Slides Chapter 1. Measure Theory and Probability

1.4. PROBABILITY MEASURES
Proposition 1.6 (Sequential continuity of the probability)
Let(Ω,A,P)beaprobabilityspace. Then,foranysequence{A n }
of events from A it holds
P
(
lim
n→∞ A n
)
= lim
n→∞
P(A n ).
Example 1.12 Considertherandomexperimentofselectingrandomly
a number from [0,1]. Then the sample space is Ω = [0,1].
Consider also the Borel σ-algebra restricted to [0,1] and define the
function
g(x) = P([0,x]), x ∈ (0,1).
Proof that g is always right continuous, for each probability measure
P that we choose.
Example 1.13 (Construction of a probability measure
for countable Ω)
If Ω is finite or countable, the σ-algebra that is typically chosen
is P(Ω). In this case, in order to define a probability function, it