  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.
