Slides Chapter 1. Measure Theory and Probability

1.3. SET FUNCTIONS
In the following we define the space over which measures, including
probability measures, will be defined. This space will be
the one whose elements will be suitable to “measure”.
Definition 1.11 (Measure space)
The pair (Ω,A), where Ω is a sample space and A is an algebra
over Ω, is called measurable space.
Problems
1. Let A,B ∈ Ω with A ∩ B = ∅. Construct the smallest σ-
algebra that contains A and B.
2. Prove that an algebra over Ω contains all finite intersections
of sets from A.
3. Prove that a σ-algebra over Ω contains all countable intersections
of sets from A.
4. Prove that a σ-algebra is an algebra that is closed under limits
of increasing sequences.
5. Let Ω = IR. Let A be the class of all finite unions of disjoint
elements from the set
C = {(a,b],(−∞,a],(b,∞);a ≤ b}
Prove that A is an algebra.
1.3 Set functions
In the following A is an algebra and we consider the extended real
line given by IR = IR∪{−∞,+∞}.
ISABEL MOLINA 8