Let $(\Omega,d)$ be a compact metric space and $\mathcal P(\Omega)$ its space of Borel probability measures. Let $D=\{ d_p\mid p\in\Omega\}$ where $d_p(x)=d(p,x)$ be the set of all "distance functionals". As usual, we can think of $D$ acting on $\mathcal P(\Omega)$ (or vice versa) via integration i.e. $\langle d_p,\mu\rangle = \int_\Omega d_p(x)\,\mathrm d\mu(x)$.

# Title Question

Does $D$ acting on $\mathcal P(\Omega)$ via integration separate points?

Or equivalently,

If $\mu,\nu \in \mathcal P(\Omega)$ and $\langle d_p,\mu\rangle = \langle d_p,\nu\rangle$ for all $p\in \Omega$, then must $\mu=\nu$?

# Alternative Formulations

There are a few other ways to frame the question as well.

### Probabilistic Formulation

Rewriting all integrals as expectations the question becomes,

If $\mathbb E_{X\sim\mu}[d_p(X)] = \mathbb E_{Y\sim\nu}[d_p(Y)]$ for all $p\in \Omega$, then must $\mu=\nu$?

In other words, does knowing the expected distance to a point for all points determine the measure?

### Geometric Formulation

Recall that the 1-Wasserstein distance is given by $W_1(\mu,\nu) = \inf_{\gamma\in\Gamma(\mu,\nu)} \int_{\Omega\times\Omega} d(x,y) \,\mathrm d\gamma(x,y)$ where $\Gamma(\mu,\nu)$ is the set of couplings between $\mu$ and $\nu$ i.e. Borel probability measures on $\Omega\times\Omega$ with marginals $\mu$ and $\nu$ respectively. Since the product measure $\delta_p\otimes\mu$ is the unique coupling between a Dirac delta measure $\delta_p$ and $\mu$, we have that

$$W_1(\delta_p,\mu)=\int_{\Omega\times\Omega} d(x,y)\,\mathrm d(\delta_p\otimes\mu)(x,y)=\int_\Omega d(p,y)\,\mathrm d\mu(y)=\langle d_p,\mu\rangle$$

Now the question can be stated geometrically as

If $W_1(\delta_p,\mu)=W_1(\delta_p,\nu)$ for all $p\in \Omega$, then must $\mu=\nu$?

In other words, does knowing the $W_1$ distance to the extreme points of $\mathcal P(\Omega)$ completely determine the probability measure?

### Integral Transform Forumlation

Define the **distance transform** of $\mu\in\mathcal P(\Omega)$ as the function $\phi_\mu:\Omega\to\mathbb R$ given by $\phi_\mu(p) = \int_\Omega d(p,x)\,\mathrm d\mu(x)$. The question can now be restated as,

Is the distance transform injective on $\mathcal P(\Omega)$?

Moreover, by the geometric formulation we have $\phi_\mu(p) = W_1(\delta_p,\mu)$. We will use the weak-$*$ topology for $\mathcal P(\Omega)$ (which coincides with the $W_1$ topology). Since the map $p\mapsto \delta_p$ is an embedding of $\Omega$ into $\mathcal P(\Omega)$, it follows that $\phi_\mu:\Omega\to\mathbb R$ is continuous. Denote the distance transform by $\Phi(\mu)=\phi_\mu$. Since $\mathcal P(\Omega)$ is compact Hausdorff and $C(\Omega)$ is Hausdorff we can restate the question as

If $\Phi:\mathcal P(\Omega)\to C(\Omega)$ is continuous, is it an embedding?

# Final Thoughts

Are any of these equivalent statements true? I have unfortunately only been able to reformulate the question and have not identified any clear proof, though I wouldn't be surprised if there is an easy one I'm overlooking. The geometric formulation of the problem leads me to believe that $D$ does indeed separate points in $\mathcal P(\Omega)$. However, if the answer is affirmative then I feel the resulting nice properties of $\Phi$ would make it something that would be easy to look up. Any insight would be appreciated.

**Update:** In light of George Lowther's elegant 4-point counter-example and Pietro Majer's affirmative answer for $\Omega=[0,1]$, it would be interesting to better understand what factors determine whether the underlying metric space yields an affirmative answer.

George's counter-example can be extended to counter-examples where $\Omega$ is a sphere (with intrinsic metric). Thus, requiring $\Omega$ to be positive-dimensional, a manifold, connected, path-connected, simply-connected, etc, will not make the issue go away. On the other hand, Pietro suspects that the answer is again affirmative in the case when $\Omega$ is a compact convex subset of Euclidean space.

algebragenerated by the distance functionals is dense. One needs the weak$^*$ density of the generatedsubspace. $\endgroup$ – Jochen Wengenroth Aug 17 at 9:15isa problem... $\endgroup$ – Jochen Glueck Aug 17 at 9:29