2.1 Metric spaces and maps between them

Whenever $(X,d)$ is a metric space and $A,B \subseteq X$ are nonempty subsets, we write $\operatorname {\mathrm {diam}}(A) := \sup _{a,b \in A} d(a,b) \in [0,\infty ]$ and $\operatorname {\mathrm {dist}}(A,B) := \inf _{a\in A, b \in B}d(a,b) \in [0,\infty )$ . If $B = \{b\}$ is a singleton, we condense notation and write $\operatorname {\mathrm {dist}}(A,b)$ for $\operatorname {\mathrm {dist}}(A,\{b\})$ . If $x \in X$ and $r \geq 0$ , we write $B_r(x)$ for the closed ball $\{y \in X: d(x,y) \leq r\}$ .

Let $(X,d_X),(Y,d_Y)$ be metric spaces and $f: X \to Y$ a map. Then f is L-Lipschitz (for some $L<\infty $ ) if for all $x,y\in X$ , it holds that $d_Y(f(x),f(y)) \leq Ld_X(x,y)$ . The minimal such L is called the Lipschitz constant of f and is denoted by $\operatorname {\mathrm {Lip}}(f)$ . The map f is a biLipschitz embedding of distortion D if f is injective and there exists a scaling factor $s\in (0,\infty )$ such that f is $sD$ -Lipschitz and $f^{-1}$ is $s^{-1}$ -Lipschitz. If f is also surjective, then it is a biLipschitz equivalence. BiLipschitz embeddings and equivalences of distortion 1 are isometric embeddings and isometries, respectively. The map f is a rough biLipschitz embedding (or quasi-isometric embedding) if there are $K,L<\infty $ such that $L^{-1}d_X(x,y) - K \leq d_Y(f(x),f(y)) \leq Ld_X(x,y) + K$ for all $x,y \in X$ . If, in addition, the image of f is coarsely dense, meaning there exists $C<\infty $ such that $\operatorname {\mathrm {dist}}(f(X),y) < C$ for every $y \in Y$ , then f is a rough biLipschitz equivalence (or quasi-isometry). If there exist $\alpha \in (0,\infty )$ and $L<\infty $ such that $L^{-1}d_X(x,y) \leq d_Y(f(x),f(y))^\alpha \leq Ld_X(x,y)$ for all $x,y \in X$ , then f is a snowflake embedding. Surjective snowflake embeddings are called snowflake equivalences. Note that snowflake embeddings (and thus biLipschitz embeddings) are always measurableFootnote ⁴ since they are continuous, but rough biLipschitz embeddings need not be measurable (indeed, any map between two bounded metric spaces is a rough biLipschitz equivalence).

2.2 Banach and Bochner-Lebesgue spaces

Let $T: V \to W$ be a linear map between Banach spacesFootnote ⁵ and $C<\infty $ . The map T is C-bounded if it is C-Lipschitz, T is a C-isomorphic embedding if it is biLipschitz with distortion C, and T is a C-isomorphism if it is a surjective C-isomorphic embedding. We will use the notation ‘ $V \approx W$ ’ to indicate that there exists an isomorphism between V and W. Linear maps that are 1-isomorphic embeddings or 1-isomorphisms are linear isometric embeddings and linear isometries, respectively. A subspace $U \subseteq W$ is C-complemented if there exists a C-bounded linear map $R: W \to U$ such that $R(u) = u$ for every $u \in U$ . It holds that V is $C_1$ -isomorphic to a $C_1$ -complemented subspace of $W\ \iff $ there exist $C_2$ -bounded linear maps $T: V \to W$ and $R: W \to V$ such that $R \circ T = id_V$ (we say that R retracts T) $\iff $ there exists a Banach space U such that $U \oplus V$ is $C_3$ -isomorphic to W, where in each equivalence, $C_{i+1}$ depends only on $C_{i}$ and vice versa. We will use the notation ‘ $V\overset {\mathrm {c}}{\hookrightarrow } W$ ’ to indicate that V is isomorphic to a complemented subspace of W. Here and in the sequel, whenever $\{(V_\lambda ,\|\cdot \|_\lambda )\}_{\lambda \in \Lambda }$ is an indexed family of Banach spaces, we denote the $\ell ^1$ -sum of the family by $\oplus ^1_{\lambda \in \Lambda } V_\lambda $ , or just $\oplus _{\lambda \in \Lambda } V_\lambda $ without the ‘1’ in the superscript. That is, $\oplus ^1_{\lambda \in \Lambda } V_\alpha $ consists of all $\Lambda $ -indexed generalized sequences $(\boldsymbol {v}_\lambda )_{\lambda \in \Lambda }$ , $\boldsymbol {v}_\lambda \in V_\lambda $ , with finite norm $\|(\boldsymbol {v}_\lambda )_{\lambda \in \Lambda }\| := \sum _{\lambda \in \Lambda } \|\boldsymbol {v}_\lambda \|_\lambda $ . Whenever each of two Banach spaces $V,W$ is C-isomorphic to a C-complemented subspace of the other, and one of the two, say V, is C-isomorphic to its countable $\ell ^1$ -sum $\oplus ^1_{n\in \mathbb {N}} V$ , then the Pełczyński decomposition method yields a $C'$ -isomorphism $V \approx W$ , where $C'$ depends only on C [Reference Benyamini and Lindenstrauss11, Proposition F.9].

As previously stated, a Banach space is an $L^1$ -space if it equals the Lebesgue space $L^1(\Omega ,\mathcal {F},\mu )$ for some measure space $(\Omega ,\mathcal {F},\mu )$ . When $\Omega $ equals $[0,1]$ equipped with the Borel $\sigma $ -algebra and Lebesgue measure, we abbreviate $L^1(\Omega ,\mathcal {F},\mu )$ by $L^1$ . When $\Omega $ equals $\mathbb {N}$ equipped with the power set $\sigma $ -algebra and counting measure, we abbreviate $L^1(\Omega ,\mathcal {F},\mu )$ by $\ell ^1$ . It holds that any separable subspace of an $L^1$ -space is contained in a separable $L^1$ -subspace (see, for example, [Reference Ostrovskii44, Fact 1.20]), and any separable, infinite dimensional $L^1$ -space is isomorphic to $L^1$ or to $\ell ^1$ [Reference Johnson and Lindenstrauss32, page 15].

Let $(\Omega ,\mathcal {F},\mu )$ be a measure space, $(V,\|\cdot \|)$ a Banach space, and $g: \Omega \to V$ a function. We say that g is Bochner measurable if g is the $\mu $ -almost everywhere pointwise limit of a sequence of simple functions, where a function is simple if it belongs to the linear span of the indicator functions $\{\mathsf {1}_Fv\}_{F\in \mathcal {F},v\in V}$ . The function g is weakly measurable if $\lambda \circ g: \Omega \to \mathbb {R}$ is measurable for every $\lambda \in V^*$ . The function g is essentially-separably-valued if there exist a separable subspace $S \subseteq V$ and a $\mu $ -null set N such that $g(\omega ) \in S$ for every $\omega \in \Omega \setminus N$ . Pettis’s measurability theorem [Reference Diestel, Uhl and Pettis20, Theorem 1.2] characterizes Bochner measurable functions as exactly those that are weakly measurable and essentially-separably-valued. Since Bochner measurability immediately implies measurability and measurability immediately implies weak measurability, it holds that a function is Bochner measurable if and only if it is measurable and essentially-separably-valued. This latter characterization is the one we will most frequently use throughout the article. The set of Bochner measurable functions g with finite norm $\|g\|_{L^1(\Omega ;V)} := \int _\Omega \|g\| d\mu < \infty $ forms a Banach space (the Bochner-Lebesgue space), denoted by $L^1(\Omega ,\mathcal {F},\mu ;V)$ , or just $L^1(\Omega ;V)$ if the $\sigma $ -algebra and measure are understood. Also, if $V=\mathbb {R}$ , we compress notation again and just write $L^1(\Omega ,\mathcal {F},\mu )$ or $L^1(\Omega )$ . Simple functions are dense in $L^1(\Omega ;V)$ , and thus, the integral $\int _\Omega h d\mu $ , which is defined in the obvious way for simple functions h, admits a unique continuous extension to all $h \in L^1(\Omega ;V)$ . Furthermore, if A is a dense subset of V, then the set of simple functions $\Omega \to A$ taking values in A is dense in $L^1(\Omega ;V)$ . Using this fact, it is easy to prove that whenever $(\Omega ',\mathcal {F}',\mu ')$ is a second measure space, $L^1(\Omega ,\mathcal {F},\mu ;L^1(\Omega ',\mathcal {F}',\mu '))$ is canonically linearly isometric to $L^1(\Omega \times \Omega ',\mathcal {F}\otimes \mathcal {F}',\mu \otimes \mu ')$ . This implies that if V is an $L^1$ -space, then $L^1(\Omega ;V)$ is another $L^1$ -space. If $\mu $ is a probability measure, we will often use probabilistic notation and denote $\int _\Omega h d\mu $ by $\mathbb {E}_{\omega \in \Omega }[h_\omega ]$ , $\mathbb {E}_\omega [h_\omega ]$ , or just $\mathbb {E}[h]$ . We refer the reader to [Reference Diestel, Uhl and Pettis20] for further background on Bochner-Lebesgue spaces.

2.3 Lipschitz free spaces

A pointed set is a set X equipped with some distinguished basepoint $x_0 \in X$ . A pointed metric space is a metric space $(X,d)$ such that X is a pointed set. We adopt the following important convention throughout the article:

