup_{Q\in\mathcal{G}}\operatorname{skel}(Q,n)\cap\mathcal{D}(i+1)=D\cup\left(\bigcup\mathcal{G}\cap\mathcal{D}(i+1)\right)=D^{\prime}$

such that

$d(\iota(p),\iota(p^{\prime}))\leq(5K^{2})^{n}\beta\|p-p^{\prime}\|_{\infty}=\sigma\beta\|p-p^{\prime}\|_{\infty}$

for each $p,p^{\prime}\in D^{\prime}$ with $\|p-p^{\prime}\|_{\infty}\leq
l^{i}$.

###  3.2 Constructing the extension to all scales

In Lemma 3.11 we demonstrated how to extend a function from some
$D\subset\mathcal{D}(i)$ to $D^{\prime}\subset\mathcal{D}(i+1)$. We now
iterate this extension over all $i\in\mathbb{N}$.

Note that, in items 5 and 4 of the following lemma, the radius of the ball in
item 4 and the upper bound in item 5 are of the form $(\sigma l)^{m}$, up to
some constant multiple. In (44) we chose $\alpha$ so that $(\sigma
l)=l^{\alpha}$, so both of these quantities are comparable to $l^{\alpha m}$.
In particular, item 5 implies that each $\iota_{i}$ is $\alpha$-Hölder with
the same Hölder constant. The fact that these two quantities have the same
power of $\alpha$ is an essential requirement for the construction given in
this section to work; looking ahead to Proposition 3.15, this is exactly the
condition required so that we can bound
$\mathcal{H}^{n}_{\infty}(\iota([0,1]^{n})\setminus C)$ in (74). Looking
behind, we see that the two powers agree because of Lemma 3.7.

######  Lemma 3.12.

Let $M\in\mathbb{N}$ with $M>\alpha/2$ , $0\leq L\leq\sigma^{M}$ and suppose
that

$\iota_{M}\colon\mathcal{D}(M)\to C$

is $L$-Lipschitz. For each $i\geq M$ there exist
$\mathcal{G}_{i},\mathcal{B}_{i}\subset\mathcal{Q}(i)$ such that

$[0,1]^{n}=\bigcup\mathcal{G}_{i}\cup\bigcup_{M\leq j\leq
i}\bigcup\mathcal{B}_{i}$ (62)

and a function

$\iota_{i}\colon\operatorname{cor}\left(\mathcal{G}_{i}\cup\bigcup_{M\leq
j\leq i}\mathcal{B}_{j}\right)\to C$ (63)

such that:

  1. 1.

$\iota_{i+1}$ is an extension of $\iota_{i}$;

  2. 2.

$\operatorname{child}(\mathcal{G}_{i})=\mathcal{G}_{i+1}\cup\mathcal{B}_{i+1}$

  3. 3.

For each $Q\in\mathcal{G}_{i}$, there exists $x\in G$ with

$\iota_{i}(\operatorname{cor}(Q))\subset B(x,2L\sigma^{i-M}l^{i})\mathchar
24635\relax\;$

  4. 4.

For all $p\in\operatorname{cor}(\mathcal{B}_{i})$,

$B(\iota_{i}(p),L\sigma^{i-M}l^{i})\cap G=\emptyset,$

  5. 5.

For each $p,p^{\prime}\in\operatorname{dom}\iota_{i}$ with
$\|p-p^{\prime}\|_{\infty}\leq l^{i-1}$,

$d(\iota_{i}(p),\iota_{i}(p^{\prime}))\leq
L\sigma^{i-M}\|p-p^{\prime}\|_{\infty}.$

Here and throughout, $\operatorname{dom}\iota_{i}$ denotes the domain of
$\iota_{i}$.

Proof: We prove the lemma by induction. To begin, for $\beta:=L$, let

$\mathcal{Q}(M)=\mathcal{G}_{M}\cup\mathcal{B}_{M}$

be given by Lemma 3.7 for $\iota=\iota_{M}$ and $i=M$. Then
$\iota_{M},\mathcal{G}_{M},\mathcal{B}_{M}$ satisfy the required conditions.

Now suppose that $\iota_{i},\mathcal{G}_{i}$ and
$\mathcal{B}_{M},\ldots,\mathcal{B}_{i}$ exist for some $i\geq M$. We first
apply Lemma 3.11 to $\iota_{i}$ and $\mathcal{G}_{i}$ and begin by checking
that the required hypotheses are satisfied. Since $L\leq\sigma^{M}$, (46)
implies

$\beta:=L\sigma^{i-M}\leq\sigma^{i}<\frac{1}{2\sigma l^{i}},$

and so (58) holds. By induction hypothesis, item 5 implies (59) for this
choice of $\beta$. Also, item 3 implies $\mathcal{G}_{i}$ satisfies (60). An
application of Lemma 3.11 constructs an extension of $\iota_{i}$ to

$\iota_{i+1}\colon\operatorname{cor}\left(\operatorname{child}(\mathcal{G}_{i})\cup\bigcup_{M\leq
j\leq i}\mathcal{B}_{j}\right)\to C$

such that, for each $p,p^{\prime}\in\operatorname{dom}\iota_{i+1}$ with
$\|p-p^{\prime}\|_{\infty}\leq l^{i}$,

$d(\iota_{i+1}(p),\iota_{i+1}(p^{\prime}))\leq\sigma\beta\|p-p^{\prime}\|_{\infty}=L\sigma^{i+1-M}\|p-p^{\prime}\|_{\infty}.$
(64)

Equation 64 implies that $\mathcal{Q}=\operatorname{child}(\mathcal{G}_{i})$
satisfies the hypothesis of Lemma 3.7 with $\beta:=L\sigma^{i+1-M}$. Let

$\operatorname{child}(\mathcal{G}_{i})=\mathcal{G}_{i+1}\cup\mathcal{B}_{i+1}$
(65)

be the decomposition given by Lemma 3.7, so that item 2 is satisfied and so
that $\mathcal{G}_{i+1}$ satisfies item 3 and $\mathcal{B}_{i+1}$ satisfies
item 4 for $i=i+1$. Note that (62) for $i$ combined with (65) implies (62) for
$i+1$. Thus the required $\iota_{i+1},\mathcal{G}_{i+1}$ and
$\mathcal{B}_{i+1}$ exist and so the lemma is true by induction.

We now prove, by induction on $i\in\mathbb{N}$, a local Lipschitz bound on
each $\iota_{i}$ (with a Lipschitz constant that depends on $i$). Afterwards,
this will be upgraded to uniform Hölder continuity.

######  Lemma 3.13.

Let $M\in\mathbb{N}$ with $M>\alpha/2$ , $0\leq L\leq\sigma^{M}$ and suppose
that

$\iota_{M}\colon\mathcal{D}(M)\to C$

is $L$-Lipschitz. For each $i\geq M$ let $\iota_{i},\mathcal{G}_{i}$ and
$\mathcal{B}_{i}$ be obtained from Lemma 3.12. Then for any $i\geq M$,
$j\in\mathbb{N}$ and $p,q\in\operatorname{dom}\iota_{i}$ with $p,q\in
Q\in\mathcal{Q}(j)$,

$d(\iota_{i}(p),\iota_{i}(q))\leq 2L\sigma^{-M+1}(\sigma l)^{j}.$ (66)

Proof: First note that, if $i=j$, (66) follows from Lemma 3.12 item 5 and if
$j>i$ then $p=q$, so that (66) holds. To prove (66) for the case $i>j\geq M$,
we will prove that

$d(\iota_{i}(p),\iota_{i}(q))\leq
L\sigma^{-M+1}\sum_{k=j}^{i-1}(l\sigma)^{k}\quad\forall
p,q\in\operatorname{dom}\iota_{i},\ p,q\in Q\in\mathcal{Q}(j)$ (67)

holds for all $i>M$ and all $i>j\geq M$ by induction on $i$. Equation 66 then
follows because the geometric series converges by (46).

Note that (67) holds for $i=M+1$ and $j=M$ since $\iota_{M}$ is $L$-Lipschitz.
Now let $i>M+1$, and suppose that (67) holds for $i-1$ and all $i-1>j\geq M$.
Let $i>j\geq M$ and $p,q\in\operatorname{dom}\iota_{i}$ with $p,q\in
Q\in\mathcal{Q}(j)$. If neither of $p$ or $q$ belong to
$\operatorname{cor}(\mathcal{G}_{i}\cup\mathcal{B}_{i})$, then they both
belong to $\operatorname{dom}\iota_{i-1}$. If $j=i-1$ then Lemma 3.12 item 5
for $i-1$ implies (67) for $i$ since $\iota_{i}$ extends $\iota_{i-1}$. If
$j<i-1$ then the induction hypothesis implies (67) for $i$. Therefore, we may
suppose $p\in\operatorname{cor}(Q^{*})$ for some
$Q^{*}\in\mathcal{G}_{i}\cup\mathcal{B}_{i}$.

