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Cournot's principle for measure-theoretic probability

Bruno Galvan

TL;DR

This work reframes Cournot's principle for measure-theoretic probability, defining a precise criterion that relates probability measures to experiments via the condition $\mathcal{T}(P^n,\delta)\subseteq\mathcal{C}(E^n)$ for all $n$ with some $\delta<1$, and explicitly states the product-rule requirement across $n$. It introduces practical certainty and typicality as rigorous empirical notions and proves that at most one probability measure can govern a given experiment. The result clarifies how to connect mathematical probability with empirical behavior and highlights three possible relational scenarios, proving the first is impossible. The framework provides a foundation for analyzing how long-run frequencies relate to single-trial outcomes in measure-theoretic probability, with potential implications for statistical interpretation and foundations.

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.

Cournot's principle for measure-theoretic probability

TL;DR

This work reframes Cournot's principle for measure-theoretic probability, defining a precise criterion that relates probability measures to experiments via the condition for all with some , and explicitly states the product-rule requirement across . It introduces practical certainty and typicality as rigorous empirical notions and proves that at most one probability measure can govern a given experiment. The result clarifies how to connect mathematical probability with empirical behavior and highlights three possible relational scenarios, proving the first is impossible. The framework provides a foundation for analyzing how long-run frequencies relate to single-trial outcomes in measure-theoretic probability, with potential implications for statistical interpretation and foundations.

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 . Cournot's principle was first formulated in the context of classical probability, in which the probability of any event is given, and the , 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.
Paper Structure (9 sections, 3 theorems, 15 equations)

This paper contains 9 sections, 3 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminary notions
  3. Mathematical formalism
  4. Experiments
  5. Vague sets
  6. Practically certain events
  7. Cournot's principle for measure-theoretic probability
  8. Three particular situations
  9. Conclusion

Key Result

Theorem 1

For all $n \in \mathbb{N}$, let $\mathcal{C}_n \subseteq \mathcal{A}^n$ be a class that does not contain disjoint events. If for some $\delta_1, \delta_2 < 1$ and all $n \in \mathbb{N}$, then $P_1 = P_2.$

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Definition 1: Practical certainty
  • Proposition 1
  • proof