A map between pointed sets is basepoint-preserving if it maps the basepoint of the domain to the basepoint of the codomain. Given a pointed metric space X, we denote by $\operatorname {\mathrm {Lip}}_0(X)$ the vector space of basepoint-preserving Lipschitz functions $f:X\to \mathbb {R}$ . The vector space $\operatorname {\mathrm {Lip}}_0(X)$ becomes a Banach space when equipped with the norm $\|f\|_{\operatorname {\mathrm {Lip}}_0(X)} := \operatorname {\mathrm {Lip}}(f)$ . There is a canonical isometric embedding $\delta : X \hookrightarrow \operatorname {\mathrm {Lip}}_0(X)^*$ given by the pointwise evaluation map: $\delta (x) := \delta _x$ for $x \in X$ , where $\delta _x(f) := f(x)$ for $f \in \operatorname {\mathrm {Lip}}_0(X)$ . The closed linear span of $\delta (X)$ in $\operatorname {\mathrm {Lip}}_0(X)^*$ is called the Lipschitz free space, or just free space, of X, which we denote by $\operatorname {\mathrm {LF}}(X)$ , or by $\operatorname {\mathrm {LF}}(X,d)$ when we want to emphasize the metric. The isometry type of $\operatorname {\mathrm {LF}}(X)$ is independent of the chosen basepoint, and thus, we omit it from the notation. The dual space of $\operatorname {\mathrm {LF}}(X)$ is $\operatorname {\mathrm {Lip}}_0(X)$ , and the induced weak $^*$ topology on bounded subsets of $\operatorname {\mathrm {Lip}}_0(X)$ is the topology of pointwise convergence. By the Krein-Smulian theorem, a bounded linear map $T: \operatorname {\mathrm {Lip}}_0(X) \to V^*$ is the adjoint of a linear map $T_*: V \to \operatorname {\mathrm {LF}}(X)$ if and only if $T(f_\alpha )$ weak $^*$ converges to $T(f)$ whenever $f_\alpha $ is a bounded net in $\operatorname {\mathrm {Lip}}_0(X)$ converging pointwise to f. In this case, $T_*$ is unique and called the preadjoint of T. Whenever $f: X \to E$ is a basepoint-preserving Lipschitz map into a Banach space, there exists a unique bounded linear map $T_f: \operatorname {\mathrm {LF}}(X) \to E$ satisfying $T_f(\delta _x) = f(x)$ for every $x \in X$ , and the operator norm of $T_f$ is $\operatorname {\mathrm {Lip}}(f)$ . We call $T_f$ the induced linear map of f. If $Y \subseteq X$ contains the basepoint, the formal identity map $\operatorname {\mathrm {LF}}(Y) \to \operatorname {\mathrm {LF}}(X)$ is a linear isometric embedding, and it is surjective if and only if Y is dense in X. This follows from the (Whitney-)McShane extension theorem, which states that every Lipschitz map $Y \to \mathbb {R}$ extends to a Lipschitz map $X \to \mathbb {R}$ with the same Lipschitz constant. Other standard facts we will use are that $\operatorname {\mathrm {LF}}(X)$ is separable if and only if X is separable and that biLipschitz embeddings (resp. equivalences) $X \to Y$ of distortion $C<\infty $ induce C-isomorphic embeddings (resp. equivalences) $\operatorname {\mathrm {LF}}(X) \to \operatorname {\mathrm {LF}}(Y)$ . See [Reference Weaver48, Chapter 3] for a standard reference on Lipshitz free spaces, and note that these spaces are also referred to as Arens-Eells spaces in that text and elsewhere.

The class of Lipschitz free spaces often exhibits qualitative behavior similar to that of the class of $L^1$ -spaces. For example, the following dichotomy in $L^1$ -spaces holds: either $L^1(\Omega )$ contains a complemented subspace isomorphic to $L^1$ , or it has the Radon-Nikodym property (see [Reference Diestel, Uhl and Pettis20, Chapter III, $\S$ 1, Definition 3] for the definition), with the latter property holding if and only if $\Omega $ is purely atomic. The same dichotomy holds in Lipschitz free spaces, with the property of pure 1-unrectifiability taking the place of pure atomicity.

With the previous dichotomy lemma in hand, we can obtain a positive solution to the Complemented Subspace of $L^1$ Problem – which asserts that separable, complemented subspaces of $L^1$ -spaces are isomorphic to $L^1$ -spaces (see [Reference Alspach and Odell5, page 129]) – when restricted to the class of Lipschitz free spaces.

2.4 Nagata dimension and Lipschitz extensions

Let X be a metric space. A Nagata cover of dimension $n\in \mathbb {N}$ , constant $\gamma <\infty $ , and scale $s>0$ is a cover $\mathcal {C}$ of X such that $\operatorname {\mathrm {diam}}(C) \leq \gamma s$ for every $C \in \mathcal {C}$ and, for every $A \subset X$ with $\operatorname {\mathrm {diam}}(A) \leq s$ , it holds that $|\{C \in \mathcal {C}: C \cap A \neq \emptyset \}| \leq n+1$ (where $|S|$ denotes the cardinality of a set S). It is not hard to see that such a cover exists if and only if there exists a cover $B_1,B_2,\dots B_{n+1}$ of X such that each $B_i$ is the union of a family of sets that is $\gamma s$ -bounded and s-separated, meaning $B_i = \bigcup _{\lambda \in \Lambda _i} B_i^\lambda $ , where $\operatorname {\mathrm {diam}}(B_i^{\lambda _1}) \leq \gamma s$ and $\operatorname {\mathrm {dist}}(B_i^{\lambda _1},B_i^{\lambda _2})> s$ for all $\lambda _1 \neq \lambda _2 \in \Lambda _i$ . In this case, the sets $\{B_i^\lambda \}_{i=1,\lambda \in \Lambda _i}^{n+1}$ form a Nagata cover. A metric space X has (Assouad-)Nagata dimension $n \in \mathbb {N}$ with constant $\gamma < \infty $ if, for every $s>0$ , there exists a Nagata cover of dimension n, constant $\gamma $ , and scale s. If such an n and $\gamma $ exist, we say that X has finite Nagata dimension. We also call $\gamma $ the n-Nagata-dimensional constant of X.

The article of Lang and Schlichenmeier [Reference Lang and Schlichenmaier37] (and references therein) contains some foundational results on Nagata dimension. For example, Nagata dimension is nonincreasing under snowflake embeddings ([Reference Lang and Schlichenmaier37, Lemma 2.1]), the Nagata dimension of an open subset of $\mathbb {R}^n$ is n ([Reference Lang and Schlichenmaier37, page 3626]), every doubling space has finite Nagata dimension ([Reference Lang and Schlichenmaier37, Lemma 2.3]), ultrametric spaces have Nagata dimension 0, and nontrivial $\mathbb {R}$ -trees have Nagata dimension 1 (see $\S $ 2.6 for these last two statements). The most important result of [Reference Lang and Schlichenmaier37] for us concerns extensions of Lipschitz functions. Let us say that a metric space Z is Lipschitz $\infty $ -connected with constant $L<\infty $ if for every Banach space V, every Lipschitz map $f: S_V \to Z$ extends to an $L\operatorname {\mathrm {Lip}}(f)$ -Lipschitz map $B_V \to Z$ (where, as usual, $S_V$ denotes the unit sphere and $B_V$ the unit ball of V). Theorem 1.6 of [Reference Lang and Schlichenmaier37] implies that every Lipschitz map $f: M \to Z$ from a closed subset $M \subseteq Y$ of a metric space of finite Nagata dimension into a Lipschitz $\infty $ -connected metric space Z extends to a Lipschitz map $\overline {f}: Y \to Z$ , where $\operatorname {\mathrm {Lip}}(\overline {f})$ depends only on the relevant data of $Y,Z,f$ .

We will use Lemma 2.3 in the proof of Lemma 2.5 to build uniform-bounded Lipschitz extension operators on metrics spaces of finite Nagata dimension, which will be defined shortly.

Let $K<\infty $ , $(Y,d)$ a metric space, and $M \subseteq Y$ a closed subset. We call a map $y \mapsto \mu _y: Y \to \operatorname {\mathrm {LF}}(M)$ a convex K-random projection onto M if, for every $y,z \in Y$ and $m \in M$ ,

• ○ $\mu _y \in \overline {\mathrm{conv}}(\delta (M))$ ,

• ○ $\mu _m = \delta _m$ , and

• ○ $\|\mu _y-\mu _z\|_{\operatorname {\mathrm {LF}}} \leq Kd(y,z)$ ,

where $\overline {\mathrm{conv}}(S)$ denotes the closed convex hull of a subset S of a Banach space. Convex K-random projections were defined in [Reference Ambrosio and Puglisi6] under the name K-strong random projections to distinguish them from the larger class of K-random projections which include all maps satisfying the second and third items but not necessarily the first. We use the term ‘convex’ in place of ‘strong’, as the former is a bit more descriptive.

Let $K,L<\infty $ , X a pointed metric space, and $M \subseteq X$ a closed subset containing the basepoint. We call $E: \operatorname {\mathrm {Lip}}_0(M) \to \operatorname {\mathrm {Lip}}_0(X)$ a K-Lipschitz-bounded linear extension operator if it is a K-bounded linear map and $E(f)(m) = f(m)$ for every $f \in \operatorname {\mathrm {Lip}}_0(M)$ and $m \in M$ . If additionally, $\|E(f)\|_\infty \leq L \|f\|_\infty $ for every $f \in \operatorname {\mathrm {Lip}}_0(M)$ (where, as usual, $\|f\|_\infty := \sup _{m\in M}|f(m)|$ ), then E is L-uniform-bounded. We say that X has the K-Lipschitz-bounded linear extension property if for every closed subset $M \subseteq X$ containing the basepoint, there exists a K-Lipschitz-bounded linear extension operator $E: \operatorname {\mathrm {Lip}}_0(M) \to \operatorname {\mathrm {Lip}}_0(X)$ . If E can additionally be chosen to be L-uniform-bounded, then X has the K-Lipschitz-bounded, L-uniform-bounded linear extension property.

The next lemma we present can, in many cases, be deduced by combining established results (specifically, one could apply [Reference Naor and Silberman42, Lemma 5.1], [Reference Lee and Naor39, Theorem 4.1], [Reference Lee and Naor39, Lemma 3.8], and [Reference Ambrosio and Puglisi6, Theorem 2.10]). However, we found there to be less technical overhead in citing Lemma 2.3 and then filling in the remaining details ourselves.

Proof. Let $M \subseteq Y$ be closed. By definition of convex K-random projections, we need to find a K-Lipschitz map $Y \to \overline {\mathrm{conv}}(\delta (M))$ extending the isometric embedding $\delta : M \to \operatorname {\mathrm {LF}}(M)$ , where K depends only on $n,\gamma $ . By Lemma 2.3, it is enough to verify that every closed convex subset of a Banach space is Lipschitz $\infty $ -connected with universal constant (we will get a constant of 4).

Let V be a Banach space and C a closed convex subset of some Banach space. Let $f: S_V \to C$ be a Lipschitz map. By composing with translations, it suffices to assume that $0 \in f(S_V)$ . We will define the Lipschitz extension $\overline {f}: B_V \to C$ on $B_V \setminus \{0\}$ , and then by Lipschitz continuity and completeness of C, it will extend to a map on all of $B_V$ with the same Lipschitz constant.

There is a homeomorphism from $B_V \setminus \{0\}$ to $(0,1] \times S_V$ given by $v \mapsto (\|v\|, \frac {v}{\|v\|})$ with inverse $(t,\theta ) \mapsto t\theta $ . By the reverse triangle inequality, $\|s\theta - t\eta \| \geq |t-s|$ , and then applying the reverse triangle inequality again, one can get $\|s\theta - t\eta \| \geq \max \{s,t\}\|\theta -\eta \| - |t-s|$ . Averaging these yields

(2.1) $$ \begin{align} \|s\theta - t\eta\| \geq \max\{|t-s|,\tfrac{1}{2}\max\{s,t\}\|\theta-\eta\|\}. \end{align} $$

We define $\overline {f}$ on $t\theta $ by $\overline {f}(t\theta ) := tf(\theta )$ . This is well-defined because C is convex, $t \in [0,1]$ , and $0 \in f(S_V) \subseteq C$ , and it obviously extends f. It remains to bound the Lipschitz constant of $\overline {f}$ . Let $t\theta ,s\eta \in B_V\setminus \{0\}$ . Then by the triangle inequality, and the fact that $\|f\|_\infty \leq 2\operatorname {\mathrm {Lip}}(f)$ (since $0 \in f(S_V)$ and $\operatorname {\mathrm {diam}}(S_V) = 2$ ), we get

(2.2) $$ \begin{align} \|\overline{f}(s\theta)-\overline{f}(t\eta)\| = \|sf(\theta)-tf(\eta)\| \leq (2|t-s| + \min\{s,t\}\|\theta-\eta\|)\operatorname{\mathrm{Lip}}(f). \end{align} $$

Combining (2.1) and (2.2) yields $\operatorname {\mathrm {Lip}}(\overline {f}) \leq 4\operatorname {\mathrm {Lip}}(f)$ .