Note that Lemma 3.12 item 2 implies that there exists
$Q^{\prime}\in\mathcal{G}_{i-1}$ with $Q^{*}\subset Q^{\prime}$. Let
$p^{\prime}\in\operatorname{cor}(Q^{\prime})\subset\operatorname{dom}\iota_{i-1}$,
so that $\|p-p^{\prime}\|_{\infty}\leq l^{i-1}$. By Lemma 3.12 item 5

$d(\iota_{i}(p),\iota_{i}(p^{\prime}))\leq
L\sigma^{i-M}l^{i-1}=L\sigma^{-M+1}(\sigma l)^{i-1}.$ (68)

Note also that, since $Q^{\prime}\in\mathcal{Q}(i-1)$ and $Q\in\mathcal{Q}(j)$
with $j\leq i-1$, we must have $Q^{\prime}\subset Q$. Thus, since
$p^{\prime},q\in Q$, the induction hypothesis implies

$d(\iota_{i-1}(p^{\prime}),\iota_{i-1}(q))\leq
L\sigma^{-M+1}\sum_{k=j}^{i-2}(l\sigma)^{k}.$ (69)

Using the fact that $\iota_{i}$ extends $\iota_{i-1}$, we may combine eqs. 68
and 69 using the triangle inequality to get

$d(\iota_{i}(p),\iota_{i}(q))\leq
L\sigma^{-M+1}\sum_{k=j}^{i-1}(l\sigma)^{k},$

completing the induction step.

Recall from (45) that $\sigma l=l^{\alpha}$. We now show that the limit of the
$\iota_{i}$ is $\alpha$-Hölder. As mentioned before Lemma 3.12, it is
essential that the exponent of $l^{i}$ appearing in item 3 in the following
lemma equals $\alpha$ in order to deduce (74).

######  Lemma 3.14.

Let $M\in\mathbb{N}$ with $M>2/\alpha$, $0\leq L\leq\sigma^{M}$ and let
$\iota\colon\mathcal{D}(M)\to C$ be $L$-Lipschitz. There exist a closed set
$S\subset[0,1]^{n}$ with $S\supset\mathcal{D}(M)$, an extension of $\iota$ to

$\iota\colon(S,\|\cdot\|_{\infty})\to C\cap B(\iota(0),5L)$

and, for each $i\geq M$, a set $\mathcal{B}_{i}\subset\mathcal{Q}(i)$ such
that:

  1. 1.

$\iota$ is $\alpha$-Hölder continuous with Hölder constant $5Ll^{-\alpha}$.

  2. 2.

$[0,1]^{n}=S\cup\bigcup_{i\geq M}\bigcup\mathcal{B}_{i}$

  3. 3.

For each $i\geq M$ and each $p\in\operatorname{cor}(\mathcal{B}_{i})$, $p\in
S$ and

$B(\iota(p),L\sigma^{-M}l^{\alpha i})\cap G=\emptyset.$

Proof: Set $\iota_{M}=\iota$ and for each $i\geq M$ let
$\mathcal{G}_{i},\mathcal{B}_{i}$ and $\iota_{i}$ be obtained from Lemma 3.12.
Set $S^{\prime}=\cup_{i\geq M}\operatorname{dom}\iota_{i}$ and define $\iota$
on $S^{\prime}$ by $\iota(x)=\iota_{i}(x)$ whenever
$x\in\operatorname{dom}\iota_{i}$. Let $p,p^{\prime}\in S^{\prime}$, let
$i\in\mathbb{N}$ be sufficiently large such that
$p,p^{\prime}\in\operatorname{dom}\iota_{i}$ and let $m\in\mathbb{N}$ be such
that $l^{m+1}<\|p-p^{\prime}\|_{\infty}\leq l^{m}$.

If $m<M$, let $Q,Q^{\prime}\in\mathcal{Q}(M)$ be such that $p\in Q$ and
$p^{\prime}\in Q^{\prime}$ and let $q\in\operatorname{cor}(Q)$ and
$q^{\prime}\in\operatorname{cor}(Q^{\prime})$ minimise the distance between
$\operatorname{cor}(Q)$ and $\operatorname{cor}(Q^{\prime})$, so that
$\|q-q^{\prime}\|_{\infty}\leq l^{m}$. Then by combining (66), the fact that
$\iota$ is $L$-Lipschitz, and using the triangle inequality

$\displaystyle d(\iota(p),\iota(p^{\prime}))$ $\displaystyle\leq
d(\iota_{i}(p),\iota_{i}(q))+d(\iota(q),\iota(q^{\prime}))+\,\mathrm{d}(\iota_{i}(q^{\prime}),\iota_{i}(p^{\prime}))$
$\displaystyle\leq 2L\sigma^{-M+1}(\sigma l)^{M}+Ll^{m}+2L\sigma^{-M+1}(\sigma
l)^{M}$ $\displaystyle\leq 4L\sigma l^{M}+Ll^{m}$ $\displaystyle\leq 5L(\sigma
l)^{m}.$ (70)

If $m\geq M$, let $p\in Q\in\mathcal{Q}(m)$ and $p^{\prime}\in
Q^{\prime}\in\mathcal{Q}(m)$. Since $\|p-p^{\prime}\|_{\infty}\leq l^{m}$, $Q$
and $Q^{\prime}$ have a common corner $q$ and hence
$\|p-q\|_{\infty},\|p^{\prime}-q\|_{\infty}\leq l^{m}$. Thus the triangle
inequality and (66) give

$\displaystyle d(\iota(p),\iota(p^{\prime}))$ $\displaystyle\leq
d(\iota_{i}(p),\iota_{i}(q))+d(\iota_{i}(q),\iota_{i}(p^{\prime}))$
$\displaystyle\leq 2L\sigma^{-M+1}(\sigma l)^{m}+2L\sigma^{-M+1}(\sigma
l)^{m}$ $\displaystyle\leq 4L(\sigma l)^{m},$ (71)

using the fact that $M\geq 2$ in the final inequality.

Equations 70, 71 and 45 show that

$d(\iota(p),\iota(p^{\prime}))\leq 5Ll^{\alpha m}\leq
5Ll^{-\alpha}\|p-p^{\prime}\|_{\infty}^{\alpha}.$ (72)

That is, $\iota$ is $\alpha$-Hölder with constant $5Ll^{-\alpha}$. This also
implies that $\iota$ maps into $B(\iota(0),5L)$.

Let $S$ be the closure of $S^{\prime}$ and extend $\iota$ to $S$ maintaining
the Hölder bound. Since $C$ is closed, this extension maps into $C\cap
B(\iota(0),5L)$. For each $i\geq M$, (62) implies

$[0,1]^{n}=\bigcup\mathcal{G}_{i}\cup\bigcup_{j\geq M}\bigcup\mathcal{B}_{j}.$
(73)

For each $i\in\mathbb{N}$,
$\operatorname{cor}(\mathcal{G}_{i})\subset\operatorname{dom}\iota_{i}\subset
S$ and so $\cup\mathcal{G}_{i}\subset B(S,l^{i})$. Since $S$ is compact this
implies

$\bigcap_{i\geq M}\bigcup\mathcal{G}_{i}\subset S.$

Thus, intersecting (73) over all $i\geq M$ gives item 2.

Finally, eq. 45 and Lemma 3.12 item 4 and imply item 3.

We now extend the map $\iota$ to $[0,1]^{n}$, assuming $X$ is a subset of
$\ell_{\infty}$. This will implicitly construct the metric space $\tilde{E}$
mentioned in the introduction. Since we have constructed $\iota$ to be Hölder
continuous, this can be done using Theorem 2.6. The Hölder condition is also
used (in combination with (35)) to deduce (74) from Lemma 3.14 item 3.

######  Proposition 3.15.

Let $M\in\mathbb{N}$ with $M>2/\alpha$, $0\leq L\leq\sigma^{M}$ and let
$\iota\colon\mathcal{D}(M)\to C$ be $L$-Lipschitz. Suppose that $X$ is a
subset of $\ell_{\infty}$. There exists an $\alpha$-Hölder extension
$\iota\colon[0,1]^{n}\to B(\iota(0),10L)$ with Hölder constant $5Ll^{-\alpha}$
such that

$\mathcal{H}^{n}_{\infty}(\iota([0,1]^{n})\setminus
C)\leq\frac{(50\sigma^{M}l^{-\alpha})^{n}}{\eta}\mathcal{H}^{n}(B(\iota(0),10L)\cap
X\setminus G).$ (74)

Proof: First apply Lemma 3.14 to find a closed $S\subset[0,1]^{n}$ and extend
$\iota$ to an $\alpha$-Hölder map

$\iota\colon S\to B(\iota(0),5L)\cap C.$ (75)

Then apply Theorem 2.6 to extend $\iota$ to an $\alpha$-Hölder map

$\iota\colon[0,1]^{n}\to\ell_{\infty}$

maintaining the Hölder constant of $5Ll^{-\alpha}$. In particular, since
$\mathcal{D}(M)\subset S$ and $M>2/\alpha$, this extension maps into
$B(\iota(0),10L)$.

