Slides Chapter 1. Measure Theory and Probability

1.2. STRUCTURES OF SUBSETS
(i) Ω ∈ A;
(ii) If A ∈ A, then A c ∈ A;
(iii) IfA 1 ,A 2 ...isasequenceofelementsofA,then∪ ∞ n=1A n ∈ A.
Thus, a σ-algebra is closed under countable union. It is also
closed under countable intersection. Moreover, if A is an algebra,
a countable union of sets in A can be expressed as the limit of
n⋃
an increasing sequence of sets, the finite unions A i . Thus, a
σ-algebra is an algebra that is closed under limits of increasing
sequences.
Example 1.4 The smallest σ-algebra is {∅,Ω}. The smallest
σ-algebra that contains a subset A ⊂ Ω is {∅,A,A c ,Ω}. It is
contained in any other σ-algebra containing A. The collection of
all subsets of Ω, P(Ω), is a well known algebra called the algebra
of the parts of Ω.
Definition 1.8 (σ-algebra spanned by a collection C of
events)
Given a collection of sets C ⊂ P(Ω), we define the σ-algebra
spanned by C, and we denote it by σ(C), as the smallest σ-algebra
that contains C.
Remark 1.1 For each C, the σ-algebra spanned by C, σ(C), always
exists, since A C is the intersection of all σ-algebras that contain
C and at least P(Ω) ⊃ C is a σ-algebra.
When Ω is finite or countable, it is common to work with the
σ-algebra P(Ω), so we will use this one unless otherwise stated. In
the case Ω = IR, later we will consider probability measures and
we will want to obtain probabilities of intervals. Thus, we need a
i=1
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