  The paper aim is the axiomatic justification of the theory of experience and
chance, one of the dual halves of which is the Kolmogorov probability theory.
The author's main idea was the natural inclusion of Kolmogorov's axiomatics of
probability theory in a number of general concepts of the theory of experience
and chance. The analogy between the measure of a set and the probability of an
event has become clear for a long time. This analogy also allows further
evolution: the measure of a set is completely analogous to the believability of
an event. In order to postulate the theory of experience and chance on the
basis of this analogy, you just need to add to the Kolmogorov probability
theory its dual reflection - the believability theory, so that the theory of
experience and chance could be postulated as the certainty
(believability-probability) theory on the Cartesian product of the probability
and believability spaces, and the central concept of the theory is the new
notion of co~event as a measurable binary relation on the Cartesian product of
sets of elementary incomes and elementary outcomes. Attempts to build the
foundations of the theory of experience and chance from this general point of
view are unknown to me, and the whole range of ideas presented here has not yet
acquired popularity even in a narrow circle of specialists; in addition, there
was still no complete system of the postulates of the theory of experience and
chance free from unnecessary complications. The main result of this work is the
axiom of co~event, intended for the sake of constructing a theory formed by
dual theories of believabilities and probabilities, each of which itself is
postulated by its own Kolmogorov system of axioms. Here a preference is given
to a system of postulates that is able to describe in a simple manner the
results of what I call an experienced-random experiment.
