Probability Theory/Kolmogorov and modern axioms and their meaning

Fundamental definition
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.