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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.05228 |
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Table of Contents:
- 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.