Some remarks concerning axiomatics of probability theory
Traditional version of the well-known Kolmogorov axioms for probabilities includes various properties ofthe set of events and ofprobabilities corresponding to these events.Roughly speaking this version postulates among other properties that events form an algebra of sets and that probability is a measure on this algebra. We discuss several topics connecting with empirical explanation of the axioms and emphasize the attention on the fact that such explanation leads naturally to some modiﬁcations of the axioms.Inparticular, not allproperties of the algebra can be explained empirically. The possibility of satisﬁability for operation of union and intersection evenfortwo events cannotbefullyjustiﬁed.So, we can assume(in necessary cases) that more general systems of sets will appear as the natural systems of events. Some structural properties of the mentioned systems are noted. Two examples of such generalized systems are pointed out.