Kolmogorov probability theory based on set theory belongs to the most
important results of mathematics of the 20th century. Naturally, its main advantage is the
possibility to use results of the modern measure theory. However, this fact sometimes
does not allow larger considerations. In this chapter we want to show this paradox can
be eliminated. Of course, we present only some basic ideas. Understanding them
enables one to study further results and applications.
Keywords: Probability, Measure, Measurable functions, Random variable,
Lebesgue integral, Independence, Limit theorems, Conditional expectation, Limit
laws for maxima, Peaks over threshold.
