Slides Chapter 1. Measure Theory and Probability

1.3. SET FUNCTIONS
Definition 1.12 (Additive set function)
A set function φ : A → IR is additive if it satisfies:
( n
)
⋃
For all{A i } n i=1 ∈ AwithA i ∩A j = ∅, i ≠ j, φ A i =
i=1
n∑
φ(A i ).
We will assume that +∞ and −∞ cannot both belong to the
range of φ. We will exclude the cases φ(A) = +∞ for all A ∈ A
and φ(A) = −∞ for all A ∈ A. Extending the definition to an
infinite sequence, we obtain a σ-additive set function.
Definition 1.13 (σ-additive set function)
A set function φ : A → IR is σ-additive if it satisfies:
For all {A n } n∈IN ∈ A with A i ∩A j = ∅, i ≠ j,
( ∞
)
⋃ ∞∑
φ A n = φ(A n ).
n=1
Observe that a σ-additive set function is well defined, since the
infinite union of sets of A belongs to A because A is a σ-algebra.
It is easy to see that an additive function satisfies µ(∅) = 0. Moreover,
countable additivity implies finite additivity.
n=1
Definition 1.14 (Measure)
A set function φ : A → IR is a measure if
(a) φ is σ-additive;
(b) φ(A) ≥ 0, ∀A ∈ A.
Definition 1.15 (Probability measure)
A measure µ with µ(Ω) = 1 is called a probability measure.
i=1
ISABEL MOLINA 9