Combining these two lemmas, we obtain the following:

2.5 Hyperbolic metric spaces

We mainly follow the presentation of hyperbolic metric spaces and their boundaries as given in Bonk-Schramm [Reference Bonk and Schramm12]. For more comprehensive accounts of this material, see [Reference Buyalo and Schroeder15] or [Reference Bridson and Haefliger13, Chapter III.H].

Let $(X,d)$ be a metric space and $x,y,p \in X$ . The Gromov product of $x,y$ with respect to p is defined by

$$ \begin{align*} (x|y)_p := \frac{1}{2}(d(x,p)+d(p,y)-d(x,y)). \end{align*} $$

Let $\delta \in [0,\infty )$ . Then X is said to be $\delta $ -hyperbolic if for all $x,y,z,p \in X$ ,

$$ \begin{align*} (x|z)_p \geq \min\{(x|y)_p,(y|z)_p\} - \delta. \end{align*} $$

The space X is Gromov hyperbolic, or just hyperbolic, if it is $\delta $ -hyperbolic for some $\delta \in [0,\infty )$ .

Suppose $(X,d)$ is a hyperbolic space. A sequence $\{x_i\}_{i=1}^\infty \subseteq X$ is said to converge at $\infty $ if for some (equivalently, for all) $p \in X$ , $\lim _{i,j\to \infty } (x_i|x_j)_p = \infty $ . Two sequences $\{x_i\}_{i=1}^\infty , \{y_i\}_{i=1}^\infty \subseteq X$ converging at $\infty $ are equivalent if for some (equivalently, for all) $p \in X$ , $\lim _{i\to \infty } (x_i|y_i)_p = \infty $ .

This is an equivalence relation on the set of sequences converging at $\infty $ by hyperbolicity, and the set of equivalences classes is called the Gromov boundary, or just boundary, of X, denoted by $\partial X$ . The Gromov product of $\xi ,\upsilon \in \partial X$ with respect to $p \in X$ is defined by

$$ \begin{align*} (\xi|\upsilon)_p := \sup \{\liminf_{i\to\infty} (x_i|y_i)_p: \{x_i\}_{i=1}^\infty \in \xi,\{y_i\}_{i=1}^\infty \in \upsilon\}. \end{align*} $$

A metric d on $\partial X$ is said to be visual if there exist $p \in X$ , $\varepsilon> 0$ , and $C< \infty $ such that $C^{-1}e^{-\varepsilon (\xi |\upsilon )_p} \leq d(\xi ,\upsilon ) \leq Ce^{-\varepsilon (\xi |\upsilon )_p}$ (in Bonk-Schramm, the set of visual metrics is called the canonical B-structure on $\partial X$ [Reference Bonk and Schramm12, page 282]). Visual metrics on $\partial X$ always exist and are unique up to snowflake equivalence, and it is clear from the definition that any metric snowflake equivalent to a visual metric is also visual. Thus, we always equip $\partial X$ with some visual metric, and we may unambiguously assert that $\partial X$ satisfies or fails a metric property P as long as P is invariant under snowflake equivalences (for example, having Nagata dimension $n\in \mathbb {N}$ ). The boundary of a hyperbolic space is always bounded and complete. Under the additional hypothesis of visibility, which will be described in the next paragraph, the boundary of a hyperbolic space completely determines the hyperbolic space up to rough biLipschitz equivalence. This theory was worked out in full by Bonk and Schramm [Reference Bonk and Schramm12].

Let $(X,d)$ be a metric space, $K<\infty $ , and $I = [0,b]$ or $I = [0,\infty )$ . The image of I under a map $f: I \to X$ satisfying $|s-t| - K \leq d(f(s),f(t)) \leq |s-t| + K$ is called a K-rough geodesic if $I = [0,b]$ and a K-rough geodesic ray if $I = [0,\infty )$ . If $I = [0,b]$ , the points $f(0),f(b)$ are called the endpoints of the rough geodesic, and if $I = [0,\infty )$ , the point $f(0)$ is called the origin of the rough geodesic ray. A metric space is K-roughly geodesic if every pair of points are the endpoints of some K-rough geodesic. A geodesic is a 0-rough geodesic, and similar definitions hold for geodesic rays and geodesic metric spaces. A metric space is visual if there exists (equivalently, for all) $p \in X$ and there exists $K' <\infty $ such that X is the union of all $K'$ -rough geodesic rays originating from p.

Let $(Z,d_Z)$ be a bounded metric space. The (Bonk-Schramm) hyperbolic filling of Z is the set

(2.3) $$ \begin{align} \operatorname{\mathrm{Hyp}}(Z) := Z \times (0,\operatorname{\mathrm{diam}}(Z)] \end{align} $$

(note that $\operatorname {\mathrm {Hyp}}(Z)$ is called Con $(Z)$ in [Reference Bonk and Schramm12]) equipped with the metric

(2.4) $$ \begin{align} \rho_Z((z,h),(z',h')) := 2\log\left(\dfrac{d_Z(z,z')+h \vee h'}{\sqrt{hh'}}\right), \end{align} $$

where here and in the sequel, $\log $ denotes the natural logarithm and $h \vee h'$ denotes $\max \{h,h'\}$ . Note that $\rho _Z$ induces the product topology on $Z \times (0,\operatorname {\mathrm {diam}}(Z)]$ . By [Reference Bonk and Schramm12, Theorem 7.2], $\operatorname {\mathrm {Hyp}}(Z)$ is always a visual hyperbolic metric space. If $f: Z \to Y$ is any map to another bounded metric space with $\operatorname {\mathrm {diam}}(Z) \leq \operatorname {\mathrm {diam}}(Y)$ , then there is an induced map $\operatorname {\mathrm {Hyp}}(f): \operatorname {\mathrm {Hyp}}(Z) \to \operatorname {\mathrm {Hyp}}(Y)$ defined by

$$ \begin{align*} \operatorname{\mathrm{Hyp}}(f)(z,h) := (f(z),h). \end{align*} $$

The salient feature of hyperbolic fillings is that, in the visual case, they invert Gromov boundaries up to rough biLipschitz equivalence.

Lemma 2.7 plays a crucial role in the proof of Theorem 5.7.

2.6 Ultrametric spaces and $\mathbb {R}$ -trees

A subset of a topological space is an arc if it is homeomorphic to $[0,1]$ . The points in the image of $\{0,1\}$ under any such homeomorphism are called the endpoints of the arc. A metric space X is an $\mathbb {R}$ -tree if any two points $x,y \in X$ are the endpoints of a unique arc, denoted $[x,y]$ , and this arc is a geodesic. It holds that a metric space is an $\mathbb {R}$ -tree if and only if it is geodesic and 0-hyperbolic (This follows, for example, from [Reference Evans26, Theorem 3.40]), and every 0-hyperbolic metric space isometrically embeds into an $\mathbb {R}$ -tree (see [Reference Evans26, Theorem 3.38]). For future reference, we note that $(X,d)$ is $0$ -hyperbolic if and only if it satisfies the 4-point condition

(2.5) $$ \begin{align} d(w,x) + d(y,z) \leq \max\{d(w,y)+d(x,z),d(w,z)+d(x,y)\} \end{align} $$

for all $w,x,y,z \in X$ (see [Reference Bridson and Haefliger13, page 410] or [Reference Evans26, Lemma 3.12]). The convex hull of a subset $S \subseteq T$ of an $\mathbb {R}$ -tree is the set $\cup _{s,t \in S} [s,t]$ . The convex hull of S is itself an $\mathbb {R}$ -tree, and it is separable whenever S is separable. Consequently, any separable 0-hyperbolic metric space isometrically embeds into a separable $\mathbb {R}$ -tree. The Nagata dimension of any nontrivial $\mathbb {R}$ -tree (one that contains more than one point) is 1 with constant $\gamma \leq 5$ (indeed, by the proofs of [Reference Lang and Schlichenmaier37, Lemma 3.1, Theorem 3.2], $\gamma \leq 4\gamma _{\mathbb {R}}+1$ , where $\gamma _{\mathbb {R}}$ 1-Nagata-dimensional constant of $\mathbb {R}$ , which is easily seen to be 1). The Lipschitz free space of any $\mathbb {R}$ -tree is linearly isometric to an $L^1$ -space [Reference Godard29, Corollary 3.3].

A metric d on a set U is called an ultrametric if the inequality

$$ \begin{align*} d(u,w) \leq \max\{d(u,v),d(v,w)\} \end{align*} $$

holds for every $u,v,w \in U$ . In this case, the pair $(U,d)$ (or just U if d is understood) is called an ultrametric space. A metric space is biLipschitz equivalent to an ultrametric space if and only if it is snowflake equivalent to an ultrametric space if and only if it has Nagata dimension 0 (see, for example, [Reference Brodskiy, Dydak, Higes and Mitra14, Theorem 3.3]). Every ultrametric space is 0-hyperbolic and thus isometrically embeds into an $\mathbb {R}$ -tree. The Lipschitz free space of any infinite, separable ultrametric space is isomorphic to $\ell ^1$ [Reference Cúth and Doucha18, Theorem 2].

It is well-known that the Gromov boundary of an $\mathbb {R}$ -tree is (biLipschitz equivalent to) an ultrametric space (see, for example, [Reference Ibragimov31, page 320]). In the proof of Theorem C (via Theorem 5.7), we will need a converse statement that the hyperbolic filling of a bounded, complete ultrametric space is roughly biLipschitz equivalent to an $\mathbb {R}$ -tree. This fact is also well-known (see, for example, [Reference Martínez-Pérez40, Proposition 3.2]), but we will require the rough biLipschitz map from the Bonk-Schramm filling $\operatorname {\mathrm {Hyp}}(U)$ to the $\mathbb {R}$ -tree to be measurable and to map separable sets to separable sets. This is proved in the next lemma. The $\mathbb {R}$ -tree is constructed using a common hyperbolic filling technique similar to [Reference Buyalo and Schroeder15, Chapter 6].

For this proof, we need to recall some definitions. Let $\theta>0$ . A subset A of a metric space $(X,d)$ is $\theta $ -separated if $d(a,b)> \theta $ for all $a \neq b \in A$ . A maximal $\theta $ -separated subset of a metric space is one that is $\theta $ -separated and not properly contained in any other $\theta $ -separated subset. Equivalently, a $\theta $ -separated subset $A \subseteq X$ is maximal if $\{B_\theta (a)\}_{a\in A}$ is a cover of X. By Zorn’s lemma, any point in a metric space is contained in some maximal $\theta $ -separated subset. Note that a subset of a metric space is uniformly discrete if and only if it is $\theta $ -separated for some $\theta>0$ .

Proof of Lemma 2.8.

