Slides Chapter 1. Measure Theory and Probability

1.2. STRUCTURES OF SUBSETS
σ-algebra containing all intervals. The Borel σ-algebra is based on
this idea, and it will be used by default when Ω = IR.
Definition 1.9 (Borel σ-algebra)
Consider the sample space Ω = IR and the collection of intervals
of the form
I = {(−∞,a] : a ∈ IR}.
We define the Borel σ-algebra over IR, represented by B, as the
σ-algebra spanned by I.
The Borel σ-algebra B contains all complements, countable intersections
and unions of elements of I. In particular, B contains
all types of intervals and isolated points of IR, although B is not
equal to P(IR). For example,
• (a,∞) ∈ B, since (a,∞) = (−∞,a] c , and (−∞,a] ∈ IR.
• (a,b] ∈ IR, ∀a < b, since this interval can be expressed as
(a,b] = (−∞,b]∩(a,∞), where (−∞,b] ∈ B and (a,∞) ∈
B.
∞⋂
• {a} ∈ B,∀a ∈ IR,since{a} =
(a− 1 ]
n ,a ,whichbelongs
to B.
When the sample space Ω is continuous but is a subset of IR,
we need a σ-algebra restricted to subsets of Ω.
Definition 1.10 (Restricted Borel σ-algebra )
Let A ⊂ IR. We define the Borel σ-algebra restricted to A as the
collection
B A = {B ∩A : B ∈ B}.
n=1
ISABEL MOLINA 7