Multidimensional and abstract probability

Abstract probabilities are introduced as semiring algebraic structures that retain several properties of classical probabilities taking values in the real number interval $[0,1]$. Compact probabilities and random variables with such probabilities are mainly studied. Analogs of the Borel–Cantelli lemma and of the law of large numbers are considered. New notions of superposition of probability spaces and superposition of random variables arise on the basis of the Cartesian product of abstract probabilities.