Since $\iota(S)\subset C$, we deduce (74) by estimating the Hausdorff content
of

$Y:=\iota\left(\bigcup_{i\geq M}\bigcup\mathcal{B}_{i}\right),$

where the $\mathcal{B}_{i}$ are as in Lemma 3.14 item 4. That is, for each
$i\geq M$, $\mathcal{B}_{i}\subset\mathcal{Q}(i)$ and, for each
$p\in\operatorname{cor}(\mathcal{B}_{i})$, $p\in S$ and

$B(\iota(p),L\sigma^{-M}l^{\alpha i})\cap G=\emptyset.$ (76)

Since $\iota$ is $\alpha$-Hölder with constant $5Ll^{-\alpha}$,

$Y\subset\\{B(\iota(p),5Ll^{-\alpha}l^{i\alpha}):i\geq M,\
p\in\operatorname{cor}(\mathcal{B}_{i})\\}.$

That is, if

$\mathcal{A}=\\{B(\iota(p),L\sigma^{-M}l^{\alpha i}):i\geq M,\
p\in\operatorname{cor}(\mathcal{B}_{i})\\},$

then

$Y\subset\bigcup_{B\in\mathcal{A}}5\sigma^{M}l^{-\alpha}B.$

Let $\mathcal{A}^{\prime}$ be a disjoint $5r$-sub-cover of $\mathcal{A}$, so
that the elements of $\mathcal{A}^{\prime}$ are pairwise disjoint and

$Y\subset\bigcup_{B\in\mathcal{A}^{\prime}}25\sigma^{M}l^{-\alpha}B.$

In particular,

$\mathcal{H}^{n}_{\infty}(Y)\leq\sum_{B\in\mathcal{A}^{\prime}}(50\sigma^{M}l^{-\alpha}\operatorname{rad}B)^{n}$
(77)

By eqs. 75 and 76,

$\bigcup\mathcal{A}^{\prime}\subset B(\iota(0),10L)\setminus G.$ (78)

For each $i\geq M$ and $p\in\operatorname{cor}(\mathcal{B}_{i})$, $\iota(p)\in
C$ and so Definition 3.2 item 1 implies

$\mathcal{H}^{n}(B(\iota(p),r)\cap X)\geq\eta r^{n}$

whenever $r\leq 1$. In particular, this is true for all balls in
$\mathcal{A}^{\prime}$. Therefore, by (78) and the disjointness of the balls
in $\mathcal{A}^{\prime}$,

$\displaystyle\eta\sum_{B\in\mathcal{A}^{\prime}}\operatorname{rad}(B)^{n}$
$\displaystyle\leq\sum_{B\in\mathcal{A}^{\prime}}\mathcal{H}^{n}(B\cap X)$
$\displaystyle\leq\mathcal{H}^{n}(B(\iota(0),10L)\cap X\setminus G).$

Combining this with (77) gives

$\mathcal{H}^{n}_{\infty}(Y)\leq\frac{(50\sigma^{M}l^{-\alpha})^{n}}{\eta}\mathcal{H}^{n}(B(\iota(0)),10L)\cap
X\setminus G),$

as required.

To conclude the section, we record the main conclusion without the fixed
quantities of 3.5. Recall, for $m\in\mathbb{N}$ and $R>0$, that

$\mathcal{D}(R,m)=\\{R2^{-m}(j_{1},\ldots,j_{n})\in[0,R]^{n}:j_{1},\ldots,j_{n}\in\mathbb{Z}\\}.$

######  Theorem 3.16.

For any $K\geq 1$, $0<\gamma<1$ and $\eta>0$ there exists $N\in\mathbb{N}$ and
$\gamma\leq\alpha<1$ such that the following is true. For any closed
$X\subset\ell_{\infty}$ and for any $R_{0}>0$, let $(X,C,G)$ satisfy

$GTA(\eta,K,2^{-N}/20,R_{0}).$ (79)

Then for any $M\in\mathbb{N}$ with $M>2/\alpha$, $0<r<R_{0}$, $0\leq
L\leq(5K^{2})^{M}$ and any $L$-Lipschitz

$\iota\colon\mathcal{D}(r,MN)\subset\ell_{\infty}^{n}\to C,$

there exists an $\alpha$-Hölder extension

$\iota\colon[0,r]^{n}\subset\ell_{\infty}^{n}\to B(\iota(0),10Lr)$

with Hölder constant $5L2^{N\alpha}r^{1-\alpha}$ such that

$\mathcal{H}^{n}_{\infty}(\iota([0,r]^{n})\setminus
C)\leq\Lambda\mathcal{H}^{n}(B(\iota(0),10Lr)\cap X\setminus G),$ (80)

for some $\Lambda>0$ depending only upon $K,\gamma,M,n,\eta$.

Proof: Given $K,\gamma,\eta$ as in the hypotheses, let $N\in\mathbb{N}$ and
$\gamma\leq\alpha<1$ be given by (44). For any $(X,C,G)$ satisfying (79) and
any $0<r<R_{0}$, let $C^{\prime}=C/r$ and $G^{\prime}=G/r$, considered as
subsets of $X^{\prime}=X/r\subset\ell_{\infty}$. Then
$(X^{\prime},C^{\prime},G^{\prime})$ satisfies $GTA(\eta,K,2^{-N}/20,1)$.
Define $\iota^{\prime}\colon\mathcal{D}(1,MN)\to C^{\prime}$ by
$\iota^{\prime}(q)=\iota(rq)/r$. Then $\iota^{\prime}$ is also $L$-Lipschitz.
An application of Proposition 3.15 gives an $\alpha$-Hölder extension of
$\iota^{\prime}$ with Hölder constant $5L2^{N\alpha}$ satisfying

$\mathcal{H}^{n}_{\infty}(\iota^{\prime}([0,1]^{n})\setminus
C^{\prime})\leq\Lambda\mathcal{H}^{n}(B(\iota(0),10L)\cap X^{\prime}\setminus
G^{\prime})$ (81)

for $\Lambda=(20(5K)^{2M}2^{\alpha N})^{n}/\eta$. This gives an
$\alpha$-Hölder extension of $\iota$ defined by
$\iota(q)=r\iota^{\prime}(q/r)$ with Hölder constant
$5L2^{N\alpha}r^{1-\alpha}$:

$\displaystyle\|\iota(y)-\iota(z)\|_{\infty}$
$\displaystyle=r\|\iota^{\prime}(y/r)-\iota^{\prime}(z/r)\|_{\infty}$
$\displaystyle\leq r5L2^{N\alpha}\|y/r-z/r\|_{\infty}^{\alpha}$
$\displaystyle=5L2^{N\alpha}r^{1-\alpha}\|y-z\|_{\infty}^{\alpha}.$

Equation 81 is equivalent to (80).

Finally, we conclude the proof of Theorem 3.4

Proof: [Proof of Theorem 3.4] The hypotheses of Theorem 3.4 imply the
hypotheses of Theorem 3.16. Let $N,\alpha$ be obtained from Theorem 3.16 and
let $m_{0}=2N$. If $(X,C,G)$ satisfies (40) and $x\in G$ then by 3.3, there
exists a $(K+2^{-m})$-bi-Lipschitz $\xi\colon\mathcal{D}(r,m)\to C$ with
$\xi(0)=x$. Since $(X,C,G)$ also satisfies (79), Theorem 3.16 with $M=2$ gives
an $\alpha$-Hölder extension $\iota\colon[0,r]^{n}\to B(x,20Kr)$ with Hölder
constant $10K2^{N\alpha}r^{1-\alpha}$ satisfying items 1 and 3 of Theorem 3.4.
Since $\gamma\leq\alpha<1$, $\iota$ is also $\gamma$-Hölder. Moreover, if
$y,z\in[0,r]^{n}$ with $\|y-z\|_{\infty}\leq 2^{-m}r$ then

$\displaystyle\|\iota(y)-\iota(z)\|_{\infty}$ $\displaystyle\leq
10K2^{N\alpha}r^{1-\alpha}\|y-z\|_{\infty}^{\alpha}$ $\displaystyle\leq
10K2^{(N-m)\alpha}r^{1-\alpha+\alpha}$ $\displaystyle\leq 10K2^{-m\alpha/2}r$
$\displaystyle\leq 10K2^{-m\gamma/2}r,$

giving item 2.

