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>2. STRUCTURES OF SUBSETS<br />
σ-algebra containing all intervals. The Borel σ-algebra is based on<br />
this idea, <strong>and</strong> it will be used by default when Ω = IR.<br />
Definition <strong>1.</strong>9 (Borel σ-algebra)<br />
Consider the sample space Ω = IR <strong>and</strong> the collection of intervals<br />
of the form<br />
I = {(−∞,a] : a ∈ IR}.<br />
We define the Borel σ-algebra over IR, represented by B, as the<br />
σ-algebra spanned by I.<br />
The Borel σ-algebra B contains all complements, countable intersections<br />
<strong>and</strong> unions of elements of I. In particular, B contains<br />
all types of intervals <strong>and</strong> isolated points of IR, although B is not<br />
equal to P(IR). For example,<br />
• (a,∞) ∈ B, since (a,∞) = (−∞,a] c , <strong>and</strong> (−∞,a] ∈ IR.<br />
• (a,b] ∈ IR, ∀a < b, since this interval can be expressed as<br />
(a,b] = (−∞,b]∩(a,∞), where (−∞,b] ∈ B <strong>and</strong> (a,∞) ∈<br />
B.<br />
∞⋂<br />
• {a} ∈ B,∀a ∈ IR,since{a} =<br />
(a− 1 ]<br />
n ,a ,whichbelongs<br />
to B.<br />
When the sample space Ω is continuous but is a subset of IR,<br />
we need a σ-algebra restricted to subsets of Ω.<br />
Definition <strong>1.</strong>10 (Restricted Borel σ-algebra )<br />
Let A ⊂ IR. We define the Borel σ-algebra restricted to A as the<br />
collection<br />
B A = {B ∩A : B ∈ B}.<br />
n=1<br />
ISABEL MOLINA 7