Multiple Systems in Probability Theory Applications

In applying probability theory to specific problems, it is not uncommon that many random experiments are to be considered simultaneously, each of which has a particular probability space associated with it, and that the problem under consideration requires to find probabilities defined in a particular space from known probabilities defined in other spaces. In such cases, before probability theory can be applied, it is necessary to transfer the known probabilities from the spaces in which they are defined to the space in which the required probability is defined. This transfer can be justified only by using special principles independent of the axioms of probability theory, because the latter apply only to events defined in a single probability space. In this paper, by analyzing in depth a notable example of this kind of problem, we formulate and justify one such principle, the Principle of Equivalence.