The globalization theorem for $\textrm{CD}(K, N)$ on locally finite spaces

Abstract

We establish the local-to-global property of the synthetic curvature-dimension condition for essentially non-branching locally finite metric-measure spaces, extending the work [Cavalletti and Milman in Invent Math 226(1):1–137, 2021].

1 Introduction

The curvature-dimension condition, or shortly $\textrm{CD}(K, N)$ on metric-measure spaces $(X,d,\mathfrak {m})$, was introduced by Lott-Villani and Sturm in the seminal papers [12, 16, 17].

A natural but longstanding question is whether such a synthetically defined condition can be checked locally. Cavalletti–Milman’s recent paper [6] gives a positive answer to this globalization problem under the assumption $\mathfrak {m}(X)=1$, which was conjectured to be merely technical there. In this paper, we extend this result to infinite-volume spaces.

Theorem 1.1

(Local-to-Global property) Let $(X, d, \mathfrak {m})$ be an essentially non-branching metric-measure space¹ with a locally finite Borel measure $\mathfrak {m}$. Assume that $(\textrm{supp}(\mathfrak {m}), d)$ is a length space. Then if $(X, d, \mathfrak {m})$ verifies $\textrm{CD}_{\textrm{loc}} (K, N)$ for $K \in \mathbb {R}$ and $N \in (1, \infty )$, it verifies $\textrm{CD}(K, N)$.

Here immediately follow several useful equivalence results once we apply Theorem 1.1 to Section 13.1 and 13.2 in [6].

Corollary 1.2

Let $(X,d,\mathfrak {m})$ be a metric-measure space with a locally finite Borel measure $\mathfrak {m}$. Then²

• if $(X,d,\mathfrak {m})$ is essentially non-branching, it holds $\textrm{CD}^{*}(K,N)$ if and only if it holds $\textrm{CD}(K,N)$;

• $(X,d,\mathfrak {m})$ holds $\textrm{RCD}^{*}(K,N)$ if and only if it holds $\textrm{RCD}(K,N)$;

• if $(\textrm{supp}(\mathfrak {m}),d)$ is a length space, it holds $\textrm{RCD}_{\textrm{loc}}(K,N)$ if and only if it holds $\textrm{RCD}(K,N)$.

In [6], Cavalletti and Milman introduced the $\textrm{CD}^1(K,N)$ condition on finite-volume spaces, which roughly requires transport rays of signed distance functions to hold the one-dimensional $\textrm{CD}(K,N)$. Then they showed that under suitable assumptions, $\textrm{CD}^1(K,N)$ implies $\textrm{CD}(K,N)$. Similarly, in this paper, we tailor the definition of $\textrm{CD}^1(K,N)$, adapting it to the infinite-volume situation by assuming conditional measures to be uniformly-locally finite. Then we split the problem into two independent ones: $\textrm{CD}_{\textrm{loc}}(K,N)\Rightarrow \textrm{CD}^1(K,N)$ and $\textrm{CD}^1(K,N)\Rightarrow \textrm{CD}(K,N)$.

For the first part, we normalize the reference measure as in [9] and show that the needle/ray-decomposition developed in [4, 5] still localizes the curvature-dimension condition to rays. For the second part, we show under the given definition, $\textrm{CD}^1(K,N)$ space is locally finite, geodesic and satisfying $\textrm{MCP}(K,N)$. Then we briefly present the strategy and arguments fulfilling the implication of $\textrm{CD}(K,N)$ in locally finite spaces, which is basically the same as in [6] under modifications. Indeed, the validity is ensured basically by three aspects: (1) owing to the local finiteness of conditional measures and the properness of the space, problems are reduced to the finite-volume case by taking exhaustion by compacts subsets; (2) $\textrm{CD}(K,N)$ is reduced to a path-wise inequality along Kantorovich geodesics by the non-branchingness, hence the one-dimensional analysis in [6, Part III] is not affected by the global infinity of $\mathfrak {m}$; (3) temporal derivatives of potentials, investigated in [6, Part I], do not rely on the measure structure.

Accordingly, the rest of this paper is organized as follows.

In Sect. 2, we recall central definitions and preliminary results.

In Sect. 3, we discuss the ray decomposition and define $\textrm{CD}^1(K,N)$ in the locally finite setting. We show under assumptions of Theorem 1.1, $\textrm{CD}_{\textrm{loc}}(K,N)$ implies $\textrm{CD}^1(K,N)$.

In Sect. 4, we discuss the implication $\textrm{CD}^1(K,N)\Rightarrow \textrm{CD}(K,N)$.

2 Preliminaries

2.1 Curvature-Dimension Condition

A triple $(X, d, \mathfrak {m})$ always stands for a metric measure space consisting of a Polish metric space equipped with the Borel $\sigma$-algebra and a locally finite Borel measure $\mathfrak {m}$ (i.e. for any $x \in X$, $\mathfrak {m} (B_r(x))<\infty$ for some $r > 0$). Denote $\mathcal {P}_2 (X)$ as the space of probability measures with finite variances and $\mathcal {P}_2 (X, \mathfrak {m})$ the subspace of all absolutely continuous measures w.r.t. $\mathfrak {m}$.

An optimal plan between $\mu _0, \mu _1 \in \mathcal {P}_2(X)$ is a coupling $\pi \in \mathcal {P} (X \times X)$ minimizing the cost

$$\begin{aligned} C (\omega ) = \int _{X \times X} \frac{{\,\textrm{d}}^2 (x, y)}{2} \omega ({\,\textrm{d}}x {\,\textrm{d}}y) \end{aligned}$$

among all $\omega \in \mathcal {P} (X \times X)$ having $\mu _0$ and $\mu _1$ as the first and second marginal. Denote by $\textrm{Opt}(\mu _0, \mu _1)$ the set of all optimal plans between $\mu _0$ and $\mu _1$. There is a $d^2/2$-concave function $\varphi : X \rightarrow \mathbb {R}$ called a Kantorovich potential associated to optimal plan $\pi$ satisfying

$$\begin{aligned} \varphi (x) + \varphi ^c (y) = \frac{{\,\textrm{d}}^2 (x, y)}{2}, \quad \pi -a.e. (x, y) \in X \times X \end{aligned}$$

where $\varphi ^c$ is the conjugate potential of $\varphi$ given by

$$\begin{aligned} \varphi ^c (y) := \inf _{z \in X} \left( \frac{{\,\textrm{d}}^2 (y, z)}{2} - \varphi (z) \right) . \end{aligned}$$

Define the $L^2$-Wasserstein distance between probabilities as $W_2 (\mu _0,\mu _1) := \sqrt{C (\pi )}$ for $\pi \in \textrm{Opt}(\mu _0,\mu _1)$, which makes $\mathcal {P}_2 (X)$ a Polish metric space. Denote $\textrm{Geo}(X)$ the set of all constant speed geodesic $\gamma :[0,1]\rightarrow X$. When endowed with the supremum distance, it is a Polish metric space.

If (X, d) is geodesic, so is $(\mathcal {P}_2 (X), W_2)$ (see [1, Theorem 2.10]). Let $e_t:\textrm{Geo}(X) \ni \gamma \mapsto \gamma _t \in X$ be the evaluation map, and $\ell (\gamma )$ be the length. Then for any $\mu _0, \mu _1 \in \mathcal {P}_2 (X)$, there exists a probability measure $\nu$ (referred to as an optimal dynamical plan) on $\textrm{Geo}(X)$ s.t.

• $(e_i)_{\#} \nu = \mu _i$, $i = 0, 1$ and $(e_0, e_1)_{\#} \nu \in \textrm{Opt}(\mu _0, \mu _1)$;

• $[0, 1] \ni t \mapsto \mu _t := (e_t)_{\#} \nu$ is a constant speed geodesic in $(\mathcal {P}_2 (X), W_2)$;

• $\nu$ is concentrated on the set of Kantorovich geodesics

  $$\begin{aligned} G_{\varphi } := \left\{ \gamma \in \textrm{Geo}(X): \varphi (\gamma _0) + \varphi ^c (\gamma _1) = \ell ^2 (\gamma ) / 2\right\} . \end{aligned}$$

Denote by $\textrm{OptGeo}(\mu _0,\mu _1)$ the set of all optimal dynamical plans.

Definition 2.1

Define the N-Rényi entropy $\mathcal {E}_N$ of any $\mu \in \mathcal {P}_2 (X, \mathfrak {m})$ by

$$\begin{aligned} \mathcal {E}_N (\mu ) := \int _X \rho ^{1 - 1 / N} (x) {\,\textrm{d}}\mathfrak {m}, \quad \rho := \frac{{\,\textrm{d}}\mu }{{\,\textrm{d}}\mathfrak {m}}. \end{aligned}$$

Given $N \in (1, \infty )$, define by the following two distortion coefficients

$$\begin{aligned} \sigma _{K, N}^{(t)} (\theta )&= \frac{\sin \left( t \theta \sqrt{\frac{K}{N}} \right) }{\sin \left( \theta \sqrt{\frac{K}{N}}\right) } := \left\{ \begin{array}{ll} \frac{\sin \left( t \theta \sqrt{\frac{K}{N}} \right) }{\sin \left( \theta \sqrt{\frac{K}{N}} \right) } &{} K > 0, \quad 0<\theta<\pi \sqrt{\frac{N}{K}}\\ t &{} K = 0,\quad 0< \theta<\infty \\ \frac{\sinh \left( t \theta \sqrt{\frac{- K}{N}} \right) }{\sinh \left( \theta \sqrt{\frac{- K}{N}} \right) } &{} K< 0, \quad 0<\theta <\infty \end{array} \right. ,\\ \tau _{K, N}^{(t)} (\theta )& := t^{1 / N} \sigma _{K, N - 1}^{(t)} (\theta )^{1 - 1 / N}. \end{aligned}$$

Definition 2.2

Let $(X, d, \mathfrak {m})$ be a metric-measure space.

• $(X, d, \mathfrak {m})$ is said to verify $\textrm{CD}(K, N)$, if for all $\mu _0, \mu _1 \in \mathcal {P}_2 (X, \mathfrak {m})$, there exists $\nu \in \textrm{OptGeo}(\mu _0, \mu _1)$ so that for all $t \in [0, 1]$, $\mu _t = (e_t)_{\#} \nu \ll \mathfrak {m}$, and for all $N' \ge N$:

  $$\begin{aligned}{} \mathcal {E}_{N'} (\mu _t)& \ge \int _{X \times X} \tau _{K, N^{\prime }}^{(1 - t)} (d (x_0, x_1)) \rho _0^{- 1 / N'} (x_0)\nonumber \\{} & {} \quad +\tau _{K, N'}^{(t)} (d (x_0, x_1)) \rho _1^{- 1 / N'} (x_1) \pi ({\,\textrm{d}}x_{0}, {\,\textrm{d}}x_1), \end{aligned}$$

  (2.1)

  where $\pi = (e_0, e_1)_{\#} \nu$ and $\rho _t := \frac{{\,\textrm{d}}\mu _{t}}{{\,\textrm{d}}\mathfrak {m}}$.

• $(X, d, \mathfrak {m})$ is said to verify $\textrm{CD}(K,N)$ locally, or $\textrm{CD}_{\textrm{loc}} (K, N)$ in short, if for any $o \in \textrm{supp}(\mathfrak {m})$ one can find a neighborhood $X_o \subset X$ of o, so that for all $\mu _0, \mu _1 \in \mathcal {P}_2 (X,\mathfrak {m})$ supported in $X_o$, there exists $\nu \in \textrm{OptGeo}(\mu _0, \mu _1)$ so that $\mu _t := (e_{t })_{\#} \nu \ll \mathfrak {m}$, and (2.1) holds for all $t \in [0, 1]$, $N' \ge N$.

• $(X,d,\mathfrak {m})$ is said to verify $\textrm{MCP}(K,N)$, if for any $o \in \textrm{supp}(\mathfrak {m})$ and $\mu _0:=\frac{\mathfrak {m}\llcorner A}{\mathfrak {m}(A)}$ given A a Borel subset of X with $0<\mathfrak {m}(A)<\infty$, there exists $\nu \in \textrm{OptGeo}(\mu _0,\delta _o)$ s.t.

  $$\begin{aligned} \frac{\mathfrak {m}}{\mathfrak {m}(A)}\ge (e_t)_{\#}(\tau ^{(1-t)}_{K,N}(d(\gamma _0,\gamma _1))^N\nu ({\,\textrm{d}}\gamma ))\quad \forall t\in [0,1]. \end{aligned}$$

  (2.2)

Definition 2.3

A set $G\subset \textrm{Geo}(X)$ is non-branching if for any $\gamma ^1,\gamma ^2\in G$ with $\gamma ^1=\gamma ^2$ on [0, t] for some $t\in (0,1)$, it holds $\gamma ^1=\gamma ^2$ on [0, 1].

