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Probability Theory/Kolmogorov and modern axioms and their meaning
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Fundamental definition
0 100% #FFFFFF Definition 2.1 (Kolmogorov's axioms):
Let be a set, and let be an algebra of subsets of . Let furthermore be a function satisfying
and
.
Then the triple is called a probability space.
Note in particular that
,
since .
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, shall always be a probability space.
Lemma 2.2:
For ,
.
Lemma 2.3:
For ,
.
Lemma 2.4:
For ,
.
Exercises
Exercise 2.2.1: Prove lemmas 2.2-2.4.