Cournot's principle for measure-theoretic probability

TL;DR

This paper reformulates Cournot's principle within measure-theoretic probability, providing a rigorous criterion for relating probability measures to experiments and proving that at most one measure can be associated with a given experiment.

Contribution

It introduces a measure-theoretic version of Cournot's principle, clarifying how probability measures relate to experiments and establishing the uniqueness of such a relation.

Findings

A new measure-theoretic formulation of Cournot's principle.

Proof that at most one probability measure can be related to an experiment.

Explicit statement of the product rule in this context.

Abstract

The problem of relating the mathematics of probability theory to the empirical world of experiments has been debated for centuries. One of the oldest solutions proposed for this problem is a principle that states that an event with probability close to 1 nearly certainly occurs in a single trial of an experiment. This principle is now called $\textit{Cournot' principle}$. Cournot's principle was first formulated in the context of classical probability, in which the probability of any event is given, and the $\textit{product rule}$, i.e., the rule that the probability that two events occur in two separated trials is the product of their probabilities, can be deduced. On the contrary, in the modern measure-theoretic approach to probability, probability measures and experiments are separate entities that must be related in an appropriate way, and the product rule cannot be deduced. In this paper, a version of Cournot's principle suitable for measure-theoretic probability is proposed. Therefore, the principle is reformulated as a criterion for relating probability measures and experiments, and the product rule is explicitly stated. In spite of the vagueness of the notions involved, the new version is formulated in a rigorous manner and an exact result, namely, that at most one probability measure can be related to an experiment, is rigorously proven.