A space $(X,d,\mathfrak {m})$ is called essentially non-branching if for all $\mu _0,\mu _1\in \mathcal {P}_2(X,\mathfrak {m})$, any $\nu \in \textrm{OptGeo}(\mu _0,\mu _1)$ is concentrated on a Borel non-branching set $G\subset \textrm{Geo}(X)$.

Remark 2.4

Throughout this paper, we assume that $\textrm{supp}(\mathfrak {m}) = X$ without any further specification as it will not affect the generality. Indeed, as discussed in [6, Remark 6.11], whenever $\mu _0,\mu _1\ll \mathfrak {m}$, almost every curve in the support of $\nu \in \textrm{OptGeo}(\mu _0,\mu _1)$ is contained in $\textrm{supp}(\mathfrak {m})$. So the problem on $(X,d,\mathfrak {m})$ is equivalent to the one on $(\textrm{supp}(\mathfrak {m}),d,\mathfrak {m})$.

2.2 Density Functions on Nonbranching spaces

Cavalletti-Mondino in [8] showed that optimal maps of transports with $\mu _0\ll \mathfrak {m}$ uniquely exist on e.n.b. $\textrm{MCP}(K,N)$ spaces. Such $\textrm{MCP}$-condition is always satisfied on e.n.b. $\textrm{CD}_{\textrm{loc}}(K,N)$ spaces (first by [10] on non-branching spaces, and then on e.n.b. spaces with properties developed in [8]).

In this subsection, $(X,d,\mathfrak {m})$ always stands for an e.n.b. length m.m.s. satisfying $\textrm{CD}_{\textrm{loc}}(K,N)$ or $\textrm{MCP}(K,N)$. It is well-known that any $\textrm{CD}_{\textrm{loc}}(K,N)$ length space is locally compact (see e.g. [6, Lemma 6.12]), so by Hopf-Rinow Theorem, it is proper and geodesic.

Proposition 2.5

(cf. [8]) For every $\mu _0,\mu _1\in \mathcal {P}_2(X)$ with $\mu _0\ll \mathfrak {m}$, there exists a unique $\nu \in \textrm{OptGeo}(\mu _0,\mu _1)$; such $\nu$ is induced by a map (i.e. $\nu =S_{\#}\mu _0$ for $S:X\supset \textrm{Dom}(S)\rightarrow \textrm{Geo}(X)$) and for every $t\in (0,1)$, $(e_t)_{\#}\nu \ll \mathfrak {m}$.

Lemma 2.6

([6, Corollary 6.16]) Given $\mu _0,\mu _1$ as in Proposition 2.5, the unique optimal dynamical plan $\nu$ is concentrated on a Borel set $G\subset \textrm{Geo}(X)$ s.t. the evaluation map $e_{t}:G\rightarrow X$ is injective for all $t\in [0,1)$. And in particular, any Borel $H\subset G$, we have

$$\begin{aligned} (e_t)_{\#}(\nu \llcorner H)=({e_t}_{\#}\nu )\llcorner e_t(H)\quad \forall t\in [0,1). \end{aligned}$$

The following can be regarded as an expansion of the original proof in [6].

Proof

We first assume $\mu _1\ll \mathfrak {m}$. Recall $\nu$ is induced by a map i.e. $\nu ={S_0}_{\#}\mu _0={S_1}_{\#}\mu _1$. As argued in [6], for both $i=0,1$ we can find $X_i\subset X$ of full $\mu _i$ measure s.t. for all $x\in X_i$, there exists a unique $\gamma \in G_{\varphi }$ with $\gamma _i=x$. In particular, $\nu (S_0(X_0))=\nu (S_1(X_1))=1$.

Take a Borel set $G\subset S_0(X_0)\cap S_1(X_1)$, still with full $\nu$-measure. We claim $e_t$ is injective on G for all $t\in [0,1]$. By construction, $e_0$ and $e_1$ are clearly injective on G. Assume there are $\gamma ,\tilde{\gamma }\in G$, $\gamma _t=\tilde{\gamma }_t$ for some $t\in (0,1)$. Define a curve $\eta$ by letting $\eta =\gamma$ on [0, t] and $\eta =\tilde{\gamma }$ on [t, 1]. By cyclic monotonicity, $\eta \in G_{\varphi }$. Since $\gamma \in S_0(X_0)$, $\eta \equiv \gamma$ on [0, 1] and so $\gamma _1=\tilde{\gamma }_1$. On the other hand, as $\gamma ,\tilde{\gamma }\in S_1(X_1)$, one concludes $\gamma \equiv \tilde{\gamma }$.

For general $\mu _1\in \mathcal {P}_2(X)$, we prove by taking restrictions of $\nu$. For any $t\in [0,1)$, define

$$\begin{aligned} \textrm{restr}^t_0:\textrm{supp}(\nu )\rightarrow \textrm{Geo}(X),\quad \gamma (\cdot )\mapsto \gamma (t\cdot ). \end{aligned}$$

Proposition 2.5 ensures that $\mu _t:= (e_t)_{\#}\nu \ll \mathfrak {m}$, and $(\textrm{restr}^t_0)_{\#}\nu$ is the unique optimal dynamical plan between $\mu _0$ and $\mu _t$. From the first step, we can find a Borel set $\tilde{G}_t$ where $(\textrm{restr}^t_0)_{\#}\nu$ is concentrated and evaluation maps are injective over there. Then, take a sequence $t_n\nearrow 1$ and a set

$$\begin{aligned} G:= \bigcap _{n\in \mathbb {N}}\left( \textrm{restr}^{t_n}_0\right) ^{-1}\left( \tilde{G}_{t_n}\right) . \end{aligned}$$

One can check $\nu (G)=1$ and $e_t$ is injective on G for all $t\in [0,1)$. $\square$

Since X is proper, any bounded subset has finite $\mathfrak {m}$-measure. Via a conditioning argument, we can extend [6, Proposition 9.1] to infinite-volume spaces.

Proposition 2.7

(Density characterization) For any $\mu _0 \in \mathcal {P}_2 (X, \mathfrak {m})$, $\mu _1 \in \mathcal {P}_2 (X)$, there exists a unique $\nu \in \textrm{OptGeo}(\mu _0,\mu _1)$ so that for all $t \in (0, 1)$, $(e_t)_{\#} \nu \ll \mathfrak {m}$ and

$$\begin{aligned} \rho ^{- 1 / N}_t (\gamma _t) \ge \tau ^{(1 - t)}_{K, N}(d(\gamma _0,\gamma _1)) \rho ^{- 1 / N}_0 (\gamma _0) \quad \text {for } \nu -\text {a.e. } \gamma . \end{aligned}$$

(2.3)

It verifies $\textrm{CD}(K, N)$ iff for any $\mu _0, \mu _1 \in \mathcal {P}_2 (X, \mathfrak {m})$, there exists a unique $\nu \in \textrm{OptGeo}(\mu _0, \mu _1)$ so that for all $t \in (0, 1)$, $(e_t)_{\#}\nu \ll \mathfrak {m}$ and

$$\begin{aligned} \rho ^{- 1 / N}_t (\gamma _t) \ge \tau ^{(1-t)}_{K,N} \left( d\left( \gamma _0,\gamma _1\right) \right) \rho ^{-1/N}_0(\gamma _0)+\tau ^{(t)}_{K, N} (d(\gamma _0,\gamma _1)) \rho ^{-1/N}_1(\gamma _1), \quad \text {for } \nu -\text {a.e. } \gamma . \end{aligned}$$

(2.4)

Sketch of proof

When $\mathfrak {m}(X)<\infty$, arguing by approximation as in [6, Proposition 9.1], for arbitrary boundedly supported $\mu _0\in \mathcal {P}_2(X,\mathfrak {m})$ and $\mu _1\in \mathcal {P}_2(X)$, we have

$$\begin{aligned} \mathcal {E}_N (\mu _t) \ge \int \tau ^{(1 - t)}_{K, N} (d (\gamma _0, \gamma _1)) \rho ^{- 1 / N}_0 (\gamma _0) \nu ({\,\textrm{d}}\gamma ), \quad \nu \in \textrm{OptGeo}(\mu _0, \mu _1), \end{aligned}$$

(2.5)

where $\mu _t=(e_t)_{\#}\nu$. Here the finiteness of volume is only required for showing the upper-semicontinuity of $\mathcal {E}_N$. In our case due to the choice of marginals, $(\mu _t)_t$ are confined to a fixed bounded set U. So redoing [16, Lemma 4.1] ensures that $\mathcal {E}_N$ is upper-semicontinuous w.r.t. weak convergence of measures supported inside U.

Now consider general $\mu _0 \in \mathcal {P}_2 (X, \mathfrak {m})$, $\mu _1 \in \mathcal {P}_2 (X)$ possibly with unbounded supports. Take any compact $G\subset \textrm{Geo}(X)$ with $\nu (G) > 0$. The restricted plan $\tilde{\nu } = \frac{1}{\nu (G)} \nu \llcorner _G$ is still an optimal dynamical plan. By Lemma 2.6, $\tilde{\mu }_t := (e_t)_{\#} \tilde{\nu }$ has the density $\tilde{\rho }_t = \frac{1}{\nu (G)} \rho _t\llcorner e_t (G)$, and having a uniformly bounded support. So (2.5) holds for $\tilde{\nu }$, implying

$$\begin{aligned} \int _G \rho _t^{- 1 / N} (\gamma _t) \nu ({\,\textrm{d}}\gamma ) \ge \int _G \tau ^{(1 - t)}_{K, N} \left( d \left( \gamma _0, \gamma _1\right) \right) \rho _0 (\gamma _0)^{- 1 / N} \nu ({\,\textrm{d}}\gamma ). \end{aligned}$$

The arbitrariness of G and the inner regularity of $\nu$ yield the inequality (2.3) for $\nu$-a.e. $\gamma$.

For the second assertion on $\textrm{CD}(K,N)$. The “only if” part follows by applying the similar conditioning to (2.1). The “if” part follows directly by integrating (2.4) against $\nu$. $\square$

An important consequence of the previous proposition is the following continuity of optimal dynamics, which plays a crucial role in the ray decomposition (see e.g. the proof of Theorem 3.10). Besides, the Lipschitz-regularity of densities is a starting point of the bootstrap argument in [6, Section 12].

Corollary 2.8

(Continuity of Dynamics, cf. [6, Section 9]) Let $\nu \in \textrm{OptGeo}(\mu _0, \mu _1)$ for $\mu _0,\mu _1 \in \mathcal {P}_2 (X, \mathfrak {m})$.

• (1) There exist versions of densities $\rho _t = \frac{{\,\textrm{d}}\mu _t}{{\,\textrm{d}}\mathfrak {m}}$, $t \in [0, 1]$, so that for $\nu$-a.e. $\gamma \in \textrm{Geo}(X)$ and all $0 \le s < t \le 1$:

  $$\begin{aligned} \rho _s (\gamma _s) > 0, \quad \left( \tau ^{\left( \frac{s}{t} \right) }_{K, N} (d (\gamma _0, \gamma _t)) \right) ^N \le \frac{\rho _t (\gamma _t)}{\rho _s (\gamma _s)} \le \left( \tau ^{\left( \frac{1 - t}{1 - s} \right) }_{K, N} (d (\gamma _s, \gamma _1)) \right) ^{- N}. \end{aligned}$$

  (2.6)

  In particular, for $\nu$-a.e. $\gamma$, the map $t \mapsto \rho _t (\gamma _t)$ is locally Lipschitz on (0, 1) and upper semi-continuous at $t = 0,1$.

• (2) For any compact $G \subset \textrm{Geo}(X)$ with $\nu (G) > 0$ s.t. (2.6) holds for all $\gamma \in G$ and $0 \le s \le t \le 1$, we have $\mathfrak {m} (e_s (G)) > 0$ for all $s \in [0, 1]$ and

  $$\begin{aligned} \left( \frac{1 - t}{1 - s} \right) ^N e^{- d (G) (t - s) \sqrt{(N - 1) K^-}} \le \frac{\mathfrak {m} (e_t (G)) \textsf {}}{\mathfrak {m} (e_s (G))} \le \left( \frac{t}{s} \right) ^N e^{d (G) (t - s) \sqrt{(N - 1) K^-}}, \end{aligned}$$

  (2.7)

  where $d (G) := \max \{ \ell (\gamma ): \gamma \in G \}$ and $K^- := \max \{ - K, 0 \}$. In particular, the map $t\mapsto \mathfrak {m} (e_t (G))$ is locally Lipschitz on (0, 1) and lower semi-continuous at $t = 0,1$.

