A new mathematical preprint gives a sharp answer to when an atomless probability distribution, one in which no single value carries positive probability, can describe the distance between two independently chosen points in a metric space. The answer is that zero must belong to the distribution's support, meaning the law places probability arbitrarily close to zero.
The question is abstract but precise. The candidate is a Borel probability measure on the nonnegative real numbers. The proposed setting is a complete separable metric space equipped with its own Borel probability measure. Two independent points are sampled from that space, and the distance between them must follow the candidate law.
The rule reaches beyond ordinary densities
The theorem applies to every atomless law that meets the support condition. It explicitly includes atomless measures that are singular with respect to Lebesgue measure, so the result is not confined to distributions described by a conventional density.
It also covers absolutely continuous laws, those represented by a density relative to ordinary length measure. If zero is in the support, the law can be realized without requiring the density to meet an additional regularity condition or to be bounded.
Within the stated class of complete separable metric probability spaces, this makes the result a necessary-and-sufficient feasibility test. It identifies exactly which atomless distance laws can be built and which fail the support condition.
The proof builds the space in layers
The proof is constructive. It replaces isolated points with compact metric components, uses distances between components to reproduce portions of the target law, and assigns distances inside each component to smaller scales. Those layers are then combined so that the full two-point distance distribution matches the measure being tested.
A local part of the construction uses a Borel metric on a sufficiently short interval. That metric is totally bounded, meaning it can be covered by finitely many small pieces at any fixed scale, and it dominates ordinary Euclidean distance. Its internal distance law is controlled by a prescribed atomless measure, which lets the larger construction keep assigning leftover contributions to finer scales.
When the target law has unbounded support, the argument handles its distant tail by dividing it beyond a bounded threshold into ordered, disjoint intervals. Their masses are controlled, while their locations move farther and farther out.
The realizing spaces have a distinctive shape
For a law with bounded support, the construction can produce a compact space with the topological form known as Cantor space. Its diameter is no greater than the bound on the law's support. In this bounded case, the distances realized in the space are exactly the values in the law's support, and the diameter equals the largest supported distance.
The metric can also be made arbitrarily close to an ultrametric, a distance rule in which each triangle is controlled by its longest side. For every comparison factor greater than one, a realization with full measure support can be chosen so that its distance is no smaller than an ultrametric and no more than that factor times the ultrametric.
For an unbounded target law, the construction may instead produce a proper space, one whose closed bounded parts are compact, with the topological form of a countable disjoint union of Cantor spaces. The result therefore describes both whether a law is realizable and what the resulting space can look like.
Two boundaries remain clear
The near-ultrametric result cannot be sharpened to an exact ultrametric for an atomless law in a separable space. The exact comparison factor of one is excluded because a separable ultrametric has only countably many positive distance values, which cannot produce an atomless distance law.
The central characterization stops at atomless measures. The paper leaves feasibility for laws with atoms unresolved and says that having zero in the support is not sufficient for those laws in general.
The theorem also addresses one two-point distance law: the distance between two independent samples. Its question is therefore narrower than describing all distances generated by several sampled points at once.
A theorem, not an empirical estimate
The supplied document is an arXiv version-one preprint dated 28 Aug 2026. The main conclusion is a theorem about abstract probability measures and metric spaces, so it carries no statistical margin of error or estimated effect.
Paper data and sources
Original title: Realising atomless laws as distance distributions on metric measure spaces
Authors: Christoph Thäle, Philipp Tuchel
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text