Probability Theory/Kolmogorov and modern axioms and their meaning

Fundamental definition

Note in particular that

$P(\mathrm{\varnothing })=0$,

since $P(\mathrm{\Omega })=P(\mathrm{\Omega }+\mathrm{\varnothing })=P(\mathrm{\Omega })+P(\mathrm{\varnothing })$.

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, $(\mathrm{\Omega },ℱ,P)$ shall always be a probability space.

Lemma 2.2:

For $A\in ℱ$,

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

Lemma 2.3:

For $A_1,\dots ,A_n\in ℱ$,

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

Lemma 2.4:

For $A,B\in ℱ$,

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

Exercises

• Exercise 2.2.1: Prove lemmas 2.2-2.4.