2.3 Intermediate-time Kantorovich Potentials

We first recall the notion of intermediate-time Kantorovich potentials.

Definition 2.9

Given a Kantorovich potential $\varphi : X \rightarrow \mathbb {R}$, the intermediate-time Kantorovich potential $\varphi _t$ at time $t \in [0, 1]$ is defined by $\varphi _0 = \varphi$, $\varphi _1 = -\varphi ^c$ and

$$\begin{aligned} \varphi _t (x) := - \underset{y \in X}{\inf }\ \left[ \frac{d^2 (x, y)}{2 t} - \varphi (y) \right] . \end{aligned}$$

Denote the domain of the dynamics and its section through x as:

$$\begin{aligned} D ({G}_{\varphi })&:= \{(x, t) \in X \times (0, 1) : \exists \gamma \in G_{\varphi }, x = \gamma _t \}, \\ G_{\varphi }(x)&:= \{t\in (0,1):(x,t)\in D(G_{\varphi })\}. \end{aligned}$$

Based on Lemma 2.6, when $(X,d,\mathfrak {m})$ is e.n.b., for simplicity, we will assume $e_{t}:G_{\varphi }\rightarrow \mathbb {R}$ is injective for all $t\in [0,1]$ as otherwise, it suffices to restrict $\nu$ to some Borel $G\subset G_{\varphi }$. Then the length function is defined by

$$\begin{aligned} \ell :D(G_{\varphi })\ni (x,t)\mapsto \ell (e_t^{-1}(x)) := \textrm{Length}(e_t^{-1}(x)) \end{aligned}$$

and we also use the notation $\ell _t(\cdot ) := \ell (\cdot , t)$ on $e_t (G_{\varphi })$ for every t.

Definition 2.10

Given a Kantorovich potential $\varphi : X \rightarrow \mathbb {R}$ and $s, t \in (0, 1)$, define the t-propagated s-Kantorovich potential $\Phi _s^t$ on $e_t (G_{\varphi })$ by

$$\begin{aligned} \Phi _s^t := \varphi _s \circ e_s \circ e_t^{- 1}=\varphi _t+(t-s)\frac{\ell ^2_t}{2}. \end{aligned}$$

For every fixed $s\in (0,1)$, according to the value of $\varphi _s$, $G_{\varphi }$ can be partitioned into closed levels

$$\begin{aligned} G_{\varphi } = \sqcup _{a_s \in \textrm{Im}(\varphi _s \circ e_s)}G_{\varphi , a_s}, \quad G_{\varphi , a_s} := (\varphi _s \circ e_s)^{-1} (a_s) = \{\gamma \in G_{\varphi }: \varphi _s (\gamma _s) = a_s \} \end{aligned}$$

which further leads to a partition of $e_t(G_{\varphi })$ (any $t\in (0,1)$) via $\Phi _s^t$ by

$$\begin{aligned} e_t (G_{\varphi }) = \sqcup _{a_s \in \textrm{Im}(\varphi _s \circ e_s)} e_t(G_{\varphi , a_s}), \quad e_t (G_{\varphi , a_s}) := (\Phi _s^t)^{- 1} (a_s) =\{\gamma _t: \varphi _s (\gamma _s) = a_s \}. \end{aligned}$$

Lemma 2.11

(Continuity of potentials, cf. [2, Sect. 3] and [6, Theorem 3.11, Proposition 4.4]) The function $X \times (0, 1) \ni (x, t) \mapsto \varphi _t (x)$ is locally Lipschitz. The length function $\ell$ is continuous on $D(G_{\varphi })$. For any $x \in X$ and $s\in (0, 1)$, functions ${G}_{\varphi }(x) \ni t \mapsto \ell _t (x)$ and ${G}_{\varphi }(x) \ni t \mapsto \Phi ^t_s (x)$ are locally Lipschitz.

3 $L^1$-disintegration

3.1 Disintegration Theorem

Proofs of assertions in this subsection can be found in [3, Appendix A]. Let $(X, \mathfrak {X}, \mathfrak {m})$, $(Q, \mathcal {Q}, \mathfrak {q})$ be measure spaces. A disintegration of $\mathfrak {m}$ over $\mathfrak {q}$ is a family of measures $(\mathfrak {m}_q)_{q \in Q}$ on X s.t. for every $E \in \mathfrak {X}$, the map $q \mapsto \mathfrak {m}_q (E)$ is $\mathfrak {q}$-measurable and $\mathfrak {m} (E) = \int _Q \mathfrak {m}_q (E) \mathfrak {q} ({\,\textrm{d}}q)$. By [11, Proposition 452F], for any $\mathfrak {m}$-measurable $\xi : X \rightarrow \mathbb {R}$, we have

$$\begin{aligned} \int _Q \int _X \xi (x) \mathfrak {m}_q ({\,\textrm{d}}x) \mathfrak {q} ({\,\textrm{d}}q) = \int _X \xi (x) \mathfrak {m} ({\,\textrm{d}}x), \end{aligned}$$

provided $\int \xi (x) \mathfrak {m} ({\,\textrm{d}}x)$ is well-defined in $\mathbb {R} \cup \{\pm \infty \}$.

Given measurable $f: (X, \mathfrak {X}) \rightarrow (Q, \mathcal {Q})$, a disintegration $(\mathfrak {m}_q)_{q \in Q}$ of $\mathfrak {m}$ over $\mathfrak {q}$ is called consistent with f if for each $I\in \mathcal {Q}$,

$$\begin{aligned} \mathfrak {m} (E \cap f^{- 1} (I)) = \int _I \mathfrak {m}_q (E) \mathfrak {q} ({\,\textrm{d}}q). \end{aligned}$$

(3.1)

And $(\mathfrak {m}_q)_{q \in Q}$ is called strongly consistent with f, if for $\mathfrak {q}$-a.e. $q \in Q$, $\mathfrak {m}_q$ is concentrated on $f^{- 1}(\{q\})$. Clearly, strong consistency implies consistency.

Remark 3.1

(Uniqueness of Disintegration) If $\mathfrak {X}$ is countably generated with a $\sigma$-finite measure $\mathfrak {m}$, and disintegrations $(\mathfrak {m}_q)$, $(\tilde{\mathfrak {m}}_q)$ of $\mathfrak {m}$ over $\mathfrak {q}$ are consistent with f, then $\mathfrak {m}_q = \tilde{\mathfrak {m}}_q$ for $\mathfrak {q}$-a.e. $q \in Q$, or in short, consistent disintegrations are $\mathfrak {q}$-unique.

Indeed, by [13, Proposition 3.3], there is a countable subalgebra $\{B_n \in \mathfrak {X}, n \in \mathbb {N}\}$ generating $\mathfrak {X}$. After putting $E = B_n$ into (3.1), we know up to a $\mathfrak {q}$-negligible set $N \subset Q$, $\mathfrak {m}_q (B_n) = \tilde{\mathfrak {m}}_q (B_n)$ for all n and q. So when $\mathfrak {m}$ is finite, with Dynkin’s theorem, $\mathfrak {m}_q =\tilde{\mathfrak {m}}_q$ for all $q \in Q {\setminus } N$. For the case where $\mathfrak {m}$ is $\sigma$-finite, we can repeat the previous argument on any subset E of finite $\mathfrak {m}$-measure to show $\mathfrak {m}_q\llcorner E=\tilde{\mathfrak {m}}_q\llcorner E$ for a.e. q. The argument is complete after taking an exhausting sequence $E_n$ of X.

In particular, strongly consistent disintegrations of a locally finite measure are $\mathfrak {q}$-unique.

If X has a partition $\Pi = \{X_q \}_{q \in Q}$, define $\mathfrak {Q}: X \rightarrow Q$ by mapping each point in $X_q$ to q. Endowed with the quotient $\sigma$-algebra $\mathcal {Q}$ and the quotient measure $\mathfrak {q}=\mathfrak {Q}_{\#} \mathfrak {m}$, $(Q, \mathcal {Q},\mathfrak {q})$ is a measure space.

Definition 3.2

A cross section of a partition $\Pi$ is a subset S of X so that $S \cap A$ is a singleton for each $A \in \Pi$. A section is a map $\mathfrak {S}: X \rightarrow X$ such that for each $x \in X$, the image of [x] under $\mathfrak {S}$ is a singleton in [x], where [x] is the equivalence class of x under $\Pi$.

A subset $S_{\mathfrak {m}}$ is called an $\mathfrak {m}$-section if there exists a Borel set $\Gamma \subset X$ s.t. $\mathfrak {m}(X\setminus \Gamma )=0$ and the partition $\Pi _{\Gamma } = \{X_q \cap \Gamma \}_{q \in Q}$ has $S_{\mathfrak {m}}$ as a cross section.

Theorem 3.3

(Disintegration Theorem) Assume $(X, \mathfrak {X}, \mathfrak {p})$ is a countably generated probability space, having a partition $\Pi = \{X_q \}_{q \in Q}$. Let $\mathfrak {Q}: X \rightarrow Q$ and $(Q, \mathcal {Q}, \mathfrak {q})$ be the quotient map and quotient space resp. There exists a unique disintegration $q \mapsto \mathfrak {p}_q \in \mathcal {P} (X)$ of $\mathfrak {p}$ over $\mathfrak {q}$ consistent with $\mathfrak {Q}$. Moreover, this disintegration is strongly consistent with $\mathfrak {Q}$ iff there exists a Borel $\mathfrak {p}$-section $S_{\mathfrak {p}}\subset Q$ s.t. the quotient $\sigma$-algebra $\mathcal {Q} \cap S_{\mathfrak {p}}$ contains $\mathcal {B} (S_{\mathfrak {p}})$.

Remark 3.4

(Disintegration over level sets) If (X, d) is Polish and the partition $\Pi$ given as level sets of a continuous function $\mathfrak {Q}:X \rightarrow \mathbb {R}$, then, by [15, Theorem 5.4.3], $\Pi$ admits a Borel cross-section S and Borel section map $\mathfrak {S}$. In particular, there is a unique disintegration of $\mathfrak {p}$ strongly consistent with $\mathfrak {S}$.

3.2 Transport Ray and $\textrm{CD}^1(K,N)$

For any 1-Lipschitz function $u:(X,d)\rightarrow \mathbb {R}$, define the transport relation $R_u$ and the transport set $\mathcal {T}_u$ as

$$\begin{aligned} R_u := \{(x, y) \in X \times X: |u (x) - u (y) | = d (x, y) \}, \qquad \mathcal {T}_u := P_1 (R_u \setminus \{x = y\}), \end{aligned}$$

where $P_i$ is the projection onto the i-th component. Denote $R_u (x) := \{y \in X: (x, y) \in R_u \}$ as the section of $R_u$ through x in the first coordinate.

Notice $R_u$ is not necessarily an equivalence relation as the transitivity may be violated. To remedy this, define the non-branched transport set by removing those branched points:

$$\begin{aligned} \mathcal {T}^{b}_u := \left\{ x \in \mathcal {T}_u: \forall z, w \in R_u (x), (z, w) \in R_u \right\} \end{aligned}$$

and hence the corresponding non-branched transport relation

$$\begin{aligned} R^b_u := R_u \cap \left( \mathcal {T}^{b}_u \times \mathcal {T}^{b}_u\right) . \end{aligned}$$

Remark 3.5

We refer to [4, 5] and [6, Section 7] for following statements:

• When (X, d) is proper, $\mathcal {T}_u$ is $\sigma$-compact, and $\mathcal {T}_u^b$, $R^{b}_u$ are Borel;

• $R^b_u$ is an equivalence relation on $\mathcal {T}^b_u$ which induces a partition $\sqcup _x R^b_u (x)$ of $\mathcal {T}^b_u$;

• When (X, d) is geodesic, for any $x \in \mathcal {T}^b_u$, $R_u (x)$ is a single (unparameterized) geodesic of positive length, so that $(R_u (x), d)$ is isometric to a closed interval in $(\mathbb {R}, | \cdot |)$ and $(R^b_u (x), d)$ is a subinterval.

We call R a transport ray if (R, d) is isometric to a closed interval in $(\mathbb {R},|\cdot |)$ of positive length and it is maximal under the partial order $\le _u$, where $x\le _u y$ if $u(x)-u(y)=d(x,y)$.

Definition 3.6

Given a continuous function $\phi : (X, d) \rightarrow \mathbb {R}$ so that $\{ \phi = 0 \}\ne \emptyset$, define the signed distance function (from zero-level set of $\phi$) as

$$\begin{aligned} d_{\phi }: X \rightarrow \mathbb {R}, \quad d_{\phi } (x) := \textrm{dist} (x, \{\phi = 0\}) \textrm{sign}(\phi ). \end{aligned}$$

When (X, d) is a length space, any signed distance function $d_{\phi }$ is 1-Lipschitz (see [6, Lemma 8.4]). If further $\mathfrak {m}(X)<\infty$, Theorem 3.3 gives a disintegration of $\mathfrak {m}$ on $\mathcal {T}^b_{d_{\phi }}$ w.r.t. the partition by $R^b_{d_{\phi }}$, which leads to the $\textrm{CD}^1$-condition introduced in [6]. We modify this condition by relaxing conditional measures to be only locally finite, instead of probabilities.

