An operational characterization of the notion of probability by algorithmic randomness II: Discrete probability spaces

Abstract: The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion of probability is not established yet. In this paper, based on the toolkit of algorithmic randomness we present an operational characterization of the notion of probability, called an ensemble, for general discrete probability spaces whose sample space is countably infinite. Algorithmic randomness, also known as algorithmic information theory, is a field of mathematics which enables us to consider the randomness of an individual infinite sequence. We use an extension of Martin-Loef randomness with respect to a generalized Bernoulli measure over the Baire space, in order to present the operational characterization. In our former work [K. Tadaki, arXiv:1611.06201], we developed an operational characterization of the notion of probability for an arbitrary finite probability space, i.e., a probability space whose sample space is a finite set. We then gave a natural operational characterization of the notion of conditional probability in terms of ensemble for a finite probability space, and gave equivalent characterizations of the notion of independence between two events based on it. Furthermore, we gave equivalent characterizations of the notion of independence of an arbitrary number of events/random variables in terms of ensembles for finite probability spaces. In particular, we showed that the independence between events/random variables is equivalent to the independence in the sense of van Lambalgen's Theorem, in the case where the underlying finite probability space is computable. In this paper, we show that we can certainly extend these results over general discrete probability spaces whose sample space is countably infinite.