An approach for metric space with a convex combination operation and applications

TL;DR

This paper introduces a method to embed metric spaces with convex combination operations into Banach spaces, enabling the extension of expectation and conditional expectation concepts with applications to martingales and ergodic theory.

Contribution

It provides a novel embedding theorem for convex combination metric spaces into Banach spaces, facilitating the analysis of expectations and stochastic processes in these spaces.

Findings

Embedding preserves metric and convex combination structures.

Representation of expectation via continuous affine mappings.

Established properties of conditional expectation, Jensen's inequality, and martingale convergence.

Abstract

In this paper, we embed metric space endowed with a convex combination operation, named convex combination space, into a Banach space and the embedding preserves the structures of metric and convex combination. For random element taking values in this kind of space, applications of embedding are also established. On the one hand, some nice properties of expectation such as representation of expected value through continuous affine mappings, the linearity of expectation will be given. On the other hand, the notion of conditional expectation will be also introduced and discussed. Thanks to embedding theorem, we establish some basic properties of conditional expectation, Jensen's inequality, convergences of martingales and ergodic theorem.