<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>A <italic>submetric space</italic> is a topological space with continuous metrics, generating a metric topology weaker than the original one (e.g. a separable Hilbert space with the weak topology).</p>
<p>We demonstrate that on submetric spaces there exists a theory of convergence in probability, in law etc. equally effective as the Probability Theory on metric spaces. In the theory on submetric spaces the central role is played by a version of the Skorokhod almost sure representation, proved by the author some 25 years ago and in 2010 rediscovered by specialists in stochastic partial differential equations in the form of “stochastic compactness method”.</p>
</abstract>

A submetric space is a topological space with continuous metrics, generating a metric topology weaker than the original one (e.g. a separable Hilbert space with the weak topology).
We demonstrate that on submetric spaces there exists a theory of convergence in probability, in law etc. equally effective as the Probability Theory on metric spaces. In the theory on submetric spaces the central role is played by a version of the Skorokhod almost sure representation, proved by the author some 25 years ago and in 2010 rediscovered by specialists in stochastic partial differential equations in the form of “stochastic compactness method”.

Probability on Submetric Spaces

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"Probability on Submetric Spaces"}

A submetric space is a topological space with continuous metrics, generating a metric topology weaker than the original one (e.g. a separable Hilbert space...

Probability on Submetric Spaces

Publié en ligne: 26 juil. 2023

Pages: 138 - 148

Reçu: 25 janv. 2023

Accepté: 13 juil. 2023

DOI: https://doi.org/10.2478/amsil-2023-0012

Mots clés
submetric space, sequential topologies, convergence of probability measures, convergence in probability, Skorokhod almost sure representation, stochastic compactness method

© 2023 Adam Jakubowski, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Probability on Submetric Spaces

Publié en ligne: 26 juil. 2023

Pages: 138 - 148

Reçu: 25 janv. 2023

Accepté: 13 juil. 2023

DOI: https://doi.org/10.2478/amsil-2023-0012

Mots cléssubmetric space, sequential topologies, convergence of probability measures, convergence in probability, Skorokhod almost sure representation, stochastic compactness method

© 2023 Adam Jakubowski, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Mots clés
submetric space, sequential topologies, convergence of probability measures, convergence in probability, Skorokhod almost sure representation, stochastic compactness method