First, we observe that once a rough biLipschitz embedding $f: \operatorname {\mathrm {Hyp}}(U) \to T'$ into an $\mathbb {R}$ -tree $(T',d)$ has been produced, the image will automatically be coarsely dense in its convex hull T (which is another $\mathbb {R}$ -tree) due to the fact that $\operatorname {\mathrm {Hyp}}(U)$ is coarsely path-connected. More precisely, by [Reference Bonk and Schramm12, Theorem 7.2], $\operatorname {\mathrm {Hyp}}(U)$ is roughly geodesic, and thus for each $x,y \in \operatorname {\mathrm {Hyp}}(U)$ , we can find a sequence of points $\{x_i\}_{i=0}^m \subseteq \operatorname {\mathrm {Hyp}}(U)$ such that $x_0 = x$ , $x_m = y$ , and for each $1 \leq i \leq m$ , $\rho _U(x_{i-1},x_i) \leq K'$ for some $K' < \infty $ (independent of $x,y$ ). Then since f is roughly Lipschitz, $d(f(x_{i-1}),f(x_i)) \leq K$ for some $K<\infty $ (independent of $x,y$ ). Hence, the K-neighborhood of $f(\operatorname {\mathrm {Hyp}}(U))$ contains $\cup _{i=1}^m [f(x_{i-1}),f(x_i)]$ . This letter set is a topological curve containing $\{f(x),f(y)\}$ , and hence by uniqueness of arcs in $T'$ , it contains $[f(x),f(y)]$ . Since $x,y \in \operatorname {\mathrm {Hyp}}(U)$ were arbitrary, this shows that $f(\operatorname {\mathrm {Hyp}}(U))$ is K-coarsely dense in its convex hull. It remains to produce a rough biLipschitz embedding of $\operatorname {\mathrm {Hyp}}(U)$ into an $\mathbb {R}$ -tree.

Let $d_U$ denote the metric on U and $k_0 := \min \{k\in \mathbb {Z}: \operatorname {\mathrm {diam}}(U) \leq e^k\}$ . For this proof, it will be more convenient to not work with $\operatorname {\mathrm {Hyp}}(U)$ exactly as it is defined in (2.3), but instead to replace the underlying set $U \times (0,\operatorname {\mathrm {diam}}(U)]$ with the set $U \times (0,e^{k_0}]$ . Obviously, the inclusion $U \times (0,\operatorname {\mathrm {diam}}(U)] \subseteq U \times (0,e^{k_0}]$ is an isometric embedding when each set is equipped with the metric $\rho _U$ defined in (2.4). Therefore, to prove the lemma, we can take $\operatorname {\mathrm {Hyp}}(U)$ to be the set $U \times (0,e^{k_0}]$ equipped with the metric $\rho _U$ .

For each $k \leq k_0$ , choose a maximal $e^k$ -separated subset $N_k$ of U. Note that $N_k$ is countable whenever U is separable. Since $\operatorname {\mathrm {diam}}(U) \leq e^{k_0}$ , $N_{k_0}$ is a singleton. By maximality of $N_k$ , for any $u \in U$ , there exists $w \in N_k$ such that $d_U(u,w) \leq e^k$ . Additionally, for any $u_1,u_2 \in U$ and $w_1, w_2 \in N_k$ with $d_U(u_i,w_i) \leq e^k$ , the ultrametric inequality implies

$$ \begin{align*} d_U(w_1,w_2) \leq \max\{d_U(w_1,u_1),d_U(u_1,u_2),d_U(u_2,w_2)\} \leq \max\{e^k,d_U(u_1,u_2)\}. \end{align*} $$

Now, if $w_1 \neq w_2$ , then this inequality and the $e^k$ -separation of $N_k$ imply $\max \{e^k,d_U(u_1,u_2)\}\ = d_U(u_1,u_2)$ , and thus, we get

$$ \begin{align*} d_U(w_1,w_2) \leq d_U(u_1,u_2). \end{align*} $$

Obviously, this inequality also holds if $w_1=w_2$ . In particular, if we take $u_1=u=u_2$ , this inequality shows that there is a unique $w \in N_k$ with $d_U(u,w) \leq e^k$ . Putting this all together, we get that there exists a unique map $\pi _k: U \to N_k$ satisfying

(2.6) $$ \begin{align} d_U(u,\pi_k(u)) \leq e^k, \end{align} $$

and this map is 1-Lipschitz. One can quickly deduce from this that the tower property

(2.7) $$ \begin{align} \pi_{k} \circ \pi_{j} = \pi_k \end{align} $$

holds for all $j \leq k \leq k_0$ , and additionally that

(2.8) $$ \begin{align} \pi_k(u) = u \end{align} $$

for all $u \in N_k$ .

Set $N := \cup _{k \leq k_0} (N_k \times \{e^k\}) \subseteq U \times (0,e^{k_0}] = \operatorname {\mathrm {Hyp}}(U)$ . Note that N is countable whenever U is separable. It can be quickly checked that for any two distinct elements $x=(u_x,e^{k_x}),y=(u_y,e^{k_y})$ of N, $\rho _U(x,y) \geq 1$ if $k_x \neq k_y$ and $\rho _U(x,y) \geq 2\log (2)$ if $k_x = k_y$ , and hence, in both cases we have

(2.9) $$ \begin{align} \rho_U(x,y) \geq 1. \end{align} $$

For each $x = (u_x,e^{k_x}), y = (u_y,e^{k_y}) \in N$ , define $k(x,y) := \min \{k \geq k_x \vee k_y: \pi _k(u_x) = \pi _k(u_y)\}$ (note that this minimum is over a nonempty set since the range of $\pi _{k_0}$ is a singleton). For future reference, we record a couple of facts. The definition of $k(x,y)$ , (2.6), and the ultrametric inequality imply that

(2.10) $$ \begin{align} d_U(u_x,u_y) \leq e^{k(x,y)}, \end{align} $$

and the definition of $k(x,y)$ and $e^{k(x,y)-1}$ -separation of $N_{k(x,y)-1}$ imply that

$$ \begin{align*} k(x,y) = \max\{k_x,k_y\} \hspace{.25in} \text{or} \hspace{.25in} e^{k(x,y)-1} < d_U(u_x,u_y), \end{align*} $$

which in turn implies

(2.11) $$ \begin{align} \max\{e^{k_x},e^{k_y},d_U(u_x,u_y)\}> e^{k(x,y)-1}. \end{align} $$

Define the symmetric function $\rho _U^N$ on N by

$$ \begin{align*} \rho_U^N(x,y) := 2k(x,y)-k_x-k_y. \end{align*} $$

One can use the definition of $k(x,y)$ and (2.8) to verify that $\rho _U^N(x,y) = 0$ if and only if $x=y$ . Indeed, $x=y \implies \rho _U^N(x,y) = 0$ is clear. Now suppose that $\rho _U^N(x,y) = 0$ . Then $k(x,y) = \frac {k_x+k_y}{2}$ by the definition of $\rho _U^N$ , and $k(x,y) \geq k_x\vee k_y$ by the definition of $k(x,y)$ . Together, these imply that $k(x,y) = k_x = k_y$ . Thus, $u_x,u_y \in N_{k(x,y)}$ , which by (2.8) implies that $\pi _{k(x,y)}(u_x) = u_x$ and $\pi _{k(x,y)}(u_y) = u_y$ . Together with the definition of $k(x,y)$ , this implies $u_x=u_y$ . Hence, $x=y$ . Since $\rho _U^N$ is integer-valued, we get as a consequence that

(2.12) $$ \begin{align} \rho_U^N(x,y) \geq 1 \end{align} $$

for all $x \neq y \in N$ . The remainder of the proof will proceed as follows. We will prove that

1. (i) there is a rough biLipschitz embedding $(\operatorname {\mathrm {Hyp}}(U),\rho _U) \to (N,\rho _U)$ that is measurable and maps separable subsets to separable subsets,

2. (ii) the identity map $(N,\rho _U) \to (N,\rho _U^N)$ satisfies the two-sided estimate $\frac {1}{3}\rho _U^N(x,y) \leq \rho _U(x,y) \leq (2\log (2)+1)\rho _U^N(x,y)$ for all $x,y \in N$ , and

3. (iii) $(N,\rho _U^N)$ is a 0-hyperbolic metric space.

Since 0-hyperbolic spaces isometrically embed into $\mathbb {R}$ -trees, the composition of the maps produced in these steps yields a rough biLipschitz embedding of $\operatorname {\mathrm {Hyp}}(U)$ into an $\mathbb {R}$ -tree that is measurable and maps separable subsets to separable subsets.

We begin with (i). The construction comes in two stages. We first define $U' := \cup _{k\leq k_0} (U \times \{e^k\}) \subseteq \operatorname {\mathrm {Hyp}}(U)$ and a map $\pi ': \operatorname {\mathrm {Hyp}}(U) \to U'$ defined by

$$ \begin{align*} \pi'(u,h) := (u,e^{\lfloor\log(h)\rfloor}). \end{align*} $$

Obviously, this map is measurable, maps separable subsets to separable subsets, and satisfies

$$ \begin{align*} \rho_U(\pi'(u,h),(u,h)) \leq 1. \end{align*} $$

It is easy to check that this inequality implies that $\pi '$ is a rough biLipschitz embedding. It therefore remains to construct a rough biLipschitz embedding $\pi _N: U' \to N$ that is measurable and maps separable subsets to separable subsets. We define such a map by

$$ \begin{align*} \pi_N(u,e^k) := (\pi_k(u),e^k). \end{align*} $$

Since each $\pi _k$ is 1-Lipschitz, the map $\pi _N$ is 1-Lipschitz, and hence, it is also measurable and maps separable subsets to separable subsets. Additionally,

$$ \begin{align*} \rho_U(\pi_N(u,e^k),(u,e^k)) = 2\log(e^{-k}d_U(\pi_k(u),u)+1) \overset{({2.6})}{\leq} 2\log(2). \end{align*} $$

As before, such an inequality implies that $\pi _N$ is a rough biLipschitz embedding.

We proceed to the proof of (ii). Let $x=(u_x,e^{k_x}),y=(u_y,e^{k_y})$ be distinct elements of N. Then we have the upper bound

$$ \begin{align*} \rho_U(x,y) &= 2\log\left(\frac{d_U(u_x,u_y)+e^{k_x} \vee e^{k_y}}{\sqrt{e^{k_x}e^{k_y}}}\right) \\ &\overset{({2.10})}{\leq} 2\log\left(\frac{e^{k(x,y)}+e^{k_x} \vee e^{k_y}}{\sqrt{e^{k_x}e^{k_y}}}\right) \\ &\leq 2\log\left(\frac{e^{k(x,y)}+e^{k(x,y)}}{\sqrt{e^{k_x}e^{k_y}}}\right) \\ &= 2\log(2)+\rho_U^N(x,y) \\ &\overset{({2.12})}{\leq} (2\log(2)+1)\rho_U^N(x,y). \end{align*} $$

For the lower bound, we need to split into two cases: (a) $0 < \rho _U^N(x,y) \leq 3$ and (b) $\rho _U^N(x,y) \geq 3$ . In case (a), we have

$$ \begin{align*} \rho_U(x,y) \overset{({2.9})}{\geq} 1 \geq \frac{1}{3}\rho_U^N(x,y), \end{align*} $$

and in case (b), we have

$$ \begin{align*} \rho_U(x,y) &= 2\log\left(\frac{d_U(u_x,u_y)+e^{k_x} \vee e^{k_y}}{\sqrt{e^{k_x}e^{k_y}}}\right) \\ &\overset{({2.11})}{>} 2\log\left(\frac{e^{k(x,y)-1}}{\sqrt{e^{k_x}e^{k_y}}}\right) \\ &= \rho_U^N(x,y)-2 \\ &\geq \rho_U^N(x,y)-\frac{2}{3}\rho_U^N(x,y) \\ &= \frac{1}{3}\rho_U^N(x,y). \end{align*} $$

This finishes the proof of (ii).