######  Remark 3.17.

If, in the hypotheses of Theorem 3.16 or Theorem 3.4, $X$ is assumed to be a
subset of a finite dimensional Banach space $(\mathbb{R}^{m},\|\cdot\|)$, then
the Hölder map $\iota$ can be constructed with values in
$(\mathbb{R}^{m},\|\cdot\|)$. One simply uses Theorem 2.6 to extend the map
into $(\mathbb{R}^{m},\|\cdot\|)$, increasing the Hölder constant by a factor
that only depends on the norm $\|\cdot\|$.

######  Remark 3.18.

At this point it would be reasonable to prove Theorem 1.1 using Theorem 3.4
and Corollary 6.4. However, there are various technical steps that are simply
easier to deal with using $d_{*}$. Thus we delay the proof until after the
proof of Theorem 1.2.

##  4 Metrics between metric measure spaces

###  4.1 The pointed measured Gromov–Hausdorff metric

Recall the definition of $G_{a,b}$ given in Definition 2.19. We use the
following metric to describe pointed measured Gromov–Hausdorff convergence.

######  Definition 4.1.

For pointed metric measure spaces $(X,\mu,x)$ and $(Y,\nu,y)$ define

$d_{\mathrm{pmGH}}((X,\mu,x),(Y,\nu,y))=G_{1,1}((X,\mu,x),(Y,\nu,y)).$

Note that $d_{\mathrm{pmGH}}$ is bounded below by $d_{\mathrm{pGH}}$ of the
corresponding pointed metric spaces.

######  Remark 4.2.

If one is only interested in metrising pointed measured Gromov–Hausdorff
convergence, it is more natural to use $F_{z}^{1}$ rather than $F_{z}$ in the
definition of $d_{\mathrm{pmGH}}$. Doing so would not change the results of
this subsection, except for minor superficial changes in Corollary 4.4. We use
$F_{z}$ to give the desired relationship to $d_{*}$ in the next subsection.

We have the following relationship between $\epsilon$-isometries and $F$.

######  Lemma 4.3.

Let $(Z,\zeta,z)$ be a pointed metric measure space $X,Y\subset Z$ Borel sets,
$\mu\in\mathcal{M}_{\mathrm{loc}}(X)$ and
$\nu\in\mathcal{M}_{\mathrm{loc}}(Y)$. For $0<\epsilon<1/2$ suppose that
$f\colon X\cap B(z,1/\epsilon)\to Y$ is a Borel function that satisfies
$\zeta(f(x),x)\leq\epsilon$ for all $x\in X\cap B(z,1/\epsilon)$. Then for any
$0<r\leq 1/\epsilon-\epsilon$ and $L>0$,

$|F^{L,r}_{z}(f_{\\#}\mu,\nu)-F^{L,r}_{z}(\mu,\nu)|\leq\epsilon
L\mu(B(z,r+\epsilon)).$

Proof: Let $g\colon Z\to[-1,1]$ be $L$-Lipschitz with
$\operatorname{spt}g\subset B(z,r)$. For any $x\in B(z,1/\epsilon)$,

$|g\circ f(x)-g(x)|\leq L\zeta(f(x),x)\leq L\epsilon$

and so, since $r\leq 1/\epsilon-\epsilon$,

$\left|\int_{B(z,r+\epsilon)}g\circ
f\,\mathrm{d}\mu-\int_{B(z,r+\epsilon)}g\,\mathrm{d}\mu\right|\leq
L\epsilon\mu(B(z,r+\epsilon)).$

However, $x\in X\cap B(z,1/\epsilon)\setminus B(z,r+\epsilon)$ then
$x,f(x)\not\in\operatorname{spt}g$ and so

$\left|\int g\circ f\,\mathrm{d}\mu-\int g\,\mathrm{d}\mu\right|\leq
L\epsilon\mu(B(z,r+\epsilon)).$

Therefore,

$\displaystyle\int g\,\mathrm{d}(f_{\\#}\mu-\nu)$ $\displaystyle=\int g\circ
f\,\mathrm{d}\mu-\int g\,\mathrm{d}\nu$ $\displaystyle\leq
L\epsilon\mu(B(z,r+\epsilon)+\int g\,\mathrm{d}(\mu-\nu).$

Taking the supremum over all such $g$ gives

$F^{L,r}_{z}(f_{\\#}\mu,\nu)\leq
L\epsilon\mu(B(z,r+\epsilon))+F^{L,r}_{z}(\mu,\nu).$

Similarly,

$F^{L,r}_{z}(\mu,\nu)\leq
L\epsilon\mu(B(z,r+\epsilon))+F^{L,r}_{z}(f_{\\#}\mu,\nu),$

as required.

######  Corollary 4.4.

Let $(X,\mu,x)$ and $(Y,\rho,y)$ be pointed metric measure spaces and
$0<\epsilon<1/2$. If $d_{\mathrm{pmGH}}((X,\mu,x),(Y,\nu,y))<\epsilon$, there
exists a $2\epsilon$-isometry $f\colon(X,x)\to(Y,y)$ such that, for any
$0<r\leq 1/\epsilon-\epsilon$ and $L>0$,

$F^{L,r}_{y}(f_{\\#}\mu,\nu)\leq\epsilon+\epsilon L\mu(B(x,r+\epsilon)).$ (82)

Conversely, if there exists an $\epsilon$-isometry
$f\colon(X,\mu,x)\to(Y,\nu,y)$ then

$d_{\mathrm{pmGH}}((X,\mu,x),(Y,\nu,y))$

is bounded above by the maximum of $2\epsilon$ and

$\inf\\{1/r+F^{r,r}_{y}(f_{\\#}\mu,\nu)+\epsilon r\mu(B(x,r+\epsilon)):0<r\leq
2\epsilon\\}.$

Proof: Let $(Z,\zeta,z)$ be a pointed metric space for which there exist
isometric embeddings $(X,x),(Y,y)\to(Z,z)$ with $H_{z}(X,Y)<\epsilon$ and
$F_{z}(\mu,\nu)<\epsilon$. Lemma 2.24 gives a $2\epsilon$-isometry
$f\colon(X,x)\to(Y,y)$ such that $\zeta(x^{\prime},f(x^{\prime}))\leq\epsilon$
for all $x^{\prime}\in X\cap B(x,1/\epsilon)$. For any $0<r\leq
1/\epsilon-\epsilon$, Lemma 4.3 implies (82).

Conversely, Lemma 2.24 gives a pointed metric space $(Z,\zeta,z)$ and
isometric embeddings $(X,x),(Y,y)\to(Z,z)$ such that
$\zeta(x^{\prime},f(x^{\prime}))\leq 2\epsilon$ for all $x^{\prime}\in X\cap
B(x,1/\epsilon)$ and $H_{z}(X,Y)\leq 2\epsilon$. Lemma 4.3 implies, for any
$0<r\leq 1/2\epsilon$,

$F^{r,r}_{z}(\mu,\nu)\leq F^{r,r}_{z}(f_{\\#}\mu,\nu)+\epsilon
r\mu(B(x,r+\epsilon)).$

Thus

$F_{z}(\mu,\nu)\leq 1/r+F^{r,r}_{z}(f_{\\#}\mu,\nu)+\epsilon
r\mu(B(x,r+\epsilon)).$

Taking the infimum over all $0<r\leq 1/\epsilon-\epsilon$ completes the proof.

######  Definition 4.5.

An isometry $(X,\mu,x)\to(Y,\nu,y)$ of pointed metric measure spaces is an
isometry $(X,x)\to(Y,y)$ with $\mu=\nu$.

######  Corollary 4.6.

On the set $\mathbb{M}_{pm}$ of isometry classes of proper pointed metric
measure spaces, $d_{\mathrm{pmGH}}$ is a complete and separable metric.

Proof: By Proposition 2.20 we know $d_{\mathrm{pmGH}}$ is a complete
pseudometric. It is easily verified that the set of finite pointed metric
spaces with rational distances and rational combinations of Dirac masses is
dense. By Corollary 2.25, any complete limit of proper pointed metric spaces
is also proper.

Let

$d_{\mathrm{pmGH}}((X,\mu,x),(Y,\nu,y))=0.$

By Corollary 4.4, for each $i\in\mathbb{N}$ there exists a $1/i$-isometry
$f_{i}\colon(X,x)\to(Y,y)$ such that, for any $L,r>0$,

$F_{L,r}((f_{i})_{\\#}\mu,\nu)\to 0.$ (83)

Fix $r>0$. As in the proof of Corollary 2.25, for each $j\in\mathbb{N}$ let
$\mathcal{N}_{j}$ be a finite $1/j$-net of $B(x,r)$. By taking a subsequence,
we may suppose the $f_{i}$ converge pointwise to an isometry $\iota$ when
restricted to $\cup_{j}\mathcal{N}_{j}$ whose image is a dense subset of
$B(y,r)$. We extend $\iota$ to an isometry $\iota\colon B(x,r)\to B(y,r)$.

