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Let us start by reviewing the concept of a σ-algebra.

1 1) Ω ∈ ;

2 2) is stable by complementarity: for any A ∈ , we have Ac ∈ , where Ac denotes the complement of A in Ω: Ac = Ω\A;

3 3) is stable under a countable union: for any sequence of elements (An)n∈ℕ of , we have

Elements of a σ-algebra are called events.

Among the elementary properties of σ-algebra, we can cite stability through any intersection (countable or not).

PROPOSITION 1.1.– Any intersection of σ-algebras over a set Ω is a σ-algebra over Ω.

ii∈I

 – first of all, for any i, Ω ∈ i, thus Ω ∈ ∩i∈Ii;

 – secondly, if A ∈ ∩i∈I i, then for any i, A ∈ i. As these are σ-algebras, we have that for any i, Ac ∈ i, thus Ac ∈ ∩i∈I i;

 – finally, if for any n ∈ ℕ, An ∈ ∩i∈I i, then for any i, n, An ∈ i. As these are σ-algebras, we have that for any i, ∪n∈ℕAn ∈ i, thus

It is generally difficult to make explicit all the events in a σ-algebra. We often describe it using generating events.