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Borel probability law on a polish space
Definition
Let S be Polish in the sense of Polish spaces are separable completely metrizable spaces. A Borel probability law on S is a countably additive measure on with total mass one. Here is The Borel sigma-algebra of a topological space and probability measure means Probability measures and probability spaces. A compatible complete metric may be fixed for a construction; it is not additional data in the law. The empty space admits no such law, since its measure must be both zero and one.
Depends on
Used by
- A Borel prior concentrated on a forward-map domain Definition
- Centered Borel Gaussian random element on a separable Banach space Definition
- Small-ball MAP conventions and Cameron–Martin relative modes Definition
- The Euclidean quadratic Wasserstein space Definition
- A countable norming family determines Borel laws Lemma
- A narrow probability curve defines its space-time measure Lemma
- Borel laws on a separable Hilbert space are determined by basis coordinates Lemma
- Gluing two Euclidean couplings Lemma
- Regularized continuity equations are represented by characteristic flows Lemma
- Every borel probability on a polish space is tight Theorem
- Finite-action continuity equations generate AC2 Wasserstein curves Theorem
- The quadratic Wasserstein distance is a metric Theorem
- The quadratic Wasserstein space is complete Theorem
Dependency tree · two levels
10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- van Gaans, §2, p. 3 (standard reference, not scraped)