Definition 3.7

A m.m.s. $(X,d,\mathfrak {m})$ with $\textrm{supp}(\mathfrak {m})=X$ satisfies $\textrm{CD}^1(K,N)$ if for any 1-Lipschitz signed distance function $u=d_{\phi }$, with the associated partition $\{R^b_u(q)\}_{q\in Q}$ of $\mathcal {T}^b_u$ by ray decomposition, there exist a probability space $(Q,\mathcal {Q},\mathfrak {q})$ and a $\mathfrak {q}$-unique disintegration $\mathfrak {m}\llcorner \mathcal {T}_u =\int _Q\mathfrak {m}_q\mathfrak {q}({\,\textrm{d}}q)$ on $\{\overline{R^b_u(q)}\}_{q\in Q}$ s.t.

• 1. Q is a section of the above partition so that $Q\supseteq \bar{Q}\in \mathcal {B}(\mathcal {T}^b_u)$ with $\bar{Q}$ an $\mathfrak {m}$-section with $\mathfrak {m}$-measurable quotient map and $\mathcal {Q}\supseteq \mathcal {B}({\bar{Q}})$;

• 2. for $\mathfrak {q}$-a.e. $q \in Q$, $\overline{R^b_u(q)}=R_u(q)$ as a transport ray;

• 3. for $\mathfrak {q}$-a.e. $q\in Q$, $\mathfrak {m}_q$ is non-null, supported on $\overline{R^b_u(q)}$;

• 4. for $\mathfrak {q}$-a.e. $q \in Q$, $(\overline{R^b_u(q)}, d, \mathfrak {m}_q)$ is a one-dimensional $\textrm{CD}(K, N)$ m.m.s.;

• 5. for every bounded subset $K \subset X$, there exists $C_K \in (0, \infty )$ s.t.

  $$\begin{aligned} \mathfrak {m}_q (K) \le C_K, \quad \text {for }\mathfrak {q}-\text {a.e. }. \end{aligned}$$

  (3.2)

Remark 3.8

The reference measure $\mathfrak {m}$ on any $\textrm{CD}^1(K,N)$ space must be locally finite, simply by (3.2) and taking $u=d(\cdot ,o)$ for some $o\in X$. And by Theorem 3.3, the disintegration is strongly consistent with the quotient measure because of 1.

Remark 3.9

For any u and disintegration from Definition 3.7, $\mathfrak {m}(\mathcal {T}_u)=\mathfrak {m}\left( \mathcal {T}^b_u\right)$. Indeed, denoting by ${\tilde{Q}}\subset Q$ the set of q that 2–4 hold, then

$$\begin{aligned} \mathfrak {m}(\mathcal {T}_u)&=\int _Q \mathfrak {m}_q(\mathcal {T}_u)\mathfrak {q}({\,\textrm{d}}q)=\int _{{\tilde{Q}}} \mathfrak {m}_q\left( \overline{R^b_u(q)}\right) \mathfrak {q}({\,\textrm{d}}q) \\ {}&=\int _{{\tilde{Q}}} \mathfrak {m}_q\left( R^b_u(q)\right) \mathfrak {q}({\,\textrm{d}}q)=\int _{{\tilde{Q}}} \mathfrak {m}_q\left( \mathcal {T}^b_u\right) \mathfrak {q}({\,\textrm{d}}q)=\mathfrak {m}\left( \mathcal {T}^b_u\right) , \end{aligned}$$

where we have used the fact that a measure carrying $\textrm{CD}(K,N)$ does not charge points.

3.3 $\textrm{CD}_{\textrm{loc}}(K,N)$ implies $\textrm{CD}^1(K,N)$

Theorem 3.10

Let $(X, d, \mathfrak {m})$ be an e.n.b. $\textrm{CD}_{\textrm{loc}}(K, N)$ length m.m.s. such that $\mathfrak {m}$ is locally finite with full-support, and $u:(X, d)\rightarrow \mathbb {R}$ be a 1-Lipschitz function. Then there exists a disintegration of $\mathfrak {m}\llcorner \mathcal {T}_u$ satisfying 1–5 of Definition 3.7. In particular, under these assumptions, $\textrm{CD}_{\textrm{loc}}(K,N)$ implies $\textrm{CD}^1(K,N)$.

Such disintegration, also called ray/needle decomposition, is extensively studied in e.g. [4, 5, 7] under the assumption $\mathfrak {m}(X)=1$. However in our case, $\mathcal {T}^b_u$ could be unbounded with infinite volume, so we can not directly apply Theorem 3.3. Therefore, we normalize the measure by adding a weight function following the approach in [9]. After such re-weighting, $\textrm{CD}$-information can be passed to rays exactly as in the finite-volume case.

Proof

As every $\textrm{CD}_{\textrm{loc}} (K, N)$ geodesic m.m.s. is proper, Remark 3.5 applies. From [5, Proposition 4.5] (together with the comments above [6, Corollary 7.3]), $\mathfrak {m}(\mathcal {T}_u\setminus \mathcal {T}^b_u)=0$. Hence it suffices to disintegrate $\mathfrak {m}\llcorner \mathcal {T}^b_u$ w.r.t. the partition $\mathcal {T}^b_u=\sqcup _q R^b_u (q)$.

Normalize $\mathfrak {m}$ to apply the disintegration theorem. Without loss of generality we assume $\mathfrak {m}(\mathcal {T}^b_u) = \infty$. Then, for any fixed $x_0 \in X$, we can find an increasing sequence $(r_n)_{n\ge 1}$ of positive numbers, so that

$$\begin{aligned} \mathcal {T}^{b, n}_u := \left\{ \begin{array}{ll} \mathcal {T}^b_u \cap \left\{ x \in X: r_n \le d (x, x_0)< r_{n+1} \right\}, & n \in \mathbb {N}_+\\ \mathcal {T}^b_u \cap \left\{ x \in X: d (x, x_0) < r_1\right\}, &{} n = 0 \end{array}\right. \end{aligned}$$

has positively finite $\mathfrak {m}$-measure for each $n\ge 0$. Define f by

$$\begin{aligned} f (x) = \sum _{n \in \mathbb {N}} \left( 2^{n + 1} \mathfrak {m}\left( \mathcal {T}^{b, n}_u\right) \right) ^{- 1} \mathbbm {1}_{\mathcal {T}^{b, n}_u} (x). \end{aligned}$$

Clearly,

$$\begin{aligned} \inf _{K\cap \mathcal {T}^b_u} f > 0, \text { for any compact } K \subset X; \quad \int _{\mathcal {T}^b_u} f (x) \mathfrak {m} ({\,\textrm{d}}x) = 1. \end{aligned}$$

(3.3)

Hence, $\mathfrak {n} := f\mathfrak {m} \llcorner \mathcal {T}^b_u$ is a probability measure and Theorem 3.3 can be applied to $\mathfrak {n}$.

On the strong consistency. First, from [4, Proposition 4.4],³ there exists an $\mathfrak {m}$-measurable section $\mathfrak {Q}: \mathcal {T}^b_u \rightarrow \mathcal {T}^b_u$ associated to the partition $\{R^b_u (q)\}_{q \in Q}$. From now on, we fix Q as the image of $\mathfrak {Q}$ and endow Q with $\sigma$-algebra $\mathcal {Q} := \mathfrak {Q}_{\#}\mathcal {B}\left( \mathcal {T}^b_u\right)$. Take $\mathfrak {q}=\mathfrak {Q}_{\#} \mathfrak {n} \textsf {}$ to be the quotient probability on Q. By the $\mathfrak {m}$-measurability of $\mathfrak {Q}$, $\mathfrak {q}$ is Borel on $\mathcal {T}^b_u$. Thus $\mathfrak {q}$ is inner regular and we can find a $\sigma$-compact set $S \subset Q$, $\mathfrak {q}(Q {\setminus } S) = 0$. Define a section $\mathfrak {S} := \mathfrak {Q} \llcorner \left( \mathfrak {Q}^{- 1} (S)\right)$ where $\mathfrak {Q}^{- 1} (S)$ has full $\mathfrak {n}$-measure. Then

$$\begin{aligned} \textrm{graph} (\mathfrak {S}) = \left\{ (x, s) \in \mathcal {T}^b_u\times S:(x,s)\in R_u \right\} \end{aligned}$$

is Borel, implying that $\mathfrak {Q}^{- 1} (S) = P_1 (\textrm{graph}(\mathfrak {S}))$ is analytic and $\mathfrak {S}$ is Borel measurable by [15, Theorem 4.5.2]. That is to say, S is a Borel $\mathfrak {n}$-section with Borel measurable section $\mathfrak {S}$ and hence $\mathcal {Q}\supset \mathcal {B}(S)$. Theorem 3.3 applies to conclude that $q \mapsto \mathfrak {n}_q$ is the $\mathfrak {q}$-unique disintegration of $\mathfrak {n}$ strongly consistent with $\mathfrak {Q}$. In particular, 1 is verified as $\mathfrak {n}$ and $\mathfrak {m}\llcorner \mathcal {T}^b_u$ sharing same measurable and null sets.

Back to $\mathfrak {m} \llcorner \mathcal {T}^b_u$, owing to the everywhere positivity of f on $\mathcal {T}^b_u$, we have

$$\begin{aligned} \mathfrak {m} \llcorner \mathcal {T}^b_u = \int _Q \mathfrak {n}_q / f\mathfrak {q} ({\,\textrm{d}}q). \end{aligned}$$

(3.4)

Define $\mathfrak {m}_q := \mathfrak {n}_q /f$. As measurability (w.r.t. $q \in Q$) is guaranteed, $q\mapsto \mathfrak {m}_q$ gives the unique disintegration of $\mathfrak {m}\llcorner \mathcal {T}^b_u$ strongly consistent with $\mathfrak {Q}$ (recall Remark 3.1). From (3.3), $\mathfrak {m}_q$ is uniformly-locally finite as (3.2). Further, we can repeat [6, Theorem 7.10](which mainly needs Proposition 2.5 but not finiteness of $\mathfrak {m}(X)$, as it is proved by contradiction and localization) for (3.4) to show that for $\mathfrak {q}$-a.e. $q \in Q$, $R_u (q) = \overline{R^b_u (q)}$ and $\textrm{supp}(\mathfrak {m}_q) = R_u (q)$.

Localize $\textrm{CD}(K,N)$ to transport rays. Let S be the $\sigma$-compact cross section in the previous step. Define the ray map $g: \textrm{Dom}(g) \subset S \times \mathbb {R} \rightarrow \mathcal {T}^b_u$ via

$$\begin{aligned} \textrm{graph} (g) := \left\{ (q, t, x) \in S \times \mathbb {R} \times \mathcal {T}^b_u: (q, x) \in R_u, u (x) - u (q) = t \right\} . \end{aligned}$$

(3.5)

Remark 3.5 ensures that each $x \in \mathcal {T}^b_u$ uniquely corresponds a pair $(\mathfrak {Q} (x), d) \in S \times \mathbb {R}$, with $d = u (x) - u \left( \mathfrak {Q} (x)\right)$ and $| d | = d (x, \mathfrak {Q}(x))$. Hence g is well-defined, bijective and Borel measurable because of its Borel graph. For any $q \in S$, $I_q := \textrm{Dom} (g (q, \cdot ))$ is an interval in $\mathbb {R}$, and $I_q \ni t \mapsto g (q, t) \in R^b_u (q)$ is an isometry, meaning $\mathcal {H}^1 \llcorner \left\{ R^b_u (q) \right\} = g (q, \cdot )_{\#} \left( \mathcal {L}^1 \llcorner I_q\right)$.

