Title: Nonlinear probability. A theory with 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.