Probability Theory, 2nd Edition

1 Probabilities

1.1 Probability spaces

The basic concern of Probability Theory is to model experiments involving randomness, that is, experiments with nondetermined outcome, shortly called random experiments. The Russian mathematician A. N. Kolmogorov established the modern Probability Theory in 1933 by publishing his book (cf. [Kol33]) Grundbegriffe der Wahrscheinlichkeitsrechnung. In it, he postulated the following:

Random experiments are described by probability spaces $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathcal{A}\mathrm{,}\mathbb{P}\mathrm{)}$.

The triple $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathcal{A}\mathrm{,}\mathbb{P}\mathrm{)}$ comprises a sample space Ω, a σ-field $\mathcal{A}$ of events, and a mapping $\mathbb{P}$ from $\mathcal{A}$ to [0,1], called probability measure or probability distribution.

Let us now explain the three different components of a probability space in detail. We start with ...