It remains to show that for $\mathfrak {q}$-a.e. $q \in Q$, $\mathfrak {m}_q \ll g(q,\cdot )_{\#}\mathcal {L}^1$ and for those q, by denoting $\mathfrak {m}_q = g(q, \cdot )_{\#}\left( h_q \cdot \mathcal {L}^1 \llcorner I_q\right)$, $h_q$ is a $\textrm{CD}(K, N)$ density.⁴ Such regularity problem for conditional measures can be solved by combining arguments in [4, Theorem 5.7] and [7, Theorem 4.2]. We refer to the appendix for more detailed demonstrations. $\square$

Observe that (3.4) depends on the chosen normalization of the reference measure. However, this affects the disintegration only by a constant factor on each ray. Namely, given two disintegrations with a weight function f and g respectively

$$\begin{aligned} \mathfrak {m}\llcorner \mathcal {T}^b_u=\int _Q \mathfrak {n}^f_q/f\mathfrak {q}^f({\,\textrm{d}}q)=\int _Q \mathfrak {n}^g_q/g\mathfrak {q}^g({\,\textrm{d}}q) \end{aligned}$$

as constructed in the proof, where the quotient space $(Q,\mathcal {Q})$ does not rely on normalizations. By the positivity of weight functions, $\mathfrak {q}^f$ and $\mathfrak {q}^g$ are mutually absolutely continuous. Hence the essential uniqueness of consistent disintegration yields an equality between $(\mathfrak {m}^f_q)_q$ and $(\mathfrak {m}_q^g)_q$.

Nevertheless, existence result of the disintegration is sufficient for our purpose.

4 From $\textrm{CD}^1(K,N)$ to $\textrm{CD}(K,N)$

This section is devoted to the following main theorem, which together with Theorem 3.10 concludes the local-to-global property of $\textrm{CD}(K,N)$.

Theorem 4.1

Let $(X,d,\mathfrak {m})$ be an e.n.b. m.m.s. with $\mathfrak {m}$ locally finite having full support. If it holds $\textrm{CD}^1(K,N)$, then it holds $\textrm{CD}(K,N)$.

It turns out that the approach developed in [6] is powerful enough to work on locally finite spaces with very mild modifications once the $\textrm{CD}^1$-condition is given by Definition 3.7. In subsequent sections, we sketch the proof with the absence of the finiteness of $\mathfrak {m}$, following closely [6], highlighting necessary modifications. Note that the following part is by no means self-contained, so a parallel reading on the paper [6] is recommended for readers looking for details.

4.1 $\textrm{CD}^1(K,N)$ implies $\textrm{MCP}(K,N)$

We begin with recovering the $\textrm{MCP}$-condition.

Proposition 4.2

If a m.m.s. $(X,d,\mathfrak {m})$ verifies $\textrm{CD}^1(K,N)$, then it verifies $\textrm{MCP}(K,N)$.

Proof

By definition, we need to show that for any $o\in X$ and $\mu _0 := \frac{\mathfrak {m}\llcorner A}{\mathfrak {m}(A)}$, there exists $\nu \in \textrm{OptGeo}(\mu _0,\delta _o)$ such that (2.2) is satisfied, where $A\subset X$ is an arbitrary Borel set with $0<\mathfrak {m}(A)<\infty$. We can further assume A to be bounded by [14, Remark 5.1].

Choosing $u=d(\cdot ,o)$, the $\textrm{CD}^1$-condition provides a disintegration of $\mathfrak {m}$ on $\mathcal {T}_u=X$ s.t. for $\mathfrak {q}$-a.e. $q\in Q$, $(R_u(q),d,\mathfrak {m}_q)$ verifies $\textrm{CD}(K,N)$, and in particular $\textrm{MCP}(K,N)$. Based on the uniform-local finiteness (3.2), the function $Q\ni q\mapsto \mathfrak {m}_q(A)$ is $\mathfrak {q}$-measurable and almost everywhere finite. For all q in

$$\begin{aligned} {\bar{Q}} := \left\{ q\in Q:\mathfrak {m}_q(A)\in (0,\infty ),\textrm{supp}(\mathfrak {m}_q)=R_u(q)=\overline{R_u^b(q)}\right\} , \end{aligned}$$

define $\mu ^q_0:= \frac{\mathfrak {m}_q\llcorner A}{\mathfrak {m}_q(A)}$. By the maximality of $R_u(q)$, $o\in \textrm{supp}(\mathfrak {m}_q)$ and there exists a unique $\nu ^q\in \textrm{OptGeo}(\mu ^q_0, \delta _o)$ for $q\in {\bar{Q}}$. Take $\nu =\int _{{\bar{Q}}}\nu ^q\frac{\mathfrak {m}_q(A)}{\mathfrak {m}(A)}\mathfrak {q}({\,\textrm{d}}q)$ and all curves in its support are contained in a common bounded subset of X. Then going in lines of the proof of [6, Proposition 8.9] validates that $\nu$ is a required optimal dynamical plan from $\mu _0$ to $\delta _o$. $\square$

As a consequence, all statements in Sect. 2.2 now hold on e.n.b. $\textrm{CD}^1(K,N)$ spaces and the underlying metric space must be Polish, proper and geodesic.

Let $\mu _0$ and $\mu _1$ be two arbitrary elements in $\mathcal {P}_2(X,\mathfrak {m})$ and $\nu \in \textrm{OptGeo}(\mu _0,\mu _1)$. Fix a Kantorovich potential $\varphi$ of the quadratic optimal transport from $\mu _0$ to $\mu _1$ and denote by $(\varphi _t)_{t \in [0, 1]}$ the family of intermediate-time Kantorovich potentials.

By Proposition 2.7, it is sufficient to show that the density $\rho _t$ of $\mu _t := (e_t)_{\#}\nu$ w.r.t. $\mathfrak {m}$ satisfies the distortion inequality

$$\begin{aligned} \rho ^{- 1 / N}_t (\gamma _t) \ge \tau ^{(1 - t)}_{K, N} (\ell (\gamma )) \rho ^{- 1 / N}_0 \left( \gamma _0\right) + \tau ^{(t)}_{K, N} (\ell (\gamma )) \rho ^{- 1 / N}_1 \left( \gamma _1\right) \end{aligned}$$

(4.1)

for $\nu$-a.e. $\gamma \in \textrm{Geo}(X)$.

For this aim, we will localize the whole problem to transport paths, by coupling different disintegrations. Since we only care the almost-everywhere statement of (4.1), by Sect. 2.2, we can work under the following convenient convention.

Convention 4.3

In the sequel, we restrict ourselves to a Borel subset of Kantorovich geodesics of full $\nu$-measure, still denoted by $G_{\varphi }$ with a slight abuse of notation, such that

• (1) $e_t$ is injective on $G_{\varphi }$ for all $t\in [0,1]$;

• (2) $(\rho _t)_t$ can be chosen that statements in (1) of Corollary 2.8 hold for each $\gamma \in G_{\varphi }$.

4.2 $L^2$-decomposition of transports.

Based on discussions in Sect. 2.3, for fixed $s,t\in [0,1]$, we have two families of partitions given by level sets of continuous functions as follows

$$\begin{aligned} G_{\varphi }=\sqcup _{a_s\in \mathbb {R}}G_{\varphi ,a_s},\quad e_t (G_{\varphi }) = \sqcup _{a_s\in \mathbb {R}} e_t(G_{\varphi , a_s}) \end{aligned}$$

(4.2)

where $G_{\varphi ,a_s} := \left\{ \gamma \in G_{\varphi }: \varphi _s(\gamma _s)=a_s\right\}$.

Replace $G_{\varphi }$ by any compact subset G with $\nu (G)>0$ and by Remark 3.4, there exist disintegrations of finite measures $\nu \llcorner G$ and $\mathfrak {m}\llcorner e_t(G)$ strongly consistent with partitions (4.2) respectively. Notice that all arguments in [6, Section 10.2] can be repeated without any change so quotient measures are absolutely continuous to the one-dimensional Lebesgue measure $\mathcal {L}^1$ for both disintegrations induced. More precisely, we can find $(\nu _{a_s})$ and $(\mathfrak {m}^t_{a_s})$ concentrated on $G_{a_s}( := G\cap G_{\varphi ,a_s})$ and $e_t \left( G_{a_s}\right)$ respectively so that

$$\begin{aligned} \nu = \int \nu _{a_s} \mathcal {L}^1 \left( {\,\textrm{d}}a_s\right) , \quad \mathfrak {m} \llcorner e_t (G) = \int \mathfrak {m}^t_{a_s} \mathcal {L}^1 \left( {\,\textrm{d}}a_s\right) . \end{aligned}$$

(4.3)

The two families of conditional measures in (4.3) are comparable under the relation

$$\begin{aligned} \mu _t \llcorner e_t (G)&=(e_t)_{\#}(\nu \llcorner G)=\int _{\varphi _s (e_s (G))} (e_t)_{\#} \nu _{a_s} \mathcal {L}^1 ({\,\textrm{d}}a_s) \\ {}&= \rho _t\mathfrak {m}\llcorner e_t(G)=\int _{\varphi _s \left( e_s (G)\right) } \rho _t \cdot \mathfrak {m}^t_{a_s}\mathcal {L}^1 \left( {\,\textrm{d}}a_s\right) . \end{aligned}$$

By Remark 3.1, for $\mathcal {L}^1$-a.e. $a_s \in \varphi _s (e_s (G))$, $\rho _t \cdot \mathfrak {m}_{a_s}^t =(e_t)_{\#} \nu _{a_s}$.

4.3 $L^1$-decomposition of $\mathfrak {m}$ via needle decomposition.

For any $s \in (0, 1)$ and $a_s \in \textrm{Im}(\varphi _s \circ e_s)$, denote $u := d_{\varphi _s-a_s}$ as the signed distance function from $\{ \varphi _s = a_s \}$. By [6, Lemma 10.3], for every $\gamma \in G_{\varphi , a_s}$ and $0 \le r \le t \le 1$, $(\gamma _r, \gamma _t) \in R_u$. In particular, $e_{[0, 1]} (G_{\varphi , a_s}) \subset \mathcal {T}_u$.

Again, when we restrict the $L^1$-disintegration to a compact subset, and with the uniform boundedness of conditional measures given by (3.2), a repetition of [6, Propositon 10.4] can be performed as follows.

Proposition 4.4

For any compact subset $G\subset G^+_{\varphi }$ with positive measure, $s\in (0,1)$ and $a_s\in \varphi _s(e_s(G))$, we have the following disintegration:

$$\begin{aligned} \mathfrak {m} \llcorner e_{[0, 1]} \left( G_{a_s}\right) = \int _{[0, 1]} \mathfrak {m}^{a_s}_t \mathcal {L}^1 ({\,\textrm{d}}t), \end{aligned}$$

(4.4)

where

$$\begin{aligned} \mathfrak {m}^{a_s}_t = g^{a_s} (\cdot , t)_{\#} \left( h^{a_s}_{\cdot } (t)\cdot \mathfrak {m}^{a_s}_s\right) \end{aligned}$$

(4.5)

so that

• (1) $g^{a_s}:e_s\left( G_{a_s}\right) \times [0,1]\rightarrow X$ is Borel measurable, mapping $(\beta ,t)$ to $e_t\left( e^{-1}_s(\beta )\right)$;

• (2) $(0,1)\ni t\mapsto \mathfrak {m}^{a_s}_t$ is continuous under weak convergence, and for each t, $\mathfrak {m}^{a_s}_t$ is concentrated on $e_t\left( G_{a_s}\right)$;

• (3) For $\mathfrak {m}^{a_s}_s$-a.e. $\beta$, $h^{a_s}_{\beta }$ is a continuous $\textrm{CD}\left( \ell ^2_s(\beta )K,N\right)$ density on (0, 1) satisfying $h^{a_s}_{\beta }(s)=1$;

• (4) There exists a constant C depending only on K, N and $\max \left\{ \ell (\gamma ):\gamma \in G\right\}$,

  $$\begin{aligned} \Vert \mathfrak {m}^{a_s}_t \Vert \le C\mathfrak {m} \left( e_{[0, 1]} \left( G_{a_s}\right) \right) ,\quad \forall t\in (0,1). \end{aligned}$$

  (4.6)

Proof

Restrict $L^1$-disintegration to curves in $G_{a_s}$. By definition, one has a probability measure ${\hat{\mathfrak {q}}}^{a_s}$ and a disintegration

$$\begin{aligned} \mathfrak {m}\llcorner \mathcal {T}_{u}=\int _Q {\hat{\mathfrak {m}}}^{a_s}_q{\hat{\mathfrak {q}}}^{a_s}({\,\textrm{d}}q). \end{aligned}$$

(4.7)

Since $e_{[0, 1]}\left( G_{a_s}\right) \subset \mathcal {T}_u$, we can restrict (4.7) to $e_{[0, 1]}\left( G_{a_s}\right)$ so that

$$\begin{aligned} \mathfrak {m}\llcorner e_{[0, 1]}\left( G_{a_s}\right) =\int _Q {\hat{\mathfrak {m}}}^{a_s}_q\llcorner e_{[0, 1]}\left( G_{a_s}\right) {\hat{\mathfrak {q}}}^{a_s}({\,\textrm{d}}q). \end{aligned}$$

If we denote

$$\begin{aligned} G^{1}_{a_s} := \left\{ \gamma \in G_{a_s}:\mathcal {T}^b_u\cap e_{[0,1]}(\gamma )\ne \emptyset \right\} ,\quad Q^1:= \left\{ q\in Q: R^b_u(q)\cap e_{[0,1]}\left( G_{a_s}\right) \ne \emptyset \right\} , \end{aligned}$$

then following exactly same arguments in Part 1–3 of the proof of [6, proposition 10.4] on the ray decomposition and measurability we know $Q^1$ is ${\hat{\mathfrak {q}}}^{a_s}$-measurable, $G^1_{a_s}$ is analytic, and there exists a Borel isomorphism $\eta : \left( Q^1, \mathcal {B}\left( Q^1\right) \right) \rightarrow \left( G_{a_s}^1,\mathcal {B}\left( G_{a_s}^1\right) \right)$, mapping q to $\gamma ^q$.

