Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
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<strong>1.</strong>3. SET FUNCTIONS<br />
Example <strong>1.</strong>5 (Counting measure)<br />
Let Ω be any set <strong>and</strong> consider the σ-algebra of the parts of Ω,<br />
P(Ω). Define µ(A) as the number of points of A. The set function<br />
µ is a measure known as the counting measure.<br />
Example <strong>1.</strong>6 (<strong>Probability</strong> measure)<br />
Let Ω = {x 1 ,x 2 ,...} be a finite or countably infinite set, <strong>and</strong> let<br />
p 1 ,p 2 ,..., be nonnegative numbers. Consider the σ-algebra of the<br />
parts of Ω, P(Ω), <strong>and</strong> define<br />
µ(A) = ∑ x i ∈Ap i .<br />
Thesetfunctionµisaprobabilitymeasureif<strong>and</strong>onlyif ∑ ∞<br />
i=1 p i =<br />
<strong>1.</strong><br />
Example <strong>1.</strong>7 (Lebesgue measure)<br />
A well known measure defined over (IR,B), which assigns to each<br />
element of B its length, is the Lebesgue measure, denoted here as<br />
λ. For an interval, either open, close or semiclosed, the Lebesgue<br />
measure is the length of the interval. For a single point, the<br />
Lebesgue measure is zero.<br />
Definition <strong>1.</strong>16 (σ-finite set function)<br />
A set function φ : A → IR is σ-finite if ∀A ∈ A, there exists a<br />
sequence {A n } n∈IN of disjoint elements of A with φ(A n ) < ∞ ∀n,<br />
whose union covers A, that is,<br />
∞⋃<br />
A ⊆ A n .<br />
n=1<br />
Definition <strong>1.</strong>17 (<strong>Measure</strong> space)<br />
The triplet (Ω,A,µ), where Ω is a sample space, A is an algebra<br />
<strong>and</strong> µ is a measure defined over (Ω,A), is called measure space.<br />
ISABEL MOLINA 10