For a moment fix $i,j\in\mathbb{N}$. For any $w\in B(y,r)$ let
$z\in\mathcal{N}_{j}$ with $w\in B(z,1/j)$. Since $f_{i}$ is an
$\epsilon_{i}$-isometry and $\iota$ is an isometry,

$\displaystyle\ \rho(\iota(w),f_{i}(w))$
$\displaystyle\leq\rho(\iota(w),\iota(z))+\rho(\iota(z),f_{i}(z))+\rho(f_{i}(z),f_{i}(w))$
$\displaystyle\leq d(w,z)+\rho(\iota(z),f_{i}(z))+d(w,z)+\epsilon_{i}$
$\displaystyle\leq\rho(\iota(z),f_{i}(z))+2/j+\epsilon_{i}.$

Thus, for any $r$-Lipschitz $g\colon Y\to[-1,1]$ with
$\operatorname{spt}g\subset B(y,r)$,

$\int g\,\mathrm{d}((f_{i})_{\\#}\mu-\iota_{\\#}\mu)\leq
r\left(\sup_{z\in\mathcal{N}_{j}}\rho(f_{i}(z),\iota(z))+2/j+\epsilon_{i}\right).$

Consequently,

$F^{r,r}_{y}((f_{i})_{\\#}\mu,\iota_{\\#}\mu)\leq
r\left(\sup_{z\in\mathcal{N}_{j}}\rho(f_{i}(z),\iota(z))+2/j+\epsilon_{i}\right)$

and so

$\limsup_{i\to\infty}F^{r,r}_{y}((f_{i})_{\\#}\mu,\iota_{\\#}\mu)\leq 2r/j.$

Since this is true for all $j\in\mathbb{N}$, (83) gives
$F^{r,r}_{y}(\iota_{\\#}\mu,\nu)=0$ for all $r>0$. Thus Lemma 2.14 implies
$\iota_{\\#}\mu=\nu$. Consequently, $d_{\mathrm{pmGH}}$ is a metric on
$\mathbb{M}_{pm}$.

######  Corollary 4.7.

A set $\mathcal{S}\subset(\mathbb{M}_{pm},d_{\mathrm{pmGH}})$ is pre-compact
if and only if

  1. 1.

$\\{(X,x):(X,\mu,x)\in\mathcal{S}\\}\subset(\mathbb{M}_{p},d_{\mathrm{pGH}})$
is pre-compact and

  2. 2.

For every $r>0$, $\\{\mu(B(x,r)):(X,\mu,x)\in\mathcal{S}\\}$ is bounded.

Proof: Suppose that $\mathcal{S}$ satisfies items 1 and 2 and, for each
$i\in\mathbb{N}$, let $(X_{i},\mu_{i},x_{i})\in\mathcal{S}$. After passing to
a subsequence, there exists a proper pointed metric space $(X,x)$ such that

$d_{\mathrm{pGH}}((X_{i},x_{i}),(X,x))<1/(2i)$

for each $i\in\mathbb{N}$. By Lemma 2.24, for each $i\in\mathbb{N}$ there
exists a $1/i$-isometry $f_{i}\colon(X_{i},x_{i})\to(X,x)$. Item 2 and Theorem
2.11 imply, after passing to a further subsequence, that
$(f_{i})_{\\#}\mu_{i}$ converges to some measure $\mu$ on $X$. Corollary 4.4
then implies

$d_{\mathrm{pmGH}}((X_{i},\mu_{i},x_{i}),(X,\mu,x))\to 0.$

The converse statement is immediate.

###  4.2 Gromov–Hausdorff convergence of large subsets

Gromov–Hausdorff convergence of pointed metric measure spaces is used
throughout analysis. However, the requirement that the underlying metric
spaces must Gromov–Hausdorff converge is too rigid for some uses in geometric
measure theory. For example, it requires that every sequence of null sets must
Gromov–Hausdorff converge to some limit.

For a concrete example, consider the following.

######  Example 4.8.

Let $X=\\{0,1\\}$ with $d(0,1)=1$ and for each $i\in\mathbb{N}$ let
$\mu_{i}(\\{0\\})=1$ and $\mu(\\{1\\})=1/i$. Then the sequence
$(X,d,\mu_{i},0)$ Gromov–Hausdorff converges to $(X,d,\delta_{0},0)$ but _not_
to $(\\{0\\},\delta_{0},0)$, which is the more natural choice. That is,
Gromov–Hausdorff convergence may require one to consider a larger metric space
than simply the support of the limit measure.

For each $i\in\mathbb{N}$ let $X_{i}$ be an $i$-point metric space including a
point $0$, equipped with the discrete metric $d_{i}$. Then the sequence
$(X_{i},d_{i},\delta_{0},0)$ has no Gromov–Hausdorff limit, even though a one
point metric space with a Dirac measure is the natural candidate from a
measure theoretic point of view.

To construct a 1-rectifiable metric space with no Gromov–Hausdorff tangents at
any point, one takes an isometric copy of an interval and scatters “dust”
around it at all scales as in the previous example.

Rather than Gromov–Hausdorff convergence we will use the following.

######  Definition 4.9.

For pointed metric measure spaces $(X,\mu,x)$ and $(Y,\nu,y)$ define

$d_{*}((X,\mu,x),(Y,\nu,y))=G_{0,1}((X,\mu,x),(Y,\nu,y)).$

Note that $d_{*}\leq d_{\mathrm{pmGH}}$. Also, for any Borel $x\in S\subset
X$, (18) implies

$d_{*}((X,\mu,x),(S,\mu|_{S},x))\leq\inf\\{r>0:\mu(B(x,1/r)\setminus S)<r\\}.$
(84)

Although $d_{*}$ does not require the underlying sets to be close in the sense
of $d_{\mathrm{pGH}}$, we can recover large subsets which are.

######  Lemma 4.10.

Let $(Z,z)$ be a complete pointed metric space and
$\mu,\nu\in\mathcal{M}_{\mathrm{loc}}(Z)$. For any $r,\epsilon,\delta>0$ there
exist compact sets

$K_{\mu}\subset B(z,r)\cap\operatorname{spt}\mu\quad\text{and}\quad
K_{\nu}\subset B(z,r)\cap\operatorname{spt}\nu$

with

$\max\\{\mu(B(z,r)\setminus K_{\mu}),\ \nu(B(z,r)\setminus
K_{\nu})\\}<(1+\delta)F^{1/\epsilon,r}_{z}(\mu,\nu),$ (85)

and

$H_{z}(K_{\mu},K_{\nu})\leq\max\left\\{\frac{1}{r-\epsilon},\epsilon\right\\}.$
(86)

Proof: There exist compact

$K^{\prime}_{\mu}\subset B(z,r)\cap\operatorname{spt}\mu\quad\text{and}\quad
K^{\prime}_{\nu}\subset B(z,r)\cap\operatorname{spt}\nu$

with

$\max\\{\mu(B(z,r)\setminus K^{\prime}_{\mu}),\ \nu(B(z,r)\setminus
K^{\prime}_{\nu})\\}<\delta F^{1/\epsilon,r}_{z}(\mu,\nu)$ (87)

Let

$\tilde{K}_{\mu}=(K^{\prime}_{\mu}\cap U(z,r-\epsilon))\setminus
B(K^{\prime}_{\nu},\epsilon),$

a relatively open set in $K^{\prime}_{\mu}$, and define $g\colon Z\to[0,1]$ by

$g(x)=\begin{cases}1&z\in\tilde{K}_{\mu}\\\
1-\frac{1}{\epsilon}\min\\{d(z,\tilde{K}_{\mu}),\epsilon\\}&\text{otherwise}.\end{cases}$

Then $g$ is $1/\epsilon$-Lipschitz with $\operatorname{spt}g\subset B(z,r)$.
By construction,

$\mu(\tilde{K}_{\mu})\leq\int g\,\mathrm{d}\mu$

and

$\int g\,\mathrm{d}\nu\leq\nu(B(z,r)\setminus K^{\prime}_{\nu}).$

Therefore,

$\displaystyle\mu(\tilde{K}_{\mu})$ $\displaystyle\leq
F_{z}^{1/\epsilon,r}(\mu,\nu)+\nu(B(z,r)\setminus K^{\prime}_{\nu})$
$\displaystyle\leq(1+\delta)F_{z}^{1/\epsilon,r}(\mu,\nu),$ (88)

by (87). Define

$K_{\mu}:=K^{\prime}_{\mu}\setminus\tilde{K}_{\mu}=(K^{\prime}_{\mu}\cap
B(K^{\prime}_{\nu},\epsilon))\cup(K^{\prime}_{\mu}\setminus U(z,r-\epsilon)),$
(89)

a compact set, and, by eqs. 87 and 4.2,

$\mu(B(z,r)\setminus K_{\mu})<(1+2\delta)F_{z}^{1/\epsilon,r}(\mu,\nu).$

Also, by (89),

$K_{\mu}\cap U(z,r-\epsilon)\subset B(K^{\prime}_{\nu},\epsilon).$ (90)