On the other hand, there exists ${\tilde{Q}}\subset Q$ of full ${\hat{\mathfrak {q}}}^{a_s}$-measure s.t. for each $q\in {\tilde{Q}}$, ${\hat{\mathfrak {m}}}^{a_s}_q$ is non-null, supported on $R_u(q)=\overline{R^b_u(q)}$ and $\left( R_u(q),d,{\hat{\mathfrak {m}}}^{a_s}_q\right)$ verifies $\textrm{CD}(K,N)$. Since each $\gamma ^q\in G_{a_s}$ is contained in $R_u(q)$, $\{\gamma _q\}_{q\in {\tilde{Q}}}$ have disjoint interiors. By (3.2) and the fact that a non-null measure carrying $\textrm{CD}(K,N)$ gives positive mass to open sets, we have

$$\begin{aligned} 0<{\hat{\mathfrak {m}}}^{a_s}_q\left( e_{[0,1]}\left( G_{a_s}\right) \right) ={\hat{\mathfrak {m}}}^{a_s}_q\left( e_{[0,1]}\left( \gamma ^q\right) \right) <C_G,\quad \forall q\in Q^1\cap {\tilde{Q}}, \end{aligned}$$

for some constant $C_{G}>0$. Therefore, summarizing above discussions with Remark 3.9 gives

$$\begin{aligned} \mathfrak {m} \llcorner e_{[0, 1]} \left( G_{a_s}\right)&= \int _{Q^1\cap {\tilde{Q}}} {\hat{\mathfrak {m}}}^{a_s}_q \llcorner \left\{ \mathcal {T}^b_u \cap e_{[0, 1]} \left( G_{a_s}\right) \right\} {\hat{\mathfrak {q}}}^{a_s}({\,\textrm{d}}q) \nonumber \\&= \int _{Q^1\cap {\tilde{Q}}} \frac{{\hat{\mathfrak {m}}}^{a_s}_q \llcorner e_{[0, 1]}\left( \gamma ^q\right) }{{\hat{\mathfrak {m}}}^{a_s}_q \left( e_{[0, 1]} \left( \gamma ^q\right) \right) } {\hat{\mathfrak {m}}}^{a_s}_q\left( e_{[0, 1]} \left( \gamma ^q\right) \right) {\hat{\mathfrak {q}}}^{a_s}({\,\textrm{d}}q). \end{aligned}$$

(4.8)

Denoting ${\bar{\mathfrak {m}}}^{a_s}_q := \frac{{\hat{\mathfrak {m}}}^{a_s}_q \llcorner e_{[0, 1]}\left( \gamma ^q\right) }{{\hat{\mathfrak {m}}}^{a_s}_q \left( e_{[0, 1]} \left( \gamma ^q\right) \right) }$, and ${\bar{\mathfrak {q}}}^{a_s} ={\hat{\mathfrak {m}}}^{a_s}_q \left( e_{[0, 1]} \left( \gamma ^q\right) \right) {\hat{\mathfrak {q}}}^{a_s}\llcorner Q^1$, (4.8) can be rewritten as

$$\begin{aligned} \mathfrak {m} \llcorner e_{[0, 1]} \left( G_{a_s}\right) =\int _{Q^1\cap {\tilde{Q}}} \bar{\mathfrak {m}}^{a_s}_q \bar{\mathfrak {q}}^{a_s} ({\,\textrm{d}}q). \end{aligned}$$

(4.9)

Change the variable and conditional measures. Pushing-forward via the Borel measurable bijection $e_{s}\circ \eta :Q^1\rightarrow e_s\left( G^1_{a_s}\right)$ induces a space $\left( e_s\left( G^1_{a_s}\right) ,\mathcal {S},\check{\mathfrak {q}}^{a_s}\right)$, with $\mathcal {S}:= (e_s\circ \eta )_{\#}\left( \mathcal {Q}\cap Q^1\right)$ and $\check{\mathfrak {q}}^{a_s}=(e_s\circ \eta )_{\#}{\bar{\mathfrak {q}}}^{a_s}$. Correspondingly, (4.9) can be expressed on the new measurable space:

$$\begin{aligned} \mathfrak {m} \llcorner e_{[0, 1]} \left( G_{a_s}\right) = \int _{e_s \left( G_{a_s}\right) }\mathfrak {m}^{a_s}_{\beta } \check{\mathfrak {q}}^{a_s} ({\,\textrm{d}}\beta ), \end{aligned}$$

(4.10)

where $\mathfrak {m}^{a_s}_{\beta } ={\bar{\mathfrak {m}}}^{a_s}_{(e_s\circ \eta )^{-1}(\beta )}$ has unit mass and $\check{\mathfrak {q}}^{a_s}$ is concentrated on $e_s\circ \eta \left( Q^1\cap {\tilde{Q}}\right)$ since ${\bar{\mathfrak {q}}}^{a_s}$ and ${\hat{\mathfrak {q}}}^{a_s}\llcorner Q^1$ are mutually absolutely continuous.

With the new cross section $e_s \left( G_{a_s}\right)$, we define a ray map $g^{a_s}$ as in (3.5) but now with the time variable fixed on [0, 1]:

$$\begin{aligned} g^{a_s}: e_s \left( G_{a_s}\right) \times [0, 1] \rightarrow X, \quad (\beta , t) \mapsto e_t (e_s^{- 1} (\beta )). \end{aligned}$$

Clearly, $g^{a_s}$ is Borel measurable. For any $\beta \in e_s \left( G_{a_s}\right)$, $t \mapsto g^{a_s} (\beta , t)$ is an isometry between $([0, 1], | \cdot |)$ and $(\gamma ^{\beta }:= e_s^{- 1} (\beta ), d /\ell _s (\beta ))$. By assumption, for $\check{\mathfrak {q}}^{a_s}$-a.e. $\beta$, $(\gamma ^{\beta }, d, \mathfrak {m}^{a_s}_{\beta })$ verifies $\textrm{CD}(K, N)$. After rescaling the metric, $(\gamma ^{\beta }, d /\ell _s (\beta ), \mathfrak {m}^{a_s}_{\beta })$ verifying $\textrm{CD}(\ell _s^2(\beta ) K, N)$. For those $\beta$, there exists a continuous function $\check{h}^{a_s}_{\beta }$ as a $\textrm{CD}(\ell _s^2 (\beta ) K, N)$ probability density on (0, 1) s.t.

$$\begin{aligned} \mathfrak {m}^{a_s}_{\beta } = g^{a_s} (\beta , \cdot )_{\#} \left( \check{h}^{a_s}_{\beta } \cdot \mathcal {L}^1 \llcorner [0, 1]\right) \end{aligned}$$

and

$$\begin{aligned} \mathfrak {m} \llcorner e_{[0, 1]} \left( G_{a_s}\right) = \int _{e_s\left( G_{a_s}\right) } g^{a_s} (\beta , \cdot )_{\#} \left( \check{h}^{a_s}_{\beta } \cdot \mathcal {L}^1 \llcorner [0, 1]\right) \check{\mathfrak {q}}^{a_s}({\,\textrm{d}}\beta ). \end{aligned}$$

Reformulate the disintegration on [0, 1]. The item 1 of Definition 3.7 allows us to repeat the step 8 of the proof of [6, Proposition 10.4] to obtain the $\check{\mathfrak {q}}^{a_s}\otimes \mathcal {L}^1$-measurability of $e_s\left( G_{a_s}\right) \times [0,1]\ni (\beta , t)\mapsto \check{h}^{a_s}_{\beta }(t)$, where we also follow the convention that $\check{h}^{a_s}_{\beta }$ vanishes at endpoints. By Fubini, we can exchange the order of (4.10) s.t. (4.4) is achieved with

$$\begin{aligned} \mathfrak {m}^{a_s}_t = g^{a_s} (\cdot , t)_{\#} \left( \check{h}^{a_s}_{\cdot } (t)\cdot \check{\mathfrak {q}}^{a_s}\right) . \end{aligned}$$

(4.11)

The map $\beta \mapsto \check{h}^{a_s}_{\beta }(s)$ is $\check{\mathfrak {q}}^{a_s}$-measurable and for $\check{\mathfrak {q}}^{a_s}$-a.e. $\beta$, $\check{h}^{a_s}_{\beta }$ is strictly positive on (0, 1). Let $h^{a_s}_{\beta }:= \frac{\check{h}^{a_s}_{\beta }}{\check{h}^{a_s}_{\beta }(s)}$ for those $\beta$ and $\mathfrak {q}^{a_s}:= \check{h}^{a_s}_{\beta }(s)\cdot \check{\mathfrak {q}}^{a_s}$. Now $h^{a_s}_{\beta }(s)=1$ for $\mathfrak {q}^{a_s}$-a.e. $\beta$, and $\check{\mathfrak {q}}^{a_s}$, $\mathfrak {q}^{a_s}$ are mutually absolutely continuous (both of them are finite measures) sharing same measurable and null sets. And $g^{a_s}(\cdot ,t)_{\#}( h^{a_s}_{\cdot } (t)\cdot \mathfrak {q}^{a_s})$ equals to $\mathfrak {m}^{a_s}_t$ still. Hence, $\mathfrak {m}^{a_s}_s=\mathfrak {q}^{a_s}$ and the translation relation (4.5) is satisfied.

The continuity of $t\mapsto \mathfrak {m}^{a_s}_t$ follows from the continuity of $t\mapsto h^{a_s}_{\beta }(t),g^{a_s}(\beta ,t)$. Finally, by [6, Lemma A.8], probability densities $\check{h}^{a_{s}}_{\beta }(t)$ are bounded uniformly for $\beta ,t$. The uniform volume bound (4.6) of $\mathfrak {m}^{a_s}_t$ is given by (4.11) and the finiteness of $\check{\mathfrak {q}}^{a_s}$:

$$\begin{aligned} \check{\mathfrak {q}}^{a_s}(e_{[0,1]}\left( G_{a_s}\right) ={\bar{\mathfrak {q}}}^{a_s}(Q^1)=\int _{Q^1}{\hat{\mathfrak {m}}}^{a_s}_q \left( e_{[0, 1]} (\gamma ^q)\right) {\hat{\mathfrak {q}}}^{a_s}({\,\textrm{d}}q)=\mathfrak {m}\left( e_{[0,1]}\left( G_{a_s}\right) \right) .&\end{aligned}$$

$\square$

4.4 Comparison between conditional measures

This section is to recover the comparison between $L^2$ and $L^1$ disintegrations based on [6, Section 11].

Recall that the t-propagated s-Kantorovich potential defined on $D({G_{\varphi }})$, by $\Phi ^t_s (x) := \varphi _t (x) + \frac{t - s}{2} \ell _t^2 (x)$, is jointly continuous and locally Lipschitz on t. The following differential properties will be crucial in the comparison argument. Moreover, they are statements of metric spaces without any reference measure.

Lemma 4.5

(cf. [6, Proposition 4.4]) Fix any $s \in (0, 1)$.

