This is my invited talk at the 9th International Symposium on Domain Theory, Singapore (online), July 5th, 2022.
A famous 1956 theorem due to Prohorov states that the space of probability measures, with the weak topology, over a Polish space is itself Polish. One way of proving this is through the so-called Kantorovich-Rubinstein metric.
I will explain how the latter generalizes naturally to quasi-metric spaces. A quasi-metric space is like a metric space, without symmetry, and the notion generalizes both metric spaces and posets.
The goal of this talk is to show an analogue of Prohorov’s theorem in the quasi-metric world: given any continuous complete quasi-metric space, its space of (sub)probability continuous valuations is continuous complete under the Kantorovich-Rubinstein quasi-metric, and the induced topology is the weak topology. Rather curiously, we will find ourselves at least twice in situations where we have the impression that the proof is almost complete, but where overcoming the remaining, apparently minor, difficulty will require pretty elaborate techniques.
As a central tool, we will extensively rely on the poset of formal balls of a quasi-metric space, and its many properties.
This is based on the following paper:
- Jean Goubault-Larrecq. Kantorovich-Rubinstein quasi-metrics I: spaces of measures and of continuous valuations. Topology and its Applications 295, 2021,
… which uses other results from the following papers:
- Martín Hötzel Escardó. Properly injective spaces and function spaces. Topology and Its Applications 89(1–2):75–120.
- Jean Goubault-Larrecq and Kok Min Ng. A few notes on formal balls. Logical Methods in Comp. Sci. 13(4:18):1–34, 2017.
- Matthew de Brecht, Jean Goubault-Larrecq, Xiaodong Jia, Zhenchao Lyu (2019). Domain-complete and LCS-complete spaces. Proceedings of the 8th ISDT, Electronic Notes in Theoretical Computer Science 345:3–35, 2019.
- Jean Goubault-Larrecq. Formal ball monads. Topology and its Applications 263:372-391, 2019.
- Jean Goubault-Larrecq. Some topological properties of spaces of Lipschitz continuous maps on quasi-metric spaces. Topology and its Applications 282:107281, 2020.
A followup, briefly mentioned at the end of the talk, is:
- Jean Goubault-Larrecq. Kantorovich-Rubinstein quasi-metrics II: hyperspaces and powerdomains. Topology and its Applications 305:107885, 2022
And all of that is just a small part of the following, (very) long paper:
- Jean Goubault-Larrecq. Complete quasi-metrics for hyperspaces, continuous valuations, and previsions. arXiv 1707.03784, 2017. Last revised (v8) July 28th, 2021, 190 pages.
Slides and videos
- Introduction, the classical setting (4:15)
- Quasi-metrics and formal balls (18:09)
- Lipschitz continuous maps (3:41)
- Continuous valuations (4:46)
- The Kantorovich-Rubinstein quasi-metric (8:10)
- Lipschitz-regular quasi-metric spaces (6:57)
- Completeness (9:12)
- Final remarks (6:47)