We conclude by proving (iii). The first step toward this end is to prove that $k(\cdot ,\cdot )$ satisfies the inequality

(2.13) $$ \begin{align} k(x,z) \leq \max\{k(x,y),k(y,z)\} \end{align} $$

for every $x,y,z \in N$ . Let $x=(u_x,e^{k_x}),y=(u_y,e^{k_y}),z=(u_z,e^{k_z}) \in N$ . Let $k_* := \max \{k(x,z),k(x,y),k(y,z)\}$ . If $k_* \neq k(x,z)$ , then (2.13) is proved. Thus, we may assume that $k_* = k(x,z)$ . Suppose that (2.13) fails, and hence, $k(x,y),k(y,z) \leq k_*-1$ . This implies that $k_x,k_y,k_z \leq k_*-1$ and, by (2.7), that $\pi _{k_*-1}(u_x) = \pi _{k_*-1}(u_y) = \pi _{k_*-1}(u_z)$ , which in turn implies that $k(x,z) \leq k_*-1$ . This contradicts $k_* = k(x,z)$ . Hence, (2.13) must hold.

We are now in a position to verify (2.5) for $\rho _U^N$ . Note that (2.5) already implies the triangle inequality, and thus, the proof of (iii) is complete upon verification of (2.5). It is immediate from the definition of $\rho _U^N$ and some simple cancellations that (2.5) is equivalent to

(2.14) $$ \begin{align} k(w,x)+k(y,z) \leq \max\{k(w,y)+k(x,z),k(w,z)+k(x,y)\} \end{align} $$

for all $w,x,y,z \in N$ . Notice that (2.14) is invariant under each of the permutations $w \leftrightarrow x$ and $(w,x) \leftrightarrow (y,z)$ . We will use these symmetries in the ensuing argument to reduce the number of cases needing verification.

Let $w,x,y,z \in N$ . By the $(w,x) \leftrightarrow (y,z)$ symmetry, we may assume that

(2.15) $$ \begin{align} k(y,z) \leq k(w,x). \end{align} $$

Applying (2.13), we have that $k(w,x) \leq \max \{k(w,y),k(x,y)\}$ . Using this and the $w \leftrightarrow x$ symmetry (which preserves (2.15)), we may assume that

(2.16) $$ \begin{align} k(w,x) \leq k(w,y). \end{align} $$

Applying (2.13) again, we have

(2.17) $$ \begin{align} k(w,x) &\leq k(x,z) \hspace{.25in} \text{or} \end{align} $$

(2.18) $$ \begin{align} k(w,x) &\leq k(w,z).\qquad \end{align} $$

In the case where (2.17) holds, we have

$$ \begin{align*} k(w,x)+k(y,z) \overset{({2.15})}{\leq} 2k(w,x) \overset{({2.16}),({2.17})}{\leq} k(w,y) + k(x,z), \end{align*} $$

which verifies (2.14). We may now assume that (2.18) holds. Applying (2.13) yet again, we have that

(2.19) $$ \begin{align} k(y,z) &\leq k(x,y) \hspace{.25in} \text{or} \end{align} $$

(2.20) $$ \begin{align} k(y,z) &\leq k(x,z).\ \qquad \end{align} $$

In the case where (2.19) holds, we have

$$ \begin{align*} k(w,x)+k(y,z) \overset{({2.18}),({2.19})}{\leq} k(w,z) + k(x,y), \end{align*} $$

and in the case where (2.20) holds, we have

$$ \begin{align*} k(w,x)+k(y,z) \overset{({2.16}),({2.20})}{\leq} k(w,y) + k(x,z). \end{align*} $$

In both cases, (2.14) is proved.

3.1 Random maps and stochastic embeddings

A random map from a pointed set $(X,x_0)$ to a pointed topological space $(Y,y_0)$ is a $Y^X$ -valued random variable that is pointwise measurable: a measurable function from some probability space $(\Omega ,\mathcal {F},\mathbb {P})$ to the topological space $Y^X$ (equipped with the product topology). Sometimes we will denote a random map by $\{\phi _\omega : X \to Y\}_{\omega \in \Omega }$ , and other times we will suppress the underlying probabilistic data and just write $\phi : X \to Y$ . There are two additional properties we require as part of the definition of random maps $\phi $ . The first is that $\phi $ is pointwise essentially-separably-valued: for every $x \in X$ , there exists a separable subset $S \subseteq Y$ and a $\mathbb {P}$ -null net N such that $\phi _\omega (x) \in S$ for every $\omega \in \Omega \setminus N$ . This crucial technical requirement ensures Bochner measurability of associated Banach-space-valued functions defined in the next paragraph, and it is also needed for other basic constructions in this section. Of course, this condition is automatic if Y is itself separable, which will often be the case. The second is that $\phi $ is almost surely basepoint-preserving, meaning that for $\mathbb {P}$ -almost every $\omega \in \Omega $ , it holds that $\phi _\omega (x_0) = y_0$ . This condition will be necessary to define an induced bounded linear map $T_\phi $ on $\operatorname {\mathrm {LF}}(X)$ when X and Y are pointed metric spaces. Whenever $W,X$ are pointed sets, $Y,Z$ are pointed topological spaces, $f_1: W \to X$ is a basepoint-preserving map, $f_2: Y \to Z$ is a basepoint-preserving continuous map, and $\{\phi _\omega : X \to Y\}_{\omega \in \Omega }$ is a random map, it holds that $\{f_2 \circ \phi _\omega \circ f_1: W \to Z\}_{\omega \in \Omega }$ is a random map.

Let $\{\phi _\omega : X \to Y\}_{\omega \in \Omega }$ be a random map between pointed metric spaces $(X,d_X)$ , $(Y,d_Y)$ and $D< \infty $ . The random map is D-Lipschitz in expectation if for every $x,y\in X$ , the inequality $\mathbb {E}_\omega [d_Y(\phi _\omega (x),\phi _\omega (y))] \leq D d_X(x,y)$ holds. In this case, there is an associated basepoint-preserving D-Lipschitz map $T_\phi : X \to L^1(\Omega ;\operatorname {\mathrm {LF}}(Y))$ defined by $T_\phi (x) := \{\delta _{\phi _\omega (x)}\}_{\omega \in \Omega }$ . The pointwise measurability and essential-separable-valuedness of $\phi $ , together with Pettis’s measurability theorem [Reference Diestel, Uhl and Pettis20, Theorem 1.2], ensure that $T_\phi (x)$ is Bochner measurable. We also write $T_\phi $ to denote the induced D-bounded linear map $\operatorname {\mathrm {LF}}(X) \to L^1(\Omega ;\operatorname {\mathrm {LF}}(Y))$ . The random map is a stochastic biLipschitz embedding of distortion D and scaling factor $s\in (0,\infty )$ if $\phi $ is $sD$ -Lipschitz in expectation and, for all $x,y \in X$ , it holds that $d_Y(\phi _\omega (x),\phi _\omega (y)) \geq sd_X(x,y)$ for $\mathbb {P}$ -almost every $\omega \in \Omega $ . We refer to this latter property as almost-sure noncontractivity. We will see in Theorem 3.1 that this property implies that the induced map $T_\phi $ is a D-isomorphic embedding. If there are $D,D',K<\infty $ such that, for every $x,y \in X$ , we have $\mathbb {E}_\omega [d_Y(\phi _\omega (x),\phi _\omega (y))] \leq D d_X(x,y) + K$ and $d_Y(\phi _\omega (x),\phi _\omega (y)) \geq (D')^{-1}d_X(x,y)-K$ for $\mathbb {P}$ -a.e. $\omega \in \Omega $ , then $\phi $ is a stochastic rough biLipschitz embedding. If we can choose $D=D'=1$ , then $\phi $ is a stochastic rough isometric embedding.

Whenever X is countable and $\{\phi _\omega : X \to Y\}_{\omega \in \Omega }$ is a stochastic rough biLipschitz embedding, then by passing to a $\mathbb {P}$ -full-measure subset, we may assume that there exists a separable subset $S \subseteq Y$ such that, for every $\omega \in \Omega $ , it holds that $\phi _\omega (X) \subseteq S$ , $\phi _\omega $ is basepoint-preserving, and $d_Y(\phi _\omega (x),\phi _\omega (y)) \geq (D')^{-1}d_X(x,y) - K$ for every $x,y \in X$ . We will use this simple reduction in the proof of Lemma 3.5.

Whenever $W,X,Y,Z$ are pointed metric spaces, $f_1: W \to X$ is a basepoint-preserving biLipschitz embedding, $f_2: Y \to Z$ is a basepoint-preserving biLipschitz embedding, and $\{\phi _\omega : X \to Y\}_{\omega \in \Omega }$ is a stochastic biLipschitz embedding, it holds that $\{f_2 \circ \phi _\omega \circ f_1: W \to Z\}_{\omega \in \Omega }$ is a stochastic biLipschitz embedding. However, the situation is more delicate for rough biLipschitz embeddings. Suppose instead that $f_1,f_2$ are rough biLipschitz embeddings and that $\{\phi _\omega \}_{\omega \in \Omega }$ is a stochastic rough biLipschitz embedding. We still have that $\{\phi _\omega \circ f_1\}_{\omega \in \Omega }$ is a stochastic rough biLipschitz embedding. However, $\{f_2 \circ \phi _\omega \}_{\omega \in \Omega }$ need not be a stochastic rough biLipschitz embedding. This is due to the fact that rough biLipschitz embeddings are generally not measurable, nor do they map separable subsets to separable subsets, so $\{f_2 \circ \phi _\omega \}_{\omega \in \Omega }$ can fail to be a random map. However, these are the only obstructions; if $f_2$ is measurable and maps separable subsets to separable subsets, then $\{f_2 \circ \phi _\omega \}_{\omega \in \Omega }$ is indeed a random map and thus a stochastic rough biLipschitz embedding. We will be using these facts throughout the next few sections.

