# 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 ...

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