By the symmetric argument, the compact set

$K_{\nu}:=(K^{\prime}_{\nu}\cap
B(K^{\prime}_{\mu},\epsilon))\cup(K^{\prime}_{\nu}\setminus U(z,r-\epsilon))$

satisfies

$\nu(B(z,r)\setminus K_{\nu})<(1+2\delta)F_{z}^{1/\epsilon,r}(\mu,\nu)$

and

$K_{\nu}\cap U(y,r-\epsilon)\subset B(K^{\prime}_{\mu},\epsilon).$ (91)

Finally, if $x\in K_{\mu}\cap U(z,r-\epsilon)$, let $y\in K^{\prime}_{\nu}$
with $x\in B(y,\epsilon)$. Then $y\in B(K^{\prime}_{\mu},\epsilon)$ and so
$y\in K_{\nu}$. Thus

$K_{\mu}\cap U(x,r-\epsilon)\subset B(K_{\nu},\epsilon).$

Similarly,

$K_{\nu}\cap U(y,r-\epsilon)\subset B(K_{\mu},\epsilon).$

Thus $H_{z}(K_{\mu},K_{\nu})\leq\max\\{1/(r-\epsilon),\epsilon\\}$. Therefore,
$K_{\mu},K_{\nu}$ have the required properties with $\delta$ replaced by
$2\delta$. Since $\delta>0$ is arbitrary, this suffices.

######  Theorem 4.11.

For $\epsilon>0$ let $(X,\mu,x)$ and $(Y,\nu,y)$ be pointed metric measure
spaces with

$d_{*}((X,\mu,x),(Y,\nu,y))<\epsilon.$

There exist compact

$x\in K_{\mu}\subset\operatorname{spt}\mu\cap
B(x,1/\epsilon)\quad\text{and}\quad y\in
K_{\nu}\subset\operatorname{spt}\nu\cap B(y,1/\epsilon)$ (92)

such that

$\max\\{\mu(B(x,1/\epsilon)\setminus K_{\mu}),\nu(B(y,1/\epsilon)\setminus
K_{\nu})\\}<\epsilon$ (93)

and

$d_{\mathrm{pmGH}}((K_{\mu},\mu|_{K_{\mu}},x),(K_{\nu},\nu|_{K_{\nu}},y))<3\epsilon.$
(94)

Conversely, if there exist compact $K_{\mu},K_{\nu}$ as in (92) satisfying
(93) and

$d_{*}((K_{\mu},\mu|_{K_{\mu}},x),(K_{\nu},\nu|_{K_{\nu}},y))<\epsilon$ (95)

then

$d_{*}((X,\mu,x),(Y,\nu,y))<3\epsilon.$

That is, $d_{*}$ is bi-Lipschitz equivalent to the metric defined by taking
the infimum over all $0<\epsilon<1/2$ for which eqs. 93 and 94 hold.

Proof: If $\epsilon\geq 1/2$ there is nothing to prove. Otherwise, by the
definition of $d_{*}$, there exists a pointed metric space $(Z,z)$ and
isometric embeddings $(X,x),(Y,y)\to(Z,z)$ with $F_{z}(\mu,\nu)<\epsilon$.

Let $\delta>0$ be such that $(1+\delta)F_{z}(\mu,\nu)<\epsilon$ and let
$K_{\mu}$ and $K_{\nu}$ be obtained from Lemma 4.10, so that (93) is
satisfied. Then (86) implies

$H_{z}(K_{\mu},K_{\nu})\leq 2\epsilon.$

Finally, eqs. 93 and 18 and the triangle inequality imply

$\displaystyle F_{z}^{1/\epsilon,1/\epsilon}(\mu|_{K_{\mu}},\nu|_{K_{\nu}})$
$\displaystyle\leq
F_{z}^{1/\epsilon,1/\epsilon}(\mu|_{K_{\mu}},\mu)+F_{z}^{1/\epsilon,1/\epsilon}(\mu,\nu)+F_{z}^{1/\epsilon,1/\epsilon}(\nu,\nu|_{K_{\nu}})$
$\displaystyle<3\epsilon.$

Thus (94) is established.

Conversely, by (84) and the triangle inequality,

$\displaystyle d_{*}((X,\mu,x),(Y,\nu,y))$ $\displaystyle\leq
d_{*}((X,\mu,x),(K_{\mu},\mu|_{K_{\mu}},x))$
$\displaystyle\quad+d_{*}((K_{\mu},\mu|_{K_{\mu}},x),(K_{\nu},\nu|_{K_{\nu}},y))$
$\displaystyle\qquad+d_{*}((K_{\nu},\nu|_{K_{\nu}},y),(Y,\nu,y))$
$\displaystyle<3\epsilon.$

Working with measures allows us to take limits of $\epsilon$-isometries (see
also (MR3477230, , Proposition 3.2), which shows the compactness of the set of
isometries).

######  Lemma 4.12.

Let $(X,d,\mu,x)$ and $(Y,\rho,\nu,y)$ be pointed metric measure spaces.
Suppose that, for each $i\in\mathbb{N}$, there exists a Borel set $x\in
S_{i}\subset X$ with

$\mu(B(x,i)\setminus S_{i})\to 0$ (96)

and a $1/i$-isometry $f_{i}\colon S_{i}\to S^{\prime}_{i}\subset Y$such that

$F_{y}((f_{i})_{\\#}(\mu|_{S_{i}}),\nu)\to 0.$ (97)

Then there exists an isometry
$\iota\colon(\operatorname{spt}\mu\cup\\{x\\},\mu,x)\to(\operatorname{spt}\nu\cup\\{y\\},\nu,y)$.

Proof: By the inner regularity of $\mu$, we may suppose each $S_{i}$ is
compact. For each $i\in\mathbb{N}$ and $w\in S_{i}$, let
$g_{i}(w)=(w,f_{i}(w))\in X\times Y$ and define

$\lambda_{i}=(g_{i})_{\\#}(\mu|_{S_{i}})\in\mathcal{M}_{\mathrm{loc}}(X\times
Y).$

By Theorem 2.11, after passing to a subsequence, we may suppose that
$(g_{i})_{\\#}(\mu|_{S_{i}})$ converges to some measure
$\lambda\in\mathcal{M}_{\mathrm{loc}}(X\times Y)$. Note that the pushforward
of $\lambda_{i}$ to $X$ under the projection map is $\mu|_{S_{i}}$. By (96)
the projection of $\operatorname{spt}\lambda$ to $X$ has full $\mu$ measure
and by (97) the projection of $\operatorname{spt}\lambda$ to $Y$ has full
$\nu$ measure.

If $(x^{1},y^{1})\in\operatorname{spt}\lambda$ then there exist $x^{1}_{i}\in
S_{i}$ with $g_{i}(x^{1}_{i})\to(x^{1},y^{1})$. Indeed, if not then there
exist $r>0$ and $i_{k}\to\infty$ such that

$\lambda_{i_{k}}(B((x^{1},y^{1}),r))=\mu_{i_{k}}(S_{i_{k}}\cap
g_{i_{k}}^{-1}(B((x^{1},y^{1}),r)))=0\quad\forall k\in\mathbb{N},$

contradicting the convergence of $\lambda_{i}$ to $\lambda$ using (14). In
particular, if $(x^{1},y^{1}),(x^{2},y^{2})\in\operatorname{spt}\lambda$,
there exist $x^{1}_{i},x_{i}^{2}\in S_{i}$ with
$(x^{1}_{i},f_{i}(x^{1}_{i}))\to(x^{1},y^{1})$ and
$(x^{2}_{i},f_{i}(x_{i}^{2}))\to(x^{2},y^{2})$. Since each $f_{i}$ is a
$1/i$-isometry, this implies

$\rho(y^{1},y^{2})=\lim_{i\to\infty}\rho(f_{i}(x^{1}_{i}),f_{i}(x_{i}^{2}))=\lim_{i\to\infty}d(x_{i}^{1},x_{i}^{2})=d(x^{1},x^{2}).$

That is, $\operatorname{spt}\lambda$ lies on the graph of an isometry. This
isometry is defined $\mu$ almost everywhere and maps onto a set of full
$\nu$-measure. Since $Y$ is complete, $\iota$ can be extended to an isometry
of $\operatorname{spt}\mu$ to $\operatorname{spt}\nu$, as required.

