Realization of atomless laws on connected or geodesic spaces

Determine which atomless probability measures on [0,∞) admit a realization by a connected or geodesic metric probability space.

Background

The realizing spaces constructed in the paper are zero-dimensional and are modeled on Cantor-type components or countable disjoint unions of such components. The paper therefore does not address whether atomless distance laws can be realized on spaces with connected or geodesic geometry. A necessary restriction is identified: if the measure has full support and the space is connected, then the support of the distance law must be an interval, because it is the closure of the continuous image of the product space under the distance map.

References

Which atomless laws admit a realisation on a connected or geodesic metric space? If \mu has full support and S is connected, then supp\Theta must be an interval, since it is the closure of the continuous image d(S\times S).

— Realising atomless laws as distance distributions on metric measure spaces  (2608.28330 - Thäle et al., 28 Aug 2026) in Section 5, item (ii) of the enumerated future-research questions