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Probability Theory/Kolmogorov and modern axioms and their meaning

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Fundamental definition

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Definition 2.1 (Kolmogorov's axioms):

Let be a set, and let be an algebra of subsets of . Let furthermore be a function satisfying

  1. and
  2. .

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

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In the following, shall always be a probability space.

Lemma 2.2:

For ,

.

Lemma 2.3:

For ,

.

Lemma 2.4:

For ,

.

Exercises

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  • Exercise 2.2.1: Prove lemmas 2.2-2.4.