Probability Theory/Kolmogorov and modern axioms and their meaning

Fundamental definition

Definition 2.1 (Kolmogorov's axioms):

Let $\Omega$ be a set, and let ${\mathcal {F}}$ be an algebra of subsets of $\Omega$. Let furthermore $P:{\mathcal {F}}\to [0,1]$ be a function satisfying

1. $P(\Omega )=1$ and
2. $P(A+B)=P(A)+P(B)$.

Then the triple $(\Omega ,{\mathcal {F}},P)$ is called a probability space.

Note in particular that

$P(\emptyset )=0$,

since $P(\Omega )=P(\Omega +\emptyset )=P(\Omega )+P(\emptyset )$.

Note that often probability spaces are defined such that the algebra of subsets is a sigma-algebra. We shall revisit these concept later, and restrict ourselves to the above definition, which seems to capture the intuitive concept of probability quite well.

Elementary theorems

In the following, $(\Omega ,{\mathcal {F}},P)$ shall always be a probability space.

Lemma 2.2:

For $A\in {\mathcal {F}}$,

$P(A)+P(A^{c})=1$.

Lemma 2.3:

For $A_{1},\ldots ,A_{n}\in {\mathcal {F}}$,

$P\left(\sum _{j=1}^{n}A_{j}\right)=\sum _{j=1}^{n}P(A_{j})$.

Lemma 2.4:

For $A,B\in {\mathcal {F}}$,

$P(A\cup B)=P(A)+P(B)-P(AB)$.

Exercises

• Exercise 2.2.1: Prove lemmas 2.2-2.4.