Nonlinear probability. A theory with incompatible stochastic variables

TL;DR

This paper introduces nonlinear measures in a generalized space, allowing for incompatible stochastic variables, and establishes a representation theorem for related functionals, extending classical measure theory with applications to quantum mechanics.

Contribution

It generalizes the concept of measures to nonlinear measures on topological spaces, enabling the representation of functionals and accommodating incompatible stochastic variables.

Findings

Defined nonlinear measures in a generalized space

Established a representation theorem for functionals

Enabled modeling of incompatible stochastic variables

Abstract

In 1991 J.F. Aarnes introduced the concept of quasi-measures in a compact topological space $\Omega$ and established the connection between quasi-states on $C (\Omega)$ and quasi-measures in $\Omega$. This work solved the linearity problem of quasi-states on $C^*$-algebras formulated by R.V. Kadison in 1965. The answer is that a quasi-state need not be linear, so a quasi-state need not be a state. We introduce nonlinear measures in a space $\Omega$ which is a generalization of a measurable space. In this more general setting we are still able to define integration and establish a representation theorem for the corresponding functionals. A probabilistic language is choosen since we feel that the subject should be of some interest to probabilists. In particular we point out that the theory allows for incompatible stochastic variables. The need for incompatible variables is well known in quantum mechanics, but the need seems natural also in other contexts as we try to explain by a questionary example. Keywords and phrases: Epistemic probability, Integration with respect to mea- sures and other set functions, Banach algebras of continuous functions, Set func- tions and measures on topological spaces, States, Logical foundations of quantum mechanics.