Preliminaries on Axiomatic Probability Theory

This work has been conceived as a purely theoretical and mathematical study dealing with the subject of its interest at a highly abstract and formalized level. Probability theory will serve as one and, as a matter of fact, the most important and the most powerful formal tool used below in order to achieve this goal. Therefore, beginning with a brief survey of the most elementary notions of probability theory, just the most elementary abstract ideas and construction of the axiomatic probability theory, as settled by Kolmogorov in Kolmogorov (1974) are presented in this chapter, intentionally leaving aside all the informal discussions, motivations, and practical examples preceding the formalized explanations of probability theory in the greatest part of textbooks and monographs dealing with this theory. The reader interested in these informal parts of probability theory is kindly invited to consult an appropriate textbook or monograph, let us mention explicitly the already classical textbooks Feller (1957) and Gnedenko (1965), where just these informal parts are explained very carefully, in detail, and with a lot of various examples. On the other side, Loeve (1960) treates probability theory at an exclusively abstract and formalized level.