Kolmogorov's axioms for probabilities with values in hyperbolic numbers

TL;DR

This paper extends Kolmogorov's probability axioms to hyperbolic numbers, establishing a new framework for probabilities that includes zero-divisors and generalizes classical probability theorems.

Contribution

It introduces hyperbolic-valued probabilistic measures satisfying generalized axioms, including hyperbolic versions of key probability theorems.

Findings

Hyperbolic probability measures satisfy classical properties.

Conditional hyperbolic probability is well-defined.

Hyperbolic analogues of multiplication, total probability, and Bayes' theorem are proved.

Abstract

We introduce the notion of a probabilistic measure which takes values in hyperbolic numbers and which satisfies the system of axioms generalizing directly Kolmogorov's system of axioms. We show that this new measure verifies the usual properties of a probability; in particular, we treat the conditional hyperbolic probability and we prove the hyperbolic analogues of the multiplication theorem, of the law of total probability and of Bayes' theorem. Our probability may take values which are zero--divisors and we discuss carefully this peculiarity.