Lexicographic probability, conditional probability, and nonstandard probability

TL;DR

This paper explores the relationships among Popper spaces, lexicographic probability systems, and nonstandard probability spaces, highlighting conditions under which they are equivalent or distinct, especially regarding countable additivity and the size of the state space.

Contribution

It clarifies the conditions for equivalence and differences among Popper spaces, LPS's, and NPS's, especially in relation to countable additivity and finite versus infinite state spaces.

Findings

Popper spaces and a subclass of LPS's are equivalent with countable additivity.

Without countable additivity, the equivalence breaks down.

NPS's are more general than LPS's in infinite state spaces.

Abstract

The relationship between Popper spaces (conditional probability spaces that satisfy some regularity conditions), lexicographic probability systems (LPS's), and nonstandard probability spaces (NPS's) is considered. If countable additivity is assumed, Popper spaces and a subclass of LPS's are equivalent; without the assumption of countable additivity, the equivalence no longer holds. If the state space is finite, LPS's are equivalent to NPS's. However, if the state space is infinite, NPS's are shown to be more general than LPS's.