Slides Chapter 1. Measure Theory and Probability

1.2. STRUCTURES OF SUBSETS
8. Let{x n }beasequenceofrealnumbersandletA n = (−∞,x n ).
What is the connection between liminf x n and limA n ? Similarly
between limsupx n and limA n .
n→∞
n→∞
1.2 Structures of subsets
AprobabilityfunctionwillbeafunctiondefinedovereventsorsubsetsofasamplespaceΩ.
Itisconvenienttoprovidea“good”structure
to these subsets, which in turn will provide “good” properties
to the probability function. In this section we study collections of
subsets of a set Ω with a good structure.
Definition 1.6 (Algebra)
An algebra (also called field) A over a set Ω is a collection of
subsets of Ω that has the following properties:
(i) Ω ∈ A;
(ii) If A ∈ A, then A c ∈ A;
(iii) If A 1 ,A 2 ,...,A n ∈ A, then
n⋃
A i ∈ A.
An algebra over Ω contains both Ω and ∅. It also contains all
finite unions and intersections of sets from A. We say that A is
closed under complementation, finite union and finite intersection.
Extending property (iii) to an infinite sequence of elements of A
we obtain a σ-algebra.
Definition 1.7 (σ-algebra)
A σ-algebra (or σ-field) A over a set Ω is a collection of subsets of
Ω that has the following properties:
i=1
ISABEL MOLINA 5