Probability Geometry and Kolmogorov Expectations via Coordinate Charts

TL;DR

This paper introduces a geometric framework for probability that reinterprets expectation through probability coordinates, linking classical means with generalized expectations and extending classical theorems.

Contribution

It develops a geometric reinterpretation of expectation using probability coordinates, connecting Kolmogorov means with generalized expectations and broadening the understanding of expectation.

Findings

Expectation can be defined beyond classical integrability in this framework.

Law of large numbers and central limit theorem hold in the probability coordinate system.

Heavy-tailed phenomena are geometrically interpreted as boundary concentration.

Abstract

This paper develops a geometric reinterpretation of probability in which expectation arises from averaging in probability coordinates rather than in value space. By interpreting the cumulative distribution functions as coordinate maps, a real-valued random variable is transported into the unit interval, where averaging becomes a linear operation in probability coordinates and is then pulled back to the value space. Within this representation, the resulting quantities coincide with Kolmogorov means, thereby linking the construction to the classical theory of generalized means associated with Kolmogorov, Nagumo, de Finetti, and Chisini. This connection clarifies that these means are not merely algebraic devices, but arise from a change of representation of probability. The framework provides a natural setting in which expectation exists beyond classical integrability assumptions. In particular, heavy-tailed phenomena are reinterpreted geometrically: divergence in value space corresponds to boundary concentration in probability coordinates. Laws of large numbers and central limit theorems are established in this coordinate system, showing that asymptotic behaviour can be recovered after this geometric representation. The perspective also connects to broader constructions based on generalized means, including R\'enyi-type formulations, and suggests that expectation should be viewed not as a fixed numerical functional, but as the outcome of a chosen probability representation.