💡 Terminology: A set $\Omega$ is called a probability base space (or sample space). An element $\omega \in \Omega$ is called an elementary event.

💡 A system of sets $\mathcal{F}\subset \mathcal{P}(\Omega )$ is called Sigma-algebra ($\sigma$-algebra), if it fulfills the following conditions:

E1. $\Omega \in \mathcal{F}$

E2. $A\in \mathcal{F}⟹A^c\in \mathcal{F}$

E3. $A_1,A_2,\dots \in \mathcal{F}⟹{\bigcup }_{i=1}^{\infty }{A}_i\in \mathcal{F}$

The elements of the $\sigma$-algebra are called events.

💡 Let $\Omega$ be a sample space and let $\mathcal{F}$ be a $\sigma$-algebra. A transformation

$$\begin{array}{rl}\text{Pr}:& \mathcal{F}⟶[0,1]\\ & A⟼\text{Pr}[A]\end{array}$$

(sometimes also denoted by $\mathbb{P}=\text{Pr}$) is called probability measure on $(\Omega ,\mathcal{F})$, if the following properties hold:

E1. $\text{Pr}[\Omega ]=1$

E2. ($\sigma$-additivity) $\text{Pr}[A]={\sum }_{i=1}^{\infty }\text{Pr}[A_i]\text{if}A={\bigcup }_{i=1}^{\infty }{A}_i$ (disjoint union)

💡 Let $\Omega$ be a sample space, $\mathcal{F}$ a $\sigma$-algebra and $\text{Pr}$ a probability measure. The triple $(\Omega ,\mathcal{F},\text{Pr})$ is called a probability space.

💡 Terminology: Let $A$ be an event and let $\omega$ be an arbitrary elementary event. We say that $A$ occurs (for $\omega$) if $\omega \in A$. We say that $A$ does not occur if $\omega \notin A$.

💡 Let $\Omega$ be a finite sample space. The Laplace model on $\Omega$ is a triple $(\Omega ,\mathcal{F},\text{Pr})$, such that

• $\mathcal{F}=\mathcal{P}(\Omega )$
• $\text{Pr}:\mathcal{F}⟶[0,1]$ is defined by $$\forall A\in \mathcal{F}\text{Pr}[A]=\frac{|A|}{|\Omega |}.$$

📖 (Closure of a $\sigma$-algebra under operations) Let $\mathcal{F}$ be a $\sigma$-algebra on $\Omega$. The following hold:

E4. $\varnothing \in \mathcal{F}$,

E5. $A_1,A_2,\dots \in \mathcal{F}⟹{\bigcap }_{i=1}^{\infty }{A}_i\in \mathcal{F}$,

E6. $A,B\in \mathcal{F}⟹A\cup B\in \mathcal{F}$,

E7. $A,B\in \mathcal{F}⟹A\cap B\in \mathcal{F}$.