Kolmogorov Axioms

probability-theory

Definition

Kolmogorov Axioms

Let $(\Omega ,\mathfrak{S},P)$ be an arbitrary probability space. The following collection hold for a probability space and are called the Kolmogorov Axioms:

1. Non-negativity: $\forall A\in \mathfrak{S}:P(A)\ge 0$
2. Unit-measure: $P(\Omega )=1$
3. $\sigma$-additivity: $\forall A_1,A_2,\cdots \in \mathfrak{S}:\left[\text{all}{A}_i\text{disjoint}⟹P\left({\bigcup }_{i=1}^{\infty }{A}_i\right)={\sum }_{i=1}^{\infty }P(A_i)\right]$

The are named after Andrey Kolmogorov.

Consequences

(Finite) Additivity

A consequence of the $\sigma$-additivity is the (finite) additivity:

$$\forall A,B\in \mathfrak{S}:\left[A\cap B=\varnothing ⟹P(A\cup B)=P(A)+P(B)\right]$$

Mutual Exclusivity

Trivially, given $A\cap B=\varnothing$ and $P(\varnothing )=0$, the following holds:

$$\forall A,B\in \mathfrak{S}:\left[A\cap B=\varnothing ⟹P(A\cap B)=0\right]$$

Complement

$$\forall A\in \mathfrak{S}:P(A^c)=1-P(A)$$

Complement

Let $A\in \mathfrak{S}$ be an arbitrary event of that sigma algebra.

$$\begin{array}{rlrl}& 1& & =P(\Omega )=P(A\cup A^c)=P(A)+P(A^c)\\ ⟺& P(A^c)& & =1-P(A)\\ ⟺& P(A)& & =1-P(A^c)\end{array}$$

Inclusion-Exclusion Principle

Definition

Inclusion-Exclusion Principle

The inclusion-exclusion principles that state the number of elements of of $n$ finite sets $A_1,\dots ,A_n$ is given by:

$$|\bigcup _{i=1}^n{A}_i|=\sum _{\varnothing \ne J\subseteq 2^I}(-1{)}^{|J|-1}\cdot |\bigcap _{j\in J}{A}_j|$$

using an index set $I=\{1,2,\dots ,n\}$ and sum over all non-empty index subsets $J\subseteq I$.

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Monotonicity

$$\forall A,B\in \mathfrak{S}:A\subseteq B⟹P(A)\le P(B)$$