3.2 Spaces of random Lipschitz functions

Let X be a pointed metric space and $(\Omega ,\mathcal {F},\mathbb {P})$ a probability space. We write $L^\infty _{pwm}(\Omega ,\mathcal {F},\mathbb {P};\operatorname {\mathrm {Lip}}_0(X))$ to denote the vector space of random maps $\{f_\omega : X \to \mathbb {R}\}_{\omega \in \Omega }$ having finite seminorm

$$ \begin{align*} \|f\|_{L^\infty_{pwm}} := \sup_{x\neq y\in X} \operatorname*{\mbox{ess sup}}_{\omega\in\Omega} \dfrac{|f_\omega(y)-f_\omega(x)|}{d_X(x,y)}, \end{align*} $$

where the essential supremum is with respect to the $\mathbb {P}$ -null sets. The subscript $pwm$ stands for ‘pointwise measurable’. Notice that the seminorm is also given by

(3.1) $$ \begin{align} \|f\|_{L^\infty_{pwm}} = \sup_{I\in\mathcal{P}(X)_{\leq\aleph_0}}\operatorname*{\mbox{ess sup}}_{\omega\in\Omega} \operatorname{\mathrm{Lip}}(f_\omega\big|_{I}), \end{align} $$

where the supremum is over all countable subsets $I\subseteq X$ . We will often abbreviate notation and simply write $L^\infty _{pwm}(\Omega ;\operatorname {\mathrm {Lip}}_0(X))$ . We could create an honest normed vector space by identifying two functions whose difference has norm 0, but this offers no advantage as we will work with $L^\infty _{pwm}(\Omega ,\mathcal {F},\mathbb {P};\operatorname {\mathrm {Lip}}_0(X))$ only through its image in $L^1(\Omega ;\operatorname {\mathrm {LF}}(X))^*$ , which we explain beginning in the next paragraph.

There is a natural linear isometric map (not claimed to be surjective) from the seminormed space $L^\infty _{pwm}(\Omega ;\operatorname {\mathrm {Lip}}_0(X))$ into the normed space $L^1(\Omega ;\operatorname {\mathrm {LF}}(X))^*$ given by the pairing

$$ \begin{align*} f(\mu) := \mathbb{E}_\omega[f_\omega(\mu_\omega)]. \end{align*} $$

The definition of this expectation requires further explanation. Let $f = \{f_\omega \}_{\omega \in \Omega } \in L^\infty _{pwm}(\Omega ;\operatorname {\mathrm {Lip}}_0(X))$ with $\|f\|_{L^\infty _{pwm}} \leq 1$ and $\mu = \{\mu _\omega \}_{\omega \in \Omega } \in L^1(\Omega ;\operatorname {\mathrm {LF}}(X))$ . First of all, since $\{\mu _\omega \}_{\omega \in \Omega }$ is Bochner-measurable, there exists a countable subset $I\subseteq X$ (which we may assume contains the basepoint) such that $\mathbb {P}_\omega (\mu _\omega \in \operatorname {\mathrm {LF}}(I)) = 1$ . By (3.1), $f_\omega \big |_I$ is 1-Lipschitz for $\mathbb {P}$ -a.e. $\omega \in \Omega $ . Therefore, for $\mathbb {P}$ -a.e. $\omega $ , the pairing $f_\omega (\mu _\omega ) = f_\omega \big |_I(\mu _\omega )$ is the usual duality pairing $\operatorname {\mathrm {Lip}}_0(I) \to \operatorname {\mathrm {LF}}(I)^*$ , and we have that $|f_\omega (\mu _\omega )| \leq \|\mu _\omega \|_{\operatorname {\mathrm {LF}}}$ . Then since the map $\omega \mapsto f_\omega (\mu _\omega )$ is measurable (it is the composition of the measurable map $\omega \mapsto (f_\omega ,\mu _\omega )$ with the jointly continuous map $(g,\nu ) \mapsto g(\nu )$ , where the intermediate space is $B_{\operatorname {\mathrm {Lip}}_0(X)} \times \operatorname {\mathrm {LF}}(X)$ equipped with the weak $^*\ \times $ norm topology), the expectation $\mathbb {E}_\omega [f_\omega (\mu _\omega )]$ is a well-defined real number with $|\mathbb {E}_\omega [f_\omega (\mu _\omega )]| \leq \mathbb {E}_\omega [\|\mu _\omega \|_{\operatorname {\mathrm {LF}}}] = \|\mu \|_{L^1(\operatorname {\mathrm {LF}})}$ . This inequality demonstrates contractivity of the map.

To see that the map is isometric, let $f \in L^\infty _{pwm}(\Omega ;\operatorname {\mathrm {Lip}}_0(X))$ with $\|f\|_{L^\infty _{pwm}}> 1$ . Then we can find $x\neq y \in X$ and $A \in \mathcal {F}$ with $\mathbb {P}(A)> 0$ such that $\dfrac {|f_\omega (y)-f_{\omega }(x)|}{d_X(x,y)}> 1$ for all $\omega \in A$ . Hence, we can find $B \in \mathcal {F}$ with $B \subseteq A$ , $\mathbb {P}(B)> 0$ , and either $f_\omega (y)-f_\omega (x)> d_X(x,y)$ for all $\omega \in B$ or $f_\omega (y)-f_\omega (x) < -d_X(x,y)$ for all $\omega \in B$ . Then $\mu := \dfrac {\delta _y-\delta _x}{d_X(x,y)}\dfrac {\mathsf {1}_{B}}{\mathbb {P}(B)}$ is a norm-1 element of $L^1(\Omega ;\operatorname {\mathrm {LF}}(X))$ with $|f(\mu )|> 1$ . Throughout this section, we will always consider any function $\{f_\omega \}_{\omega \in \Omega } \in L^\infty _{pwm}(\Omega ;\operatorname {\mathrm {Lip}}_0(X))$ as an element of $L^1(\Omega ;\operatorname {\mathrm {LF}}(X))^*$ under this pairing. There are natural questions that arise concerning the closedness or density of the image of $L^\infty _{pwm}(\Omega ;\operatorname {\mathrm {Lip}}_0(X))$ in $L^1(\Omega ;\operatorname {\mathrm {LF}}(X))^*$ with respect to different topologies, but we do not pursue them here as they are unnecessary for the focus of the article.

3.3 Stochastic embeddings into $\mathbb {R}$ -trees and isomorphisms to $L^1$ -spaces

In this subsection, we work toward our base result that stochastic biLipschitz embeddability of a pointed metric space into an $\mathbb {R}$ -tree implies that its free space is isomorphic to an $L^1$ -space (Theorem 3.3). Following that theorem, there is one final lemma of the section (Lemma 3.5) that will be used to prove Theorem C (via Theorem 5.7). This lemma asserts that a stochastic rough biLipschitz embedding into an $\mathbb {R}$ -tree can be upgraded to a stochastic biLipschitz embedding into an $\mathbb {R}$ -tree when the domain is countable and uniformly discrete. We begin with a general result and then successively specialize to cases where the target space has additional structure.

Proof. Write $d_X,x_0$ for the metric and basepoint on X, and $d_Y,y_0$ for the metric and basepoint on Y. By rescalingFootnote ⁶ $d_Y$ and observing that Lipschitz free spaces over rescaled metrics are linearly isometric, we may assume that the scaling factor s in the definition of stochastic biLipschitz embedding equals 1. Let $(\Omega ,\mathcal {F},\mathbb {P})$ denote the probability space underlying the stochastic biLipschitz embedding $\{\phi _\omega \}_{\omega \in \Omega }$ . The three statements of the theorem will be proved simultaneously in the following way: we will construct a set-theoretic map $E_\infty : \operatorname {\mathrm {Lip}}_0(X) \to L^1(\Omega ;\operatorname {\mathrm {LF}}(Y))^*$ (thought of as a random extension operator) splitting $T_\phi ^*$ (i.e., $T_\phi ^* \circ E_\infty = id_{\operatorname {\mathrm {Lip}}_0(X)}$ ). In the general case with no additional assumptions on Y or X, the map $E_\infty $ will not be linear but will satisfy $\|E_\infty (f)\|_{L^1(\operatorname {\mathrm {LF}})^*} \leq \operatorname {\mathrm {Lip}}(f)$ . In the proof, we will refer to this as ‘case (1)’, and we refer to inequalities of the form $\|E(h)\| \leq L\|h\|$ by saying that ‘E is L-bounded’, even if E is only a set-theoretic map between normed spaces and not linear. Statement (1) of the theorem then follows. In the case where Y is assumed to have the K-Lipschitz-bounded, L-uniform-bounded linear extension property and X is compact, the map $E_\infty $ will be linear, K-bounded, and weak $^*$ -weak $^*$ continuous. We will refer to this as ‘case (2)’ in the proof. In this case, the preadjoint $(E_\infty )_*$ serves as the retract R. In the case where Y is assumed to have the K-Lipschitz-bounded linear extension property, the map $E_\infty $ will be linear and K-bounded. We will refer to this as ‘case (3)’ in the proof. In this case, we then use the assumption that $\operatorname {\mathrm {LF}}(X)$ is C-complemented in $\operatorname {\mathrm {LF}}(X)^{**}$ to construct the retract R. We now proceed with the construction of the random extension operator $E_\infty : \operatorname {\mathrm {Lip}}_0(X) \to L^1(\Omega ;\operatorname {\mathrm {LF}}(Y))^*$ .

Let $\mathcal {P}_{<\infty }(X)$ denote the directed set of finite subsets of X containing the basepoint, ordered by inclusion. Let $\mathscr {A}$ denote the directed set of $\sigma $ -subalgebras of $\mathcal {F}$ , ordered by inclusion, consisting of those $\mathcal {A}$ generated by some countable partition of $\Omega $ . We consider $\mathcal {P}_{<\infty }(X) \times \mathscr {A}$ as a directed set equipped with its product preorder. Let $(F,\mathcal {A}) \in \mathcal {P}_{<\infty }(X) \times \mathscr {A}$ , and suppose that $\mathcal {A}$ is generated by the countable partition $\{A_i\}_{i=1}^\infty $ of $\Omega $ (so $\{A_i\}_{i=1}^\infty $ is the set of atoms of the purely atomic $\sigma $ -algebra $\mathcal {A}$ ). For each atom $A_i \in \mathcal {A}$ with $\mathbb {P}(A_i)> 0$ , choose $\omega _{F,i} \in A_i$ such that $\phi _{\omega _{F,i}}(x_0) = y_0$ and

(3.2) $$ \begin{align} d_Y(\phi_{\omega_{F,i}}(x),\phi_{\omega_{F,i}}(y)) \geq d_X(x,y) \end{align} $$

for every $x,y \in F$ . We can make such a choice by the almost-sure basepoint preservation and almost-sure noncontractivity of $\phi $ (recalling that $s=1$ ), and the fact that F is finite (note that if F was uncountable, we may not be able to choose any $\omega _{F,i} \in A_i$ with this property). Let $E_{F,i}: \operatorname {\mathrm {Lip}}_0(\phi _{\omega _{F,i}}(F)) \to \operatorname {\mathrm {Lip}}_0(Y)$ be an extension operator (noting that each subset $\phi _{\omega _{F,i}}(F)$ contains the basepoint $y_0$ ). In case (1) with no assumption on Y, $E_{F,i}$ is constructed using the McShane extension theorem, and it is not linear but is 1-bounded. In cases (2)–(3) where Y is assumed to have the K-Lipschitz-bounded linear extension property, $E_{F,i}$ is chosen to be K-bounded and linear. In case (2), we also choose $E_{F,i}$ to be L-uniform-bounded.

