Infinite Previsions and Finitely Additive Expectations

TL;DR

This paper extends de Finetti's coherence concept to unbounded random variables with infinite previsions, proposing a finitely additive integral and a generalized Fundamental Theorem of Prevision for conditional expectations.

Contribution

It introduces a finitely additive extension of the Daniell integral and generalizes the Fundamental Theorem of Prevision for unbounded variables and conditional previsions.

Findings

Finitely additive integral offers advantages over Lebesgue-style integrals in this setting

Extension of coherence to unbounded variables with infinite previsions

Generalized Fundamental Theorem of Prevision for conditional expectations

Abstract

We give an extension of de Finetti's concept of coherence to unbounded (but real-valued) random variables that allows for gambling in the presence of infinite previsions. We present a finitely additive extension of the Daniell integral to unbounded random variables that we believe has advantages over Lebesgue-style integrals in the finitely additive setting. We also give a general version of the Fundamental Theorem of Prevision to deal with conditional previsions and unbounded random variables.