######  Corollary 4.13.

Let $\mathbb{M}_{*}$ be the set of equivalence classes of all pointed metric
measure spaces under the relation

$(X,\mu,x)\sim(Y,\nu,y)\text{ if
}(\operatorname{spt}\mu\cup\\{x\\},\mu,x)\text{ is isometric to
}(\operatorname{spt}\nu\cup\\{y\\},\nu,y).$

Then $(\mathbb{M}_{*},d_{*})$ is a complete and separable metric space.

Proof: Proposition 2.20 implies $d_{*}$ is a complete pseudometric on the set
of all pointed metric measure spaces. It is easily verified that the set of
finite pointed metric spaces with rational distances and rational combinations
of Dirac masses is dense.

We next show that $d_{*}$ is well defined on $\mathbb{M}_{*}$. Indeed, if
$(X,d,\mu,x)\sim(Y,\rho,\nu,y)$, let

$\iota\colon\operatorname{spt}\mu\cup\\{x\\}\to\operatorname{spt}\nu\cup\\{y\\}$

be an isometry of pointed metric measure spaces. Define $\zeta$ on $X\sqcup Y$
by $\zeta=d$ on $X$, $\zeta=\rho$ on $Y$ and

$\zeta(x,y)=\inf\\{d(x,w)+\rho(f(w),y):w\in\operatorname{spt}\mu\cup\\{x\\}\\}$

whenever $x\in X$ and $y\in Y$. A similar argument to Lemma 2.24 shows that
$\zeta$ is a pseudometric on $X\sqcup Y$. Indeed, if $z,z^{\prime}\in X$ and
$t\in Y$ then for any $w,w^{\prime}\in\operatorname{spt}\mu\cup\\{x\\}$, the
triangle inequality for $d$ and $\rho$ and the fact that $f$ is an isometry
implies

$\displaystyle\zeta(z,z^{\prime})$ $\displaystyle\leq
d(z,w)+d(w,w^{\prime})+d(w^{\prime},z^{\prime})$
$\displaystyle=d(z,w)+\rho(f(w),f(w^{\prime}))+d(w^{\prime},z^{\prime})$
$\displaystyle\leq
d(z,w)+\rho(f(w),t)+\rho(t,f(w^{\prime}))+d(w^{\prime},z^{\prime}).$

Taking the infimum over $w,w^{\prime}\in\operatorname{spt}\mu\cup\\{x\\}$
gives $\zeta(z,z^{\prime})\leq\zeta(z,t)+\zeta(t,z^{\prime})$. A similar
argument for $z,z^{\prime}\in Y$ and $t\in X$ gives the triangle inequality
for $\zeta$. Note that $\zeta(x,f(x))=0$ for all
$x\in\operatorname{spt}\mu\cup\\{x\\}$.

Let $(Z,\zeta)$ be the metric space obtained from $(X\sqcup Y,\zeta)$ and let
$z\in Z$ be the point corresponding to $x=y$. Then $(X,x)$ and $(Y,y)$
isometrically embed into $(Z,z)$ and, since $f$ is an isometry of metric
measure spaces, the pushforwards of $\mu$ and $\nu$ under these embeddings
agree. In particular, $F_{z}(\mu,\nu)=0$ in $Z$ and hence

$d_{*}((X,d,\mu,x),(Y,\rho,\nu,y))=0.$

That is, $d_{*}$ is well defined on $\mathbb{M}_{*}$.

By combining Corollary 4.4, Lemma 4.12 and Theorem 4.11, any pointed metric
measure spaces satisfying

$d_{*}((X,\mu,x),(Y,\nu,y))=0$

are isometric. Hence $d_{*}$ is a metric on $\mathbb{M}_{*}$.

When considering $(X,\mu,x)\in\mathbb{M}_{*}$, the set
$X\setminus(\operatorname{spt}\mu\cup\\{x\\})$ often plays no role. In such
cases, to simplify notation, we will write $(\mu,x)$ for the element of
$\mathbb{M}_{*}$.

######  Remark 4.14.

Sturm sturm  defines a metric on the set of metric measure spaces equipped
with a probability measure with finite variance. Greven, Pfaffelhuber and
Winter MR2520129  define the _Gromov–Prokhorov_ metric on the set of compact
metric measure spaces equipped with a probability measure. The definitions of
the metrics are analogous to that of $d_{*}$, but use the $L_{2}$
transportation distance and the Prokhorov metric respectively in place of $F$
and there are no requirements imposed on a distinguished point. In this
setting, one can slightly modify the definition of $d_{*}$ to not make any
requirements on a distinguished point, or modify the definitions of sturm  or
MR2520129  to consider a distinguished point. After doing so, since $F$, the
$L_{2}$ transportation distance and the Prokhorov metric all metrise weak*
convergence in the respective settings, the metrics all define the same
topology whenever they are defined.

Gigli, Mondino and Savaré MR3477230  define a notion of convergence on the set
of pointed metric measure spaces by $(X_{i},\mu_{i},x_{i})\to(X,\mu,x)$ if
there exist a complete and separable metric space $Z$ and isometric embeddings
$\operatorname{spt}\mu_{i},\operatorname{spt}\mu\to Z$ with $x_{i}\to x$ and
$\mu_{i}\to\mu$ in $\mathcal{M}_{\mathrm{loc}}(Z)$. By Proposition 2.20 we see
that this definition agrees with convergence with respect to $d_{*}$, using
Lemma 2.22 to ensure that the distinguished points are mapped to the same
point. Several equivalent definitions of this convergence for _non-zero_
measures are given in (MR3477230, , Theorem 3.15).

Shortly after a first version of the present paper appeared I became aware of
Pasqualetto and Schultz schultz . Amongst other results, they prove a
relationship between pointed measured Gromov–Hausdorff convergence and the
convergence of MR3477230 . This relationship corresponds to the relationship
between $d_{\mathrm{pmGH}}$ and $d_{*}$ given by Theorem 4.11.

######  Theorem 4.15.

A set $\mathcal{S}\subset(\mathbb{M}_{*},d_{*})$ is pre-compact if and only
if, for every $\epsilon>0$ and $(\mu,x)\in\mathcal{S}$, there exists a compact
$K_{\mu,x}\subset B(x,1/\epsilon)$ with

$\mu(B(x,1/\epsilon)\setminus K_{\mu,x})\leq\epsilon$ (98)

such that

$\\{(K_{\mu,x},\mu|_{K_{\mu,x}},x):(\mu,x)\in\mathcal{S}\\}\subset(\mathbb{M}_{pm},d_{\mathrm{pmGH}})$

is pre-compact.

Proof: Let $(\mu_{i},x_{i})$ be a sequence in $\mathcal{S}$ and, for
$i,j\in\mathbb{N}$ let $K_{i}^{j}\subset B(x_{i},j)$ be as in (98) for
$\epsilon=1/j$. Let
$\mathcal{K}(i,j)=(K_{i}^{j},(\mu_{i})|_{K_{i}^{j}},x_{i})$. First note that
(84) implies

$d_{*}(\mathcal{K}(i,j),(\mu_{i},x_{i}))\leq 1/j\quad\forall i,j\in\mathbb{N}$
(99)

and hence

$d_{*}(\mathcal{K}(i,j),\mathcal{K}(i,j^{\prime}))\leq 2/j\quad\forall
i\in\mathbb{N},\ \forall j\leq j^{\prime}\in\mathbb{N}.$ (100)

By assumption, for each $j\in\mathbb{N}$ there exists a
$\mathcal{K}_{j}\in\mathbb{M}_{pm}$ and a subsequence $i^{j}_{k}$ such that

$d_{\mathrm{pmGH}}(\mathcal{K}(i_{k}^{j},j),\mathcal{K}_{j})\to 0\text{ as
}k\to\infty.$

By taking a diagonal subsequence, we may suppose that

$d_{\mathrm{pmGH}}(\mathcal{K}(i,j),\mathcal{K}_{j})\to 0\quad\forall
j\in\mathbb{N}.$

Equation 100 then implies

$d_{*}(\mathcal{K}_{j},\mathcal{K}_{j^{\prime}})\leq 2/j\quad\forall j\leq
j^{\prime}\in\mathbb{N}.$

Since $(\mathbb{M}_{*},d_{*})$ is complete, $\mathcal{K}_{j}$ converges to
some $(\mu,x)$. Equation 99 and the triangle inequality then imply
$(\mu_{i},x_{i})$ converges to $(\mu,x)$.

