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Wasserstein spaces have the Radon-Nikodym property

  • Charles Bertucci
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Abstract

Wasserstein spaces carry two natural geometries: a vertical one, inherited from the linear structure of signed measures, and a horizontal one, induced by couplings. We prove that a Radon-Nikodym property holds in both. In the vertical geometry, bounded subsets of Pp(Rd) are dentable in an ambient Banach space which itself fails the Radon-Nikodym property; we derive variational principles, Radon-Nikodym theorems for vector measures and martingale convergence results. In the horizontal geometry, we show that a Radon-Nikodym property is inherited from the one of spaces of random variables. We introduce horizontal martingales and prove that they are exactly the laws of Lp -valued martingales.

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Cite

BibTeXRIS
BibTeX
@misc{bertucci2026wasserstein,
  title = {{Wasserstein spaces have the Radon-Nikodym property}},
  author = {Charles Bertucci},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12028},
  eprint = {2609.12028},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05737539v1},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Wasserstein spaces have the Radon-Nikodym property
AU  - Charles Bertucci
PY  - 2026
DA  - 2026/09/03
PB  - HAL
DO  - 10.48550/arxiv.2609.12028
UR  - https://hal.science/hal-05737539v1
AB  - Wasserstein spaces carry two natural geometries: a vertical one, inherited from the linear structure of signed measures, and a horizontal one, induced by couplings. We prove that a Radon-Nikodym property holds in both. In the vertical geometry, bounded subsets of P_p(R^d) are dentable in an ambient Banach space which itself fails the Radon-Nikodym property; we derive variational principles, Radon-Nikodym theorems for vector measures and martingale convergence results. In the horizontal geometry, we show that a Radon-Nikodym property is inherited from the one of spaces of random variables. We introduce horizontal martingales and prove that they are exactly the laws of L^p -valued martingales.
ER  -