Define a map $E_{(F,\mathcal {A})}: \operatorname {\mathrm {Lip}}_0(X) \to L_{pwm}^\infty (\Omega ,\mathcal {A};\operatorname {\mathrm {Lip}}_0(Y))$ by

$$ \begin{align*} E_{(F,\mathcal{A})}(f)_\omega := \sum_{i=1}^\infty E_{F,i}(f \circ \phi_{\omega_{F,i}}^{-1})\mathsf{1}_{A_i}(\omega), \end{align*} $$

where we consider $\phi _{\omega _{F,i}}^{-1}$ as a map from $\phi _{\omega _{F,i}}(F)$ to X, which is 1-Lipschitz by (3.2). Then $E_{(F,\mathcal {A})}$ is 1-bounded in case (1) and K-bounded and linear in cases (2)–(3). After composing with the natural pairing $L_{pwm}^\infty (\Omega ,\mathcal {A};\operatorname {\mathrm {Lip}}_0(Y)) \to L^1(\Omega ,\mathcal {F};\operatorname {\mathrm {LF}}(Y))^*$ , we may consider $E_{(F,\mathcal {A})}$ as having values in the dual space $L^1(\Omega ,\mathcal {F};\operatorname {\mathrm {LF}}(Y))^*$ , and define the map $E_\infty : \operatorname {\mathrm {Lip}}_0(X) \to L^1(\Omega ,\mathcal {F};\operatorname {\mathrm {LF}}(Y))^*$ by

$$ \begin{align*} E_{\infty}(f) := \mathcal{U}\text{-}w^*\lim_{(F,\mathcal{A})\to\infty} E_{(F,\mathcal{A})}(f), \end{align*} $$

where $\mathcal {U}$ is any fixed eventuality ultrafilterFootnote ⁷ on the directed set $\mathcal {P}_{<\infty }(X)\times \mathscr {A}$ , and the limit is the $\mathcal {U}$ -ultralimit with respect to the (compact) weak $^*$ topology on the ball of radius $K\operatorname {\mathrm {Lip}}(f)$ in $L^1(\Omega ,\mathcal {F};\operatorname {\mathrm {LF}}(Y))^*$ . In case (1), $E_{\infty }$ is 1-bounded and (generally) nonlinear, and in cases (2)–(3), $E_{\infty }$ is linear and K-bounded.

Our first order of business is to prove that $E_\infty $ splits $T_\phi ^*$ . Fix $f \in \operatorname {\mathrm {Lip}}_0(X)$ and $x \in X$ . Let $\varepsilon>0$ be arbitrary. Since $\phi $ is pointwise essentially-separably-valued, there exists a separable subset $S \subseteq Y$ such that $\phi _\omega (x) \in S$ for $\mathbb {P}$ -a.e. $\omega \in \Omega $ . Hence, we may find a countable, Borel measurable partition $\{S_i\}_{i=1}^\infty $ of S such that $\operatorname {\mathrm {diam}}(S_i)<\frac {\varepsilon }{K\operatorname {\mathrm {Lip}}(f)}$ for every i. Then since the map $\omega \to \phi _\omega (x)$ is measurable and takes values in S almost surely, the preimage $\{G_i\}_{i=1}^\infty $ of $\{S_i\}_{i=1}^\infty $ under this map is a countable, $\mathcal {F}$ -measurable partition of $\Omega \setminus N$ for some $\mathbb {P}$ -null set $N\in \mathcal {F}$ . Let $\mathcal {G}$ be the $\sigma $ -subalgebra of $\mathcal {F}$ generated by $\{G_i,N\}_{i=1}^\infty $ , so that $\mathcal {G} \in \mathscr {A}$ . Then, for any $\mathcal {A} \in \mathscr {A}$ with $\mathcal {A} \supseteq \mathcal {G}$ , for any atom $A_i \in \mathcal {A}$ with $\mathbb {P}(A_i)>0$ , and for any $\omega ,\omega ' \in A_i$ , it holds that

$$ \begin{align*} d_Y(\phi_\omega(x),\phi_{\omega'}(x)) < \frac{\varepsilon}{K\operatorname{\mathrm{Lip}}(f)}. \end{align*} $$

Then we get the estimate

$$ \begin{align*} &|T^*_\phi(E_\infty(f))(x) - f(x)| \\ &\hspace{6pt}= |E_\infty(f)(T_\phi(x)) - f(x)| \\ &\hspace{6pt}= \left|\left[\mathcal{U}\text{-}w^*\lim_{(F,\mathcal{A})\to\infty} E_{(F,\mathcal{A})}(f)\right](T_\phi(x)) - f(x)\right| \\ &\hspace{6pt}= \mathcal{U}\text{-}\lim_{(F,\mathcal{A})\to\infty} |E_{(F,\mathcal{A})}(f)(T_\phi(x)) - f(x)| \\ &\hspace{6pt}= \mathcal{U}\text{-}\lim_{(F,\mathcal{A})\to\infty} \left|\sum_{i=1}^\infty \int_{A_i} E_{F,i}(f \circ \phi_{\omega_{F,i}}^{-1})(\phi_\omega(x)) \mathrm{d}\mathbb{P}(\omega) - f(x)\right| \\ &\hspace{6pt}\leq \sup_{F\ni x, \mathcal{A}\supseteq\mathcal{G}} \sum_{i=1}^\infty \int_{A_i} |E_{F,i}(f \circ \phi_{\omega_{F,i}}^{-1})(\phi_\omega(x)) - E_{F,i}(f \circ \phi_{\omega_{F,i}}^{-1})(\phi_{\omega_{F,i}}(x))| \mathrm{d}\mathbb{P}(\omega) \\ &\hspace{12pt}\ + \sum_{i=1}^\infty \int_{A_i} | E_{F,i}(f \circ \phi_{\omega_{F,i}}^{-1})(\phi_{\omega_{F,i}}(x)) - f(x)| \mathrm{d}\mathbb{P}(\omega) \\ &\hspace{6pt}\leq \sup_{F\ni x,\mathcal{A}\supseteq\mathcal{G}} \sum_{i=1}^\infty \int_{A_i} K\operatorname{\mathrm{Lip}}(f)\frac{\varepsilon}{K\operatorname{\mathrm{Lip}}(f)} \mathrm{d}\mathbb{P}(\omega) + \sum_{i=1}^\infty \int_{A_i} | (f \circ \phi_{\omega_{F,i}}^{-1})(\phi_{\omega_{F,i}}(x)) - f(x)| \mathrm{d}\mathbb{P}(\omega) \\ &\hspace{6pt}= \varepsilon.\end{align*} $$

Since $\varepsilon>0$ was arbitrary, this proves $T^*_\phi (E_\infty (f))(x) = f(x)$ . Since $f \in \operatorname {\mathrm {Lip}}_0(X)$ and $x \in X$ were arbitrary, this proves $T_\phi ^* \circ E_\infty = id_{\operatorname {\mathrm {Lip}}_0(X)}$ . The proof of statement (1) is now complete, and from here on, we work in cases (2)–(3) where $E_\infty $ is linear and K-bounded.

We focus on case (2) first, using the L-uniform-boundedness of $E_{F,i}$ and compactness of X to prove that $E_\infty $ is weak $^*$ -weak $^*$ continuous. Let $f_{\alpha } \in B_{\operatorname {\mathrm {Lip}}_0(X)}$ be a net converging pointwise to $f \in B_{\operatorname {\mathrm {Lip}}_0(X)}$ . Then since X is compact, $f_\alpha \overset {\alpha \to \infty }{\to } f$ uniformly. We need to prove that, for every $\mu = \{\mu _\omega \}_{\omega \in \Omega } \in L^1(\Omega ;\operatorname {\mathrm {LF}}(Y))$ , $E_\infty (f_\alpha )(\mu ) \overset {\alpha \to \infty }{\to } E_\infty (f)(\mu )$ . By replacing $f_\alpha $ with $f_\alpha -f$ , we may assume that $f=0$ . Since the linear functionals $E_\infty (f_\alpha )$ are uniformly (in $\alpha $ ) bounded, it suffices to prove the desired convergence for all $\mu $ in a chosen subset of $L^1(\Omega ;\operatorname {\mathrm {LF}}(Y))$ whose linear span is dense. Toward this end, we consider $\mu $ of the form $\delta _y\mathsf {1}_B$ , where $y \in Y$ and $B\in \mathcal {F}$ . Let $\varepsilon>0$ be arbitrary. Since $f_\alpha \to 0$ uniformly, we may choose $\alpha _0$ large enough so that for all $\alpha \geq \alpha _0$ , we have $\|f_\alpha \|_\infty \leq \frac {\varepsilon }{L}$ . Then, for all $\alpha \geq \alpha _0$ , we have

$$ \begin{align*} \left|E_\infty(f_\alpha)(\delta_y\mathsf{1}_B)\right| &= \left|\left[\mathcal{U}\text{-}w^*\lim_{(F,\mathcal{A})\to\infty} E_{(F,\mathcal{A})}(f_\alpha)\right](\delta_y\mathsf{1}_B)\right| \\ &= \mathcal{U}\text{-}\lim_{(F,\mathcal{A})\to\infty} |E_{(F,\mathcal{A})}(f_\alpha)(\delta_y\mathsf{1}_B)| \\ &\leq \sup_{(F,\mathcal{A})} |E_{(F,\mathcal{A})}(f_\alpha)(\delta_y\mathsf{1}_B)| \\ &= \sup_{(F,\mathcal{A})} \left|\sum_{i=1}^\infty E_{F,i}(f_\alpha \circ \phi_{\omega_{F,i}}^{-1})(y) \mathbb{P}(A_i\cap B)\right| \\ &\leq \sup_{(F,\mathcal{A})} \sum_{i=1}^\infty \|E_{F,i}(f_\alpha \circ \phi_{\omega_{F,i}}^{-1})\|_\infty \mathbb{P}(A_i\cap B) \\ &\leq \sup_{(F,\mathcal{A})} \sum_{i=1}^\infty L\|f_\alpha\|_\infty \mathbb{P}(A_i\cap B) \\ &\leq \varepsilon\mathbb{P}(B) \leq \varepsilon. \end{align*} $$

Since $\varepsilon> 0$ was arbitrary, this shows $E_\infty (f_\alpha )(\delta _y\mathsf {1}_B) \overset {\alpha \to \infty }{\to } 0$ , completing the proof of weak $^*$ -weak $^*$ continuity. Then the preadjoint $(E_\infty )_*$ is K-bounded and satisfies

$$ \begin{align*} ((E_\infty)_* \circ T_\phi)^* = T_\phi^* \circ E_\infty = id_{\operatorname{\mathrm{Lip}}_0(X)} = id_{\operatorname{\mathrm{LF}}(X)}^*, \end{align*} $$

which implies $(E_\infty )_* \circ T_\phi = id_{\operatorname {\mathrm {LF}}(X)}$ . This completes the proof of statement (2).