Conversely, suppose that $\mathcal{S}$ is pre-compact but the conclusion does
not hold. Note that, for any $r>0$, the set
$\\{\mu(B(x,r)):(\mu,x)\in\mathcal{S}\\}$ must be bounded. Therefore, there
exist $\delta,\epsilon>0$ and, for each $i\in\mathbb{N}$, a
$(\mu_{i},x_{i})\in\mathcal{S}$ such that, for any
$K\subset\operatorname{spt}\mu_{i}\cap B(x_{i},1/\epsilon)$ with

$\mu(B(x_{i},1/\epsilon)\setminus K)\leq\epsilon,$

$K$ contains $i$ points separated by distance at least $\delta$. After passing
to a subsequence, we may suppose that $(\mu_{i},x_{i})$ converges to some
$(\mu,x)$. By Proposition 2.20, there exists a complete and separable pointed
metric space $(Z,z)$ and isometric embeddings
$(\operatorname{spt}\mu_{i}\cup\\{x_{i}\\},x_{i})\to(Z,z)$ with
$\mu_{i}\to\mu$. Theorem 2.11 implies that there exists a compact $K\subset Z$
with $\mu_{i}(B(z,r)\setminus K)\leq\epsilon$ for all $i$, contradicting our
initial assumption.

######  Remark 4.16.

Theorem 4.15 implies (MR2520129, , Proposition 7.1) and (MR3477230, ,
Corollary 3.22). The properties of $d_{*}$ and its relationship to
$d_{\mathrm{pmGH}}$ allow for a simpler proof.

######  Lemma 4.17.

For $0<\delta<1$, $0<\epsilon<1/2$, and $k\geq 1$ let $R>0$ satisfy

$R>(4R(\epsilon/\delta)^{\frac{1}{k}})^{-1}.$ (101)

Suppose that $(X,\mu,x)$ is a $2^{k}$-doubling pointed metric measure space
with $\mu(B(x,R))\geq\delta$ and suppose that $S\subset X$ satisfies

$\mu(B(x,2R)\setminus S)<\epsilon.$

Then

$d_{\mathrm{pmGH}}((X,\mu,x),(S,\mu|_{S},x))\leq
4R(\epsilon/\delta)^{\frac{1}{k}}+\epsilon.$ (102)

Proof: For $0<r<R$ suppose that $w\in B(x,R)\setminus B(S,r)$. Then

$\mu(B(w,r))\leq\mu(B(x,R+r)\setminus S)<\epsilon$

and so

$\delta\leq\mu(B(x,R))\leq\mu(B(w,2R))\leq(2^{k})^{m}\mu(B(w,r))<\epsilon
2^{km},$

for $m=\lceil\log_{2}2R/r\rceil$. Therefore

$r<4R(\epsilon/\delta)^{\frac{1}{k}}$

and so

$B(x,R)\subset B(S,4R(\epsilon/\delta)^{\frac{1}{k}}).$

Combining this with (101) gives

$B(x,(4R(\epsilon/\delta)^{\frac{1}{k}})^{-1})\subset
B(S,4R(\epsilon/\delta)^{\frac{1}{k}}),$

so that

$H_{x}(X,S)\leq 4R(\epsilon/\delta)^{\frac{1}{k}}.$

Since $\mu(B(x,2R)\setminus S)<\epsilon$, (102) follows from (18).

We also note the following partial converse to the fact that $d_{*}\leq
d_{\mathrm{pmGH}}$.

######  Corollary 4.18.

For $0<\delta<1$ and $k\geq 1$ let $\mathcal{S}$ be the equivalence classes of
those $2^{k}$-doubling pointed metric measure spaces $(X,\mu,x)$ with
$\mu(B(x,1))\geq\delta$. Then, when restricted to $\mathcal{S}$,

$d_{\mathrm{pmGH}}\leq 9\delta^{-\frac{1}{k}}d_{*}^{\frac{1}{2k}}.$ (103)

Proof: For $0<\epsilon<1/2$ suppose that $(X,\mu,x),(Y,\nu,y)\in\mathcal{S}$
with

$d_{*}((X,\mu,x),(Y,\nu,y))<\epsilon.$

By Theorem 4.11, there exist compact $K_{\mu}\subset B(x,\epsilon^{-1/2k})$
and $K_{\nu}\subset B(y,\epsilon^{-1/2k})$ satisfying

$\max\\{\mu(B(x,\epsilon^{-1/2k})\setminus K_{\mu}),\
\nu(B(y,\epsilon^{-1/2k})\setminus K_{\nu})\\}<\epsilon$ (104)

and eq. 94. Equation 104 and Lemma 4.17 (applied with $R=\epsilon^{-1/2k}$,
which satisfies (101) for $\delta\leq 1$) imply that

$d_{\mathrm{pmGH}}((X,\mu,x),(K_{\mu},\mu|_{K_{\mu}},x))\text{ and
}d_{\mathrm{pmGH}}((Y,\nu,y),(K_{\nu},\nu|_{K_{\nu}},y))$

are at most $4\delta^{-\frac{1}{k}}\epsilon^{\frac{1}{2k}}+\epsilon$.
Therefore (94) and the triangle inequality imply (103).

##  5 Tangent spaces of metric measure spaces

With the notion of convergence defined in Section 4, we may define a tangent
space following Preiss MR890162 . Given $(\mu,d,x)\in\mathbb{M}_{*}$ with
$x\in\operatorname{spt}\mu$ and $r>0$, let

$T_{r}(\mu,d,x):=\left(\frac{\mu}{\mu(B(x,r))},\frac{d}{r},x\right)$

######  Definition 5.1.

A $(\nu,\rho,y)\in\mathbb{M}_{*}$ is a _tangent measure_ of
$(\mu,d,x)\in\mathbb{M}_{*}$ if there exist $r_{k}\to 0$ such that

$d_{*}(T_{r_{k}}(\mu,d,x),(\nu,\rho,y))\to 0.$

The set of all tangent measures of $(\mu,d,x)$ will be denoted by
$\operatorname{Tan}(\mu,d,x)$.

######  Remark 5.2.

If $X=\ell_{2}^{m}$ for some $m\in\mathbb{N}$, any tangent measure is also
supported on $\ell_{2}^{m}$. In this case, $\operatorname{Tan}(\mu,x)$
consists of the equivalence classes of the tangent measures of Preiss.

By Corollary 4.18, $\operatorname{Tan}(\mu,x)$ agrees with the usual
definition of Gromov–Hausdorff tangent spaces for doubling metric measure
spaces.

######  Proposition 5.3.

Let $X$ be a metric space and let $\mu\in\mathcal{M}_{\mathrm{loc}}(X)$ be
asymptotically doubling. For $\mu$-a.e. $x\in\operatorname{spt}\mu$ and every
$r_{0}>0$,

$\\{T_{r}(\mu,x):0<r<r_{0}\\}\text{ is pre-compact.}$ (105)

In particular, for any $x\in\operatorname{spt}\mu$ satisfying (105),
$\operatorname{Tan}(\mu,x)$ is a non-empty compact metric space when equipped
with $d_{*}$ and

$\forall\delta>0,\ \exists r_{x}>0\text{ s.t.
}d_{*}(T_{r}(\mu,x),\operatorname{Tan}(\mu,x))\leq\delta\ \forall 0<r<r_{x}.$
(106)

Proof: By Lemma 2.8, we know that we may cover $\mu$ almost all of
$\operatorname{spt}\mu$ by

$X_{m}:=\\{x\in\operatorname{spt}\mu:\mu(B(x,2r))\leq m\mu(B(x,r))\ \forall
0<r<1/m\\}$

with $m\in\mathbb{N}$. Fix an $m\in\mathbb{N}$. By Theorem 2.10, $\mu$ almost
every $x\in X_{m}$ is a Lebesgue density point of $X_{m}$. Fix such an $x$. It
suffices to prove the conclusion for $x$. To do so, fix $R\geq 2$.

First note that, if $R/(2m)\leq r<r_{0}$ then

$\frac{\mu(B_{d/r}(x,R))}{\mu(B_{d}(x,r))}=\frac{\mu(B_{d}(x,Rr))}{\mu(B_{d}(x,r))}\leq\frac{\mu(B_{d}(x,Rr_{0}))}{\mu(B_{d}(x,R/(2m)))}<\infty.$

Here the subscripts on the balls indicate the metrics used to define the
balls, for $d$ the original metric in $X$. Also, if $0<r<R/(2m)$ then

$\frac{\mu(B_{d/r}(x,R))}{\mu(B_{d}(x,r))}=\frac{\mu(B_{d}(x,Rr))}{\mu(B_{d}(x,r))}\leq
m^{4\log_{2}R}.$

Thus

$\\{\nu(B(x,R)):(\nu,x)\in T_{r}(\mu,x),\ 0<r<r_{0}\\}$

is bounded.

Secondly, for any $\epsilon>0$, since $x$ is a Lebesgue density point of
$X_{m}$, there exists $r_{1}>0$ such that