Slides Chapter 1. Measure Theory and Probability

1.3. SET FUNCTIONS
Example 1.5 (Counting measure)
Let Ω be any set and consider the σ-algebra of the parts of Ω,
P(Ω). Define µ(A) as the number of points of A. The set function
µ is a measure known as the counting measure.
Example 1.6 (Probability measure)
Let Ω = {x 1 ,x 2 ,...} be a finite or countably infinite set, and let
p 1 ,p 2 ,..., be nonnegative numbers. Consider the σ-algebra of the
parts of Ω, P(Ω), and define
µ(A) = ∑ x i ∈Ap i .
Thesetfunctionµisaprobabilitymeasureifandonlyif ∑ ∞
i=1 p i =
1.
Example 1.7 (Lebesgue measure)
A well known measure defined over (IR,B), which assigns to each
element of B its length, is the Lebesgue measure, denoted here as
λ. For an interval, either open, close or semiclosed, the Lebesgue
measure is the length of the interval. For a single point, the
Lebesgue measure is zero.
Definition 1.16 (σ-finite set function)
A set function φ : A → IR is σ-finite if ∀A ∈ A, there exists a
sequence {A n } n∈IN of disjoint elements of A with φ(A n ) < ∞ ∀n,
whose union covers A, that is,
∞⋃
A ⊆ A n .
n=1
Definition 1.17 (Measure space)
The triplet (Ω,A,µ), where Ω is a sample space, A is an algebra
and µ is a measure defined over (Ω,A), is called measure space.
ISABEL MOLINA 10