Foundations of abstract probability theory

TL;DR

This paper develops a generalized framework for probability theory using abstract algebra, unifying various theories of uncertainty through abstract measures, integrals, and axioms.

Contribution

It introduces an abstract algebraic foundation for probability, extending Kolmogorov's axioms and defining new concepts like abstract probability measures and conditional probabilities.

Findings

Abstract probability measures satisfy recognizable properties.

The abstract Lebesgue integral generalizes expectation.

The framework unifies different uncertainty theories.

Abstract

Using the ideas of abstract algebra, we introduce the basic concepts of abstract probability theory that generalize the Kolmogorov's probability theory, possibility theory and other theories that deal with uncertainty. Based on abstract addition and multiplication, we define an abstract measure and abstract Lebesgue integral. System of Kolmogorov's axioms is criticized, after which we introduce an abstract probability measure and abstract conditional probability, show that they have recognizable probability properties. In addition, we define an abstract expected value operator as the abstract Lebesgue integral and prove its properties.