• (1) For any $x\in X$, $t \mapsto \Phi ^t_s (x)$ is differentiable iff $t \mapsto \ell ^2_t (x)$ is differentiable on $G_{\varphi }(x)$ or $t=s\in G_{\varphi }(x)$, with derivatives

  $$\begin{aligned} \partial _t \Phi ^t_s (x) = \ell ^2_t (x) + (t - s) \frac{\partial _t \ell ^2_t (x)}{2}, \quad \partial _t |_{t = s} \partial _t \Phi ^t_s (x) = \ell ^2_s (x). \end{aligned}$$

• (2) For all $(x, t)\in D(G_{\varphi })$,

  $$\begin{aligned}&\min \left\{ \frac{s}{t},\frac{1-s}{1-t}+\frac{t-s}{t(1-t)}\right\} \ell ^2_t(x)\le \liminf _{G_{\varphi }(x)\ni \tau \rightarrow t} \frac{\Phi ^{\tau }_s(x)-\Phi ^t_s(x)}{\tau -t}\\&\quad \le \limsup _{G_{\varphi }(x)\ni \tau \rightarrow t} \frac{\Phi ^{\tau }_s(x)-\Phi ^t_s(x)}{\tau -t}\le \max \left\{ \frac{s}{t},\frac{1-s}{1-t}+\frac{t-s}{t(1-t)}\right\} \ell ^2_t(x). \end{aligned}$$

Proposition 4.6

Let G be a compact subset of $G_{\varphi }^+$ with $\nu (G)>0$. For any $s \in (0, 1)$, $\mathcal {L}^1$-a.e. $t \in (0, 1)$ including $t = s$ and $\mathcal {L}^1$-a.e. $a_s \in \varphi _s (e_s (G))$, we have

$$\begin{aligned} \mathfrak {m}^{a_s}_t = \partial _t \Phi ^t_s \cdot \mathfrak {m}^t_{a_s}, \end{aligned}$$

(4.12)

where $\partial _t \Phi ^t_s(x)$ exists and is positive for $\mathfrak {m}^{t}_{a_s}$-a.e. x.

Sketch of proof

By Lemma 4.5, for any $x \in X$, $\partial _t \Phi ^t_s (x)$ exists for $\mathcal {L}^1$-a.e. $t \in G_{\varphi }(x)$ including $t = s$, and wherever differentiable, $\partial _t \Phi ^t_s (x)>0$. Then the statement on differentiability can be concluded by applying Fubini to the set $\{(x, t): \exists \gamma \in G_{\varphi }, \gamma _t = x \} \subset X \times (0, 1)$ to rephrase the exceptional set of differentiation, together with the disintegration $\mathfrak {m}\llcorner e_t(G)=\int \mathfrak {m}^t_{a_s}\mathcal {L}^1({\,\textrm{d}}a_s)$.

Now we start to show the equivalence. The main idea is a “sum-up” of $\mathfrak {m}^{a_s}_t$ for all $a_s \in \varphi _s (e_s (G))$ to recover $\mathfrak {m} \llcorner e_t (G)$ (which has a disintegration $\int \mathfrak {m}^t_{a_s}\mathcal {L}^1({\,\textrm{d}}a_s)$) and so to connect the two families of conditional measures.

For $t_0 \in \mathbb {R}$ and $x_0 \in X$, define

$$\begin{aligned} 1^1_{t_0}: X \ni x \mapsto (t_0, x) \in \mathbb {R} \times X, \quad 1^2_{x_0}: \mathbb {R} \ni t \mapsto (t, x_0) \in \mathbb {R} \times X. \end{aligned}$$

Recall that $\mathfrak {m}^{a_s}_t$ is supported inside a common compact set (e.g. $e_{[0,1]}(G)$) for all t and $a_s$, having uniformly bounded mass by (4.6). Using the following variant of (4.4)

$$\begin{aligned} \mathfrak {m} \llcorner e_{(t-\epsilon ,t+\epsilon )}\left( G_{a_s}\right) =\int _{(t-\epsilon ,t+\epsilon )} \mathfrak {m}^{a_s}_{\tau }\mathcal {L}^1 ({\,\textrm{d}}\tau ), \end{aligned}$$

and the continuity of $\tau \mapsto \mathfrak {m}^{a_s}_{\tau }$, we have

$$\begin{aligned} \int _{\varphi _s (e_s (G))} \left( 1^1_{a_s}\right) _{\#} \mathfrak {m}^{a_s}_t\mathcal {L}^1 ({\,\textrm{d}}a_s) =\underset{\epsilon \rightarrow 0}{\lim }\ \int _{\varphi _s (e_s (G))} \frac{1}{2 \epsilon } \left( 1^1_{a_s}\right) _{\#} \mathfrak {m} \llcorner e_{(t - \epsilon , t + \epsilon )} \left( G_{a_s}\right) \mathcal {L}^1 ({\,\textrm{d}}a_s), \end{aligned}$$

under the weak topology. Manipulating the right-hand side via Fubini and functions $(\Phi ^t_s)_t$ as in the proof of [6, Theorem 11.3], one gets

$$\begin{aligned} \int _{\varphi _s (e_s (G))} \left( 1^1_{a_s}\right) _{\#} \mathfrak {m}^{a_s}_t\mathcal {L}^1 ({\,\textrm{d}}a_s)=\underset{\epsilon \rightarrow 0}{\lim }\ \int _{e_t (G)} \frac{1}{2 \epsilon } \left( 1^2_x\right) _{\#} \left( \mathcal {L}^1 \llcorner \left\{ \Phi ^{\tau }_s (x): \tau \in (t - \epsilon , t + \epsilon ) \cap G(x)\right\} \right) \mathfrak {m} ({\,\textrm{d}}x), \end{aligned}$$

where we only need the uniform boundedness of mass of measures

$$\begin{aligned} \frac{1}{2 \epsilon } \mathcal {L}^1 \llcorner \left\{ \Phi ^{\tau }_s (x): \tau \in (t -\epsilon , t + \epsilon ) \cap G(x)\right\} \end{aligned}$$

(4.13)

for all $\epsilon$, which is ensured by Lemma 4.5 and the compactness of G. Besides, since $t\mapsto \Phi ^t_s (x)$ is a strictly monotone Lipschitz function on $G(x)\cap (t-\epsilon ,t+\epsilon )$ (with a uniform Lipschitz bound for $x\in e_t(G)$), one-dimensional measures in (4.13) converge to $\partial _t \Phi ^t_s(x) \delta _{\Phi ^t_s (x)}$ when $\epsilon \rightarrow 0$, for $\mathcal {L}^1$-a.e. $t \in (0, 1)$ and $\mathfrak {m}$-a.e. $x \in e_t(G)$.

As a result, for $\mathcal {L}^1$-a.e. t and $a_s$,

$$\begin{aligned} \int _{\varphi _s (e_s (G))} \left( 1^1_{a_s}\right) _{\#} \mathfrak {m}^{a_s}_t\mathcal {L}^1 ({\,\textrm{d}}a_s) = \int _{e_t (G)} \left( 1^2_x\right) _{\#} \left( \partial _t \Phi ^t_s (x)\delta _{\Phi ^t_s (x)}\right) \mathfrak {m}({\,\textrm{d}}x). \end{aligned}$$

Testing the above equality by $1 \otimes f \in C_b (\mathbb {R} \times X)$ with the disintegration (4.3) of $\mathfrak {m}\llcorner e_t(G)$ implies

$$\begin{aligned} \int _{\varphi _s (e_s (G))} \mathfrak {m}^{a_s}_t \mathcal {L}^1({\,\textrm{d}}a_s) = \int _{\varphi _s (e_s (G))} \partial _t \Phi ^t_s \cdot \mathfrak {m}^t_{a_s} \mathcal {L}^1 ({\,\textrm{d}}a_s). \end{aligned}$$

(4.14)

Actually, disintegrations on both sides of (4.14) are strongly consistent on $\left\{ e_{t}\left( G_{a_s}\right) \right\} _{a_s}$ of $e_t(G)$, and hence (4.12). The assertion for $t=s$ can be proven exactly the same as [6, Theorem 11.3], since the underlying space verifies $\textrm{MCP}(K,N)$ by Proposition 4.2 and $\ell (\gamma )$ is uniformly bounded by the compactness of G. $\square$

4.5 Proof of the main theorem

Once all disintegrations and the comparison are produced, a so-called change-of-variable formula (cf. Equation (11.10) in [6]) can be derived. The remaining part after that, though highly technical, not related to the finiteness of $\mathfrak {m}$, will follow naturally. Here we outline the proof of the main theorem in the locally finite case, closely following Section 11.2, 12 and 13.1 of [6].

Proof of Theorem 4.1

Deriving the change-of-variable formula. First, consider any G, as a compact subset of $G^+_{\varphi }$ with positive $\nu$-measure. Fix $s\in (0,1)$. As mentioned in the end of Sect. 4.2, for every $t\in (0,1)$, $\mathcal {L}^1$-a.e. $a_s \in \varphi _s (e_s (G))$, $\rho _t \cdot \mathfrak {m}_{a_s}^t =(e_t)_{\#} \nu _{a_s}$. By evaluating both of them to $e_t(H)$ for an arbitrary Borel $H\subset G$, we have

$$\begin{aligned} \int _{e_t (H)} \rho _t (x) \cdot \mathfrak {m}^t_{a_s} ({\,\textrm{d}}x)=\nu _{a_s} (H). \end{aligned}$$

In the above integral, replacing $\mathfrak {m}^t_{a_s}$ by $\left( \partial _t \Phi ^t_s\right) ^{- 1} \mathfrak {m}^{a_s}_t$ using Proposition 4.6, and combining the translation formula (4.5), we have

$$\begin{aligned} \int _{e_s (H)} \underbrace{\left( \rho _t \cdot (\partial _t \Phi ^t_s)^{- 1}\right) \circ g^{a_s} (\beta , t) h^{a_s}_{\beta } (t)}_{ := f_{t}(\beta )}\mathfrak {m}^{a_s}_s ({\,\textrm{d}}\beta )=\nu _{a_s} (H), \end{aligned}$$

(4.15)

for $\mathcal {L}^1$-a.e. $t \in (0, 1)$ including $t=s$ and $a_s \in \varphi _s (e_s (G))$. Denote by $f_{t}(\beta )$ the integrand in (4.15). From the arbitrariness of H, there is a subset $T\subset [0,1]$ of full measure s.t. for each $t\in T$, $f_{t}=f_{s}$ for $\mathfrak {m}^{a_s}_{s}$-a.e. $\beta$ (due to the continuity of $\rho _t(\cdot )$, $h^{a_s}_{\beta }(\cdot )$ and $g^{a_s}(\beta ,\cdot )$, and the fact that $\partial _t\Phi ^t_s(x)$ converges to $\ell ^2_s(x)$ when $t\rightarrow s$ by Lemma 4.5). Recall that $h^{a_s}_{\beta }(s)=1$ for $\mathfrak {m}^{a_s}_{s}$-a.e. $\beta$ and $g^{a_s}(\cdot , s)=id$. Hence, for $\mathcal {L}^1$-a.e. t,

$$\begin{aligned} f_{t}(\beta )=\left( \rho _t \cdot (\partial _t \Phi ^t_s)^{- 1}\right) \circ g^{a_s} (\beta , t) h^{a_s}_{\beta } (t)=f_{s}(\beta )=\rho _{s}(\beta )/\ell ^2_s(\beta ), \end{aligned}$$

(4.16)

for $\mathfrak {m}^{a_s}_{s}$-a.e. $\beta \in e_s (G)$. Again by Proposition 4.6, $\mathfrak {m}^{a_s}_s$ and $\mathfrak {m}_{a_s}^s$ are mutually absolutely continuous, so (4.16) holds for $\mathfrak {m}_{a_s}^s$-a.e. $\beta$ as well. Further, the validity of (4.16) for almost each $a_s$ indicates, after recovering $\mathfrak {m}\llcorner e_s(G)$ by disintegration $\int \mathfrak {m}^s_{a_s} \mathcal {L}^1({\,\textrm{d}}a_s)$, that (4.16) holds for $\mathfrak {m}$-a.e. $\beta = \gamma _s$ with $\gamma \in G$.

In conclusion, after changing the variable $\beta$ to $\gamma _s$, for $\nu$-a.e. $\gamma \in G$, and $\mathcal {L}^1$-a.e. $t \in (0, 1)$, we have

$$\begin{aligned} \frac{\rho _s(\gamma _s)}{\rho _t(\gamma _t)}=\frac{ h_{\gamma _s}^{\varphi _s (\gamma _s)} (t)}{\partial _{\tau }|_{\tau =t}\Phi _s^{\tau }(\gamma _t)/\ell ^2(\gamma )}. \end{aligned}$$

(4.17)

Recall from the construction in Proposition 4.4 that, $\check{h}_{\beta }^{a_s} (t)$ is uniquely defined as the continuous density of ${\hat{\mathfrak {m}}}^{a_s}_{q}$ (given by the $L^1$-disintegration (4.7) of $\mathcal {T}_u$) after conditioning it on $e_{[0, 1]} (\gamma ^q)$ and pulling it back to the interval [0, 1] via the ray map $g^{a_s}$ ( which can be defined on the whole $e_s(G_{\varphi }^+)\times [0,1]$ by Convention 4.3). In particular, $h^{a_s}_{\beta }$ and hence (4.17) does not depend on the choice of G. Then by the inner regularity of $\nu$, the validity of (4.17) holds for $\nu$-a.e. $\gamma \in G^{+}_{\varphi }$.