We now find ourselves in case (3). In this final case, the lack of uniform-boundedness of $E_{F,i}$ prevents us from being able to prove weak $^*$ -weak $^*$ continuity of $E_\infty $ , and hence, we cannot use a preadjoint $(E_\infty )_*$ to retract $T_\phi $ . Naturally, the assumption that $\operatorname {\mathrm {LF}}(X)$ is complemented in $\operatorname {\mathrm {LF}}(X)^{**}$ provides an alternate route. Define the $CK$ -bounded linear map $R: L^1(\Omega ,\mathcal {F};\operatorname {\mathrm {LF}}(Y)) \to \operatorname {\mathrm {LF}}(X)$ by $R := P \circ E_\infty ^* \circ J_Y$ , where $P: \operatorname {\mathrm {LF}}(X)^{**} \to \operatorname {\mathrm {LF}}(X)$ is a C-bounded linear map retracting the canonical embedding $J_X: \operatorname {\mathrm {LF}}(X) \to \operatorname {\mathrm {LF}}(X)^{**} = \operatorname {\mathrm {Lip}}_0(X)^*$ , and $J_Y: L^1(\Omega ,\mathcal {F};\operatorname {\mathrm {LF}}(Y)) \to L^1(\Omega ,\mathcal {F};\operatorname {\mathrm {LF}}(Y))^{**}$ is the canonical embedding. It is then immediate to verify that R retracts $T_\phi $ . To see this, let us introduce the notation $J_{X^{**}}: \operatorname {\mathrm {LF}}(X)^{**} \to \operatorname {\mathrm {LF}}(X)^{****}$ and $J_{Y^{**}}: L^1(\Omega ,\mathcal {F};\operatorname {\mathrm {LF}}(Y))^{**} \to L^1(\Omega ,\mathcal {F};\operatorname {\mathrm {LF}}(Y))^{****}$ for the canonical embeddings, and, letting $J_{U}: U \to U^{**}$ denote the canonical embedding for a general Banach space U, recall that for any Banach spaces $V,W$ and bounded linear map $S: V \to W$ , we have $J_{V}^{**} \circ J_V = J_{V^{**}} \circ J_V$ and $S^{**} \circ J_V = J_W \circ S$ . Using these facts together with the definition of R and the identity $T_\phi ^* \circ E_\infty = id_{\operatorname {\mathrm {LF}}(X)^*}$ established above, we get

$$ \begin{align*} (R \circ T_\phi)^{**} \circ J_X &= P^{**} \circ E_\infty^{***} \circ J_Y^{**} \circ T_\phi^{**} \circ J_X \\ &= P^{**} \circ E_\infty^{***} \circ J_Y^{**} \circ J_Y \circ T_\phi \\ &= P^{**} \circ E_\infty^{***} \circ J_{Y^{**}} \circ J_Y \circ T_\phi \\ &= P^{**} \circ E_\infty^{***} \circ J_{Y^{**}} \circ T_\phi^{**} \circ J_X \\ &= J_X \circ P \circ E_\infty^{*} \circ T_\phi^{**} \circ J_X \\ &= J_X \circ P \circ J_X \\ &= J_X, \end{align*} $$

which implies $R \circ T_\phi = id_{\operatorname {\mathrm {LF}}(X)}$ .

We now specialize to the case where the target space has finite Nagata dimension and apply Corollary 2.6.

We now reach our base result, which essentially follows from Corollary 3.2, the fact that $\mathbb {R}$ -trees have finite Nagata dimension, and a widely used decomposition lemma of Kalton [Reference Kalton33, Lemma 4.2]. This lemma allows one, in many cases, to restrict to bounded subsets of a metric space to prove isomorphic properties of its free space.

The final result of this section states that if X is countable and uniformly discrete, then a stochastic rough biLipschitz embedding of X into an $\mathbb {R}$ -tree can be upgraded to a stochastic biLipschitz embedding. This lemma will be used in the proof of Theorem 5.7.

Proof. Let $\{\phi _\omega : X \to Y\}_{\omega \in \Omega }$ be a random map into a separable subset Y of a pointed $\mathbb {R}$ -tree $(T,d_T,q)$ and $D,D',K<\infty $ such that, for every $\omega \in \Omega $ and $x,y \in X$ , $d_T(\phi _\omega (x),\phi _\omega (y))\ \geq (D')^{-1}d_X(x,y) - K$ , $\phi _\omega (p) = q$ , and $\mathbb {E}_\omega [d_T(\phi _\omega (x),\phi _\omega (y))] \leq Dd_X(x,y) + K$ (we can ask for these things to happen for all $\omega \in \Omega $ due to countability of X). By replacing $d_T$ with $(D')^{-1}d_T$ , and noting that rescalings of $\mathbb {R}$ -trees are still $\mathbb {R}$ -trees, we may assume that $D'=1$ . Furthermore, by replacing T with the convex hull of Y, we may assume that T is separable.

Let $N_X \subseteq X$ be a maximal $2K$ -separated subset of X containing p and $N_T$ a maximal $\frac {K}{6}$ -separated subset of T containing q. Then $N_X$ and $N_T$ are countable since $X,T$ are separable. Equip each of these sets with a well-order coming from any bijection with $\mathbb {N}$ , and then define projections $\pi _X: X \to N_X$ and $\pi _T: T \to N_T$ by $\pi _X(x) = \min \{n\in N_X: d_X(n,x) \leq 2\operatorname {\mathrm {dist}}(N_X,x)\}$ and $\pi _T(t) = \min \{n\in N_T: d_T(n,t) \leq 2\operatorname {\mathrm {dist}}(N_T,t)\}$ . By maximality of $N_X$ , it holds that

(3.3) $$ \begin{align} d_X(\pi_X(x),x) \leq 4K \end{align} $$

for every $x \in X$ , and hence,

(3.4) $$ \begin{align} d_X(x,y) \leq 8K && \text{ if } \pi_X(x) = \pi_X(y), \end{align} $$

(3.5) $$ \begin{align} d_X(x,y) \leq 8K + d_X(\pi_X(x),\pi_X(y)) &\leq 5d_X(\pi_X(x),\pi_X(y)) & \text{ if } \pi_X(x) \neq \pi_X(y). \end{align} $$

By the countability of $N_T$ , it is easily seen that $\pi _T$ is measurable. The maximality of $N_T$ implies that $d_T(\pi _T(t),t) \leq \frac {K}{3}$ for every $t \in T$ . Using this, together with $2K$ -separation of $N_X$ , we have for any $n \neq m \in N_X$ and $\omega \in \Omega $ ,

(3.6) $$ \begin{align} \nonumber d_T(\pi_T(\phi_\omega(m)),\pi_T(\phi_\omega(n))) &\geq d_T(\phi_\omega(m),\phi_\omega(n)) - \frac{2K}{3} \\ \nonumber &\geq d_X(m,n) - K - \frac{2K}{3} \\ &\geq \frac{1}{6}d_X(m,n). \end{align} $$

Additionally,

(3.7) $$ \begin{align} \nonumber \mathbb{E}_\omega[d_T(\pi_T(\phi_\omega(m)),\pi_T(\phi_\omega(n)))] &\leq \mathbb{E}_\omega[d_T(\phi_\omega(m),\phi_\omega(n))] + \frac{2K}{3} \\ \nonumber &\leq Dd_X(m,n) + K + \frac{2K}{3} \\ &\leq \left(D+\frac{5}{6}\right)d_X(m,n). \end{align} $$

This shows that $\{(\pi _T \circ \phi _\omega )\big |_{N_X}: N_X \to N_T\}_{\omega \in \Omega }$ is a stochastic biLipschitz embedding. We will next use the uniform discreteness assumption to extend this to all of X by precomposing with $\pi _X$ . However, we need to enlarge our target tree. There are many ways to do this – we prioritize simplicity of the exposition over optimization of the distortion constants involved. Toward this end, we employ a star-type construction.

Define the metric space $(Z,d_Z)$ by

$$ \begin{align*} Z &:= X \times N_T, \\ d_Z((x,m),(y,n)) & := \begin{cases} 0 & (x,m)=(y,n) \\ 1+d_T(m,n) & (x,m) \neq (y,n) \end{cases}. \end{align*} $$

One can verify that Z is countable, uniformly discrete and 0-hyperbolic since X is countable and $N_T$ is countable and 0-hyperbolic. Thus, Z is isometric to a countable, uniformly discrete subset of an $\mathbb {R}$ -tree (see $\S $ 2.6). Equip Z with basepoint $(p,q)$ . We will show that the random map $\{\psi _\omega : X \to Z\}_{\omega \in \Omega }$ given by

$$ \begin{align*} \psi_\omega(x) := (x,(\pi_T \circ \phi_\omega \circ \pi_X)(x)) \end{align*} $$

is a stochastic biLipschitz embedding. First note that $\psi $ is indeed pointwise measurable and essentially-separably-valued since $\pi _T$ is measurable and Z is countable. It is also basepoint-preserving since $p\in N_X$ and $q\in N_T$ . Now let $\theta> 0$ such that $d_X(x,y)> \theta $ for every $x\neq y \in X$ . Let $x\neq y \in X$ . We treat two cases: (i) $\pi _X(x) = \pi _X(y)$ and (ii) $\pi _X(x) \neq \pi _X(y)$ . Assume case (i) holds. Then for every $\omega $ , we have

$$ \begin{align*} d_{Z}(\psi_\omega(x),\psi_\omega(y)) = 1 \overset{({3.4})}{\geq} \frac{1}{8K}d_X(x,y). \end{align*} $$

Likewise,

$$ \begin{align*} \mathbb{E}_\omega[d_{Z}(\psi_\omega(x),\psi_\omega(y))] = 1 \leq \theta^{-1}d_X(x,y). \end{align*} $$

Now assume that we are in case (ii). Then by (3.5), $d_X(x,y) \leq 5d_X(\pi _X(x),\pi _X(y))$ . Thus, for every $\omega \in \Omega $ , we have

$$ \begin{align*} d_{Z}(\psi_\omega(x),\psi_\omega(y)) &= 1+d_{T}(\pi_T(\phi_\omega(\pi_X(x))),\pi_T(\phi_\omega(\pi_X(y)))) \\ &\overset{({3.6})}{\geq} 1 + \frac{1}{6}d_X(\pi_X(x),\pi_X(y)) \\ &\geq 1 + \frac{1}{30}d_X(x,y) \\ &> \frac{1}{30}d_X(x,y). \end{align*} $$

Likewise,

$$ \begin{align*} \mathbb{E}_\omega[d_{Z}(\psi_\omega(x),\psi_\omega(y))] &= 1 + \mathbb{E}_\omega[d_{T}(\pi_T(\phi_\omega(\pi_X(x))),\pi_T(\phi_\omega(\pi_X(y))))] \\ &\overset{({3.7})}{\leq} 1 + \left(D+\frac{5}{6}\right)d_X(\pi_X(x),\pi_X(y)) \\ &\overset{({3.3})}{\leq} 1 + \left(D+\frac{5}{6}\right)(d_X(x,y) + 8K) \\ &\leq \left(\theta^{-1}+\left(D+\frac{5}{6}\right)(1+8K\theta^{-1})\right)d_X(x,y).\\[-34pt] \end{align*} $$