τ‐smooth B<scp>OREL</scp> Measures on Topological Spaces

Adamski, Wolfgang

doi:10.1002/mana.19770780108

Cited by 11 publications

(7 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result (see 3.16) leads to a common generalization of results in [16,26,13,7,8]. 1. Preliminaries.…”

mentioning

confidence: 54%

Extensions of tight set functions with applications in topological measure theory

Adamski¹

1984

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Let Xx, X2 be lattices of subsets of a set X with X, c X2. The main result of this paper states that every semifinite tight set function on Xx can be extended to a semifinite tight set function on JT2. Furthermore, conditions assuring that such an extension is uniquely determined or o-smooth at are given. Since a semifinite tight set function defined on a lattice Jf"[and being o-smooth at ] can be identified with a semifinite ^regular content [measure] on the algebra generated by X, the general results are applied to various extension problems in abstract and topological measure theory.

show abstract

“…This result (see 3.16) leads to a common generalization of results in [16,26,13,7,8]. 1. Preliminaries.…”

mentioning

confidence: 54%

Extensions of tight set functions with applications in topological measure theory

Adamski¹

1984

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Continuous valuations and measures are close cousins. Every τ -smooth Borel measure defines a continuous valuation by restricting it to OX; and every Borel measure on a hereditary Lindelöf space is τ -smooth [1]. Conversely, every continuous valuation on an LCS-complete space extends to a measure on the Borel σ-algebra [5, Theorem 1.1]-an LCS-complete space is any subspace obtained as a G δ subset of a locally compact sober space.…”

Section: Preliminariesmentioning

confidence: 99%

Probabilistic Powerdomains and Quasi-Continuous Domains

Jean¹

2020

Preprint

View full text Add to dashboard Cite

The probabilistic powerdomain VX on a space X is the space of all continuous valuations on X. We show that, for every quasi-continuous domain X, VX is again a quasi-continuous domain, and that the Scott and weak topologies then agree on VX. This also applies to the subspaces of probability and subprobability valuations on X. We also show that the Scott and weak topologies on the VX may differ when X is not quasicontinuous, and we give a simple, compact Hausdorff counterexample.

show abstract

“…He then shows that the spaces of continuous valuations (resp., probability, subprobability continuous valuations) on an algebraic dcpo X are algebraic complete quasi-metric spaces under the sup quasi-metric [29,Theorem 3.4]. Although the sup quasi-metric is very different from the d KR and d a KR metrics on metric spaces, we will see that it is exactly our quasimetric d 1 KR on posets: see Remark 3.6.…”

Section: Introductionmentioning

confidence: 95%

“…This work is part one of a study of quasi-metrics on spaces of so-called previsions [14], of which this is the linear case. The non-linear cases, which model not just probabilistic choice, but also various forms of non-deterministic choice and mixed, probabilistic and non-deterministic choice, will be explored in subsequent parts 1 .…”

Section: Introductionmentioning

confidence: 99%

Kantorovich-Rubinstein quasi-metrics I: Spaces of measures and of continuous valuations

Goubault-Larrecq

2021

Topology and its Applications

View full text Add to dashboard Cite

We show that the space of subprobability measures, equivalently of subprobability continuous valuations, on an algebraic (resp., continuous) complete quasi-metric space is again algebraic (resp., continuous) and complete, when equipped with the Kantorovich-Rubinstein quasi-metrics d KR (unbounded) or d a KR (bounded), themselves asymmetric forms of the well-known Kantorovich-Rubinstein metric. We also show that the d KR -Scott and the d a KR -Scott topologies then coincide with the weak topology. We obtain similar results for spaces of probability measures, equivalently of probability continuous valuations, with the d a KR quasi-metrics, or with the d KR quasi-metric under an additional rootedness assumption.

show abstract

τ‐smooth BOREL Measures on Topological Spaces

Cited by 11 publications

References 10 publications

Extensions of tight set functions with applications in topological measure theory

Extensions of tight set functions with applications in topological measure theory

Probabilistic Powerdomains and Quasi-Continuous Domains

Kantorovich-Rubinstein quasi-metrics I: Spaces of measures and of continuous valuations

Contact Info

Product

Resources

About