“L-Y” decomposition of the density along $\gamma \in G_{\varphi }^+$. We show that along each $\gamma$ satisfying (4.17), the density admits a decomposition $\rho _t(\gamma _t)^{-1}=L(t)Y(t)$, where L is concave and Y is a $\textrm{CD}(\ell ^2(\gamma )K,N)$ density on (0, 1).

All steps in the proof of [6, Theorem 12.3] can be repeated since it is only a matter of one-dimensional analysis on [0, 1]. Once we check that condition (C) is satisfied in the statement of [6, Theorem 12.3] (the validity of (A) and (B) is clear by Convention 4.3 and Proposition 4.4). Indeed, the condition is reduced to an estimate of the 3-rd order derivative of $t\mapsto \varphi _t(\gamma _t)$, where no difference occurs between finite and locally finite spaces.

Afterwards, an application of Hölder’s inequality (cf. [6, Theorem 13.2]) to the “L-Y" decomposition, with the upper semi-continuity of $t\mapsto \rho _t(\gamma _t)$ at $t=0,1$ from Convention 4.3, yields the desired inequality (4.1).

On null-geodesics. Denote by $G_{\varphi }^0$ the set of all curves in $G_{\varphi }$ with zero length and $X_0 := e_{[0,1]}(G^0_{\varphi })$. By [6, Corollary 9.8], as a consequence of Corollary 2.8, $\mu _t\llcorner X_0=\mu _0\llcorner X_0$ for all $t\in [0,1]$. As a result, same to the step 0 of [6, Theorem 11.4] we can always redefine $\rho _t\llcorner X_0 := \rho _0\llcorner X_0$ so that (4.17) holds automatically over $\gamma \in G^0_{\varphi }$ and $t\mapsto \rho _t(\gamma _t)$ will not be affected for all $\gamma \in G_{\varphi }^+$. $\square$

Appendix A. Proof of Ray Decomposition

Definition A.1

A non-negative function h on an interval $I\subset \mathbb {R}$ is called a $\textrm{CD}(K,N)$ density if for all $x_0, x_1 \in I$ and $t \in [0, 1]$:

$$\begin{aligned} h \left( t x_1 + (1 - t) x_0\right) ^{\frac{1}{N - 1}} \ge \sigma _{K, N - 1}^{(t)} (|x_0 - x_1 |) h (x_1)^{\frac{1}{N - 1}} + \sigma _{K, N - 1}^{(1 - t)}(|x_0 - x_1 |) h (x_0)^{\frac{1}{N - 1}}. \end{aligned}$$

(A.1)

The name comes from the fact that a 1-dimensional m.m.s. $(I,|\cdot |,\mu )$ verifies $\textrm{CD}(K,N)$ if and only if $\mu \ll \mathcal {L}^1$ and the density $h={\,\textrm{d}}\mu /{\,\textrm{d}}\mathcal {L}^1$ has a version being a $\textrm{CD}(K,N)$ density (see [6, Theorem A.2]). Moreover, if $h\in C^2_{\textrm{loc}}(I)$, then h is a $\textrm{CD}(K,N)$ density if and only if

$$\begin{aligned} \frac{((\log h)')^2}{N - 1} + (\log h)'' \le - K. \end{aligned}$$

We call a property on $I \subset \mathbb {R}$ local if once it holds on an interval $I_x$ of any point $x \in I$, then it holds globally on I. In particular, being positive, locally Lipschitz or a $\textrm{CD}(K, N)$ density are all local properties in $\mathbb {R}$ (see [10, Section 5] for the local-to-global property of $\textrm{CD}(K, N)$ densities).

Completion of the Proof of Theorem 3.10

Via the ray map g introduced in (3.5), $\mathfrak {m}\llcorner \mathcal {T}^b_u$ can be reformed as a measure on $S \times \mathbb {R}$:

$$\begin{aligned} \left( g^{- 1}\right) _{\#} \left( \mathfrak {m}\llcorner \mathcal {T}^b_u\right) = \left( g^{- 1}\right) _{\#} \mathfrak {m}\llcorner \left( \mathfrak {Q}^{- 1} (S)\right) = \int _S \left( g^{- 1} (q, \cdot )\right) _{\#} \mathfrak {m}_q \mathfrak {q} ({\,\textrm{d}}q), \end{aligned}$$

where the second equality is guaranteed by the strong consistency of disintegration $q\mapsto \mathfrak {m}_q$. As $\left( g^{- 1} (q, \cdot )\right) _{\#} \mathfrak {m}_q$ is locally-finite on $\mathbb {R}$ from (3.3), Lebesgue’s decomposition gives

$$\begin{aligned} \left( g^{- 1} (q, \cdot )\right) _{\#} \mathfrak {m}_q = h_q \mathcal {L}^1 + \omega _q, \quad \omega _q \perp \mathcal {L}^1. \end{aligned}$$

• (i) It suffices to show for $\mathfrak {q}$-a.e. $q \in S_{a, b}$, $\mathfrak {m}_q$ verifies Theorem 3.10 on [a, b], where

  $$\begin{aligned} S_{a, b} := \left\{ q \in S:[a,b]\subset I_q, a,b\in \mathbb {Q} \right\} . \end{aligned}$$

  Such $S_{a,b}$ is always $\mathfrak {q}$-measurable, since

  $$\begin{aligned} S_{a,b}= P_1\left( \left\{ (x,y,z)\in S^3:(x,y),(x,z)\in R_u,u(y)-u(x)\ge b, u(z)-u(x)\le a\right\} \right) . \end{aligned}$$

  Let $S_{a,b}^{*}$ be the set of all q in $S_{a,b}$ s.t. Theorem 3.10 is violated somewhere in [a, b]. As all statements are local, the set $Q^{*} := \left\{ q\in S:\mathfrak {m}_q\text { does not verify Theorem } 3.10\, \text { on }I_q\right\}$ is contained in $\cup _{a,b\in \mathbb {Q}}S^{*}_{a,b}$.

  It can be reduced to show each $S^*_{a,b}$ is negligible and hence for the time being, we assume S a bounded subset of $S_{a, b}$. For simplicity, we directly assume $\mathfrak {m}$ a measure on $S \times \mathbb {R}$ to avoid writing g all the time.

• (ii) Prove that $\mathfrak {m}_q \ll \mathcal {L}^1$. If otherwise, there exists a bounded set $A \subset \mathcal {T}^b_u \subset S \times \mathbb {R}$, $\mathfrak {m}(A) > 0$ but for $\mathfrak {q}$-a.e. $q \in S$

  $$\begin{aligned} \mathcal {L}^1 \left( A \cap \left( \{ q \}\times \mathbb {R}\right) \right) = 0. \end{aligned}$$

  (A.2)

  Take $\nu$ to be the unique optimal dynamical plan transporting $\mu _0 := \mathfrak {m}(A)^{- 1} \mathfrak {m}\llcorner A$ onto $S \times \{ a \}$ along vertical rays $R^b_u (q) = \{ q \} \times I_q$. Denote $A_t := e_t (\textrm{supp}(\nu ))$,

  $$\begin{aligned} A_t = \left\{ (q, (1-t)\tau + ta): (q, \tau ) \in A \right\} . \end{aligned}$$

  (A.3)

  Corollary 2.8 ensures $\mathfrak {m}(A_t) > 0$ for each $t \in [0, 1)$, so a contradiction follows:

  $$\begin{aligned} 0< & {} \int _0^{1 / 2} \mathfrak {m}(A_t) {\,\textrm{d}}t =\mathfrak {m}\otimes \mathcal {L}^1 \left( \left\{ (q, \tau , t) \in S \times \mathbb {R} \times [0, 1 / 2]: (q, \tau ) \in A_t \right\} \right) \\= & {} \int _{S \times \mathbb {R}} \mathcal {L}^1 \left( \{ t \in [0, 1 / 2]: (q, \tau ) \in A_t \}\right) \mathfrak {m}({\,\textrm{d}}q {\,\textrm{d}}\tau )\\\overset{(A.3)}{=} & {} \int _{S \times \mathbb {R}} \mathcal {L}^1 \left( \left\{ t\in [0, 1 / 2]:(q,\frac{\tau - at}{1 - t})\in A \right\} \right) \mathfrak {m}({\,\textrm{d}}q {\,\textrm{d}}\tau ) \overset{(A.2)}{=} 0. \end{aligned}$$

• (iii) Prove that $h_q$ is positive and Lipschitz. For any $[a_0, b_0] \subset [a, b]$ and $S_0 \subset S$ with $\mathfrak {q} (S_0) > 0$, take A as a bounded subset of $\mathcal {T}^b_u \cap (S_0 \times \mathbb {R})$ having positive mass and consider the transport optimally moving $\mathfrak {m}(A)^{- 1}\mathfrak {m}\llcorner A$ into $S \times \{ (a_0 + b_0)/2\}$. As in (ii), once $\mathfrak {m}(A) > 0$, $\mathfrak {m}(A_t) > 0$ for all $t \in (0, 1)$. One can easily find $A_t \subset S_0 \times [a_0, b_0]$ for some time t from the boundedness of A, ensuring $\mathfrak {m}(S_0 \times [a_0, b_0]) > 0$.

  Next, taking any such $A = S_0 \times [a_0, b_0]$, apply (2.7) to the optimal dynamical plan $\nu$ transporting $\mu _0 := \mathfrak {m}(A)^{- 1}\mathfrak {m}\llcorner A$ into $S \times \{ b \}$. By disintegration, we have

  $$\begin{aligned} \mathfrak {m}(A_t) = \int _{S_0} \int _{[a_0, b_0]_t} h_q {\,\textrm{d}}\mathcal {L}^1 {\,\textrm{d}}\mathfrak {q}, \quad [a_0, b_0]_t := [(1 - t) a_0 + t b, (1 - t) b_0 + t b]. \end{aligned}$$

  Given any $0 \le r< s < 1$, the arbitrariness of $S_0$ implies, for $\mathfrak {q}$-a.e. $q \in S$

  $$\begin{aligned} c (r, s) \int _{[a_0, b_0]_r} h_q {\,\textrm{d}}\mathcal {L}^1 \le \int _{[a_0, b_0]_s} h_q {\,\textrm{d}}\mathcal {L}^1 \le C (r, s) \int _{[a_0, b_0]_r} h_q {\,\textrm{d}}\mathcal {L}^1, \end{aligned}$$

  (A.4)

  where c(r, s), C(r, s) are locally Lipschitz functions of r, s given by (2.7). Since $h_q$ (or $\mathfrak {m}_q$) is locally finite, both sides of (A.4) continuously depend on $s, r, a_0, b_0$, and so (A.4) holds simultaneously for all $r, s, a_0, b_0$. At Lebesgue points $\tau _0, \tau _1$ of $h_q$, choosing $[a_0, b_0] = [\tau _0 - \varepsilon , \tau _0 + \varepsilon ]$, $r = 0$ and $s = \frac{\tau _1 - \tau _0}{b - \tau _0}$ in (A.4) and shrinking $\varepsilon \rightarrow 0$, a two-sided inequality between $h_q (\tau _0)$, $h_q (\tau _1)$ follows, leading to the Lipschitz continuity.

  Finally, because c(r, s), C(r, s) are positive for all s, r, the continuous density $h_q$ is either identically 0 or everywhere positive inside $I_q$. But the positivity of all $\mathfrak {m}(S_0 \times [a, b])$ excludes the former case (up to a $\mathfrak {q}$-negligible set of q).

• (iv) Prove that $h_q$ is a $\textrm{CD}(K, N)$ density. Consider, any $a< A_0< A_1 < b$ and $L_0, L_1 > 0$ with $A_0 + L_0 < A_1$ and $A_1 + L_1 < b$. Define

  $$\begin{aligned} \mu _0 := \int _S \frac{1}{L_0} \mathcal {L}^1 \llcorner \left[ A_0, A_0 + L_0\right] \mathfrak {q} ({\,\textrm{d}}q), \qquad \mu _1 := \int _S \frac{1}{L_1} \mathcal {L}^1 \llcorner \left[ A_1, A_1 + L_1\right] \mathfrak {q} ({\,\textrm{d}}q). \end{aligned}$$

  In (iii), we have shown the positivity of $h_q$, so densities of $\mu _i$ w.r.t. $\mathfrak {m}$ are

  $$\begin{aligned} \rho _i ((q, t))=\frac{1}{L_i} h_q (t)^{- 1},\quad \forall t \in [A_i, A_i + L_i],i = 0, 1. \end{aligned}$$

  When $L_0$ and $A_1 + L_1$ are close enough (up to further localizing S), we can apply $\textrm{CD}_{\textrm{loc}} (K, N)$ to the optimal dynamical plan between $\mu _0$ and $\mu _1$. Then (A.1) follows by the same argument in [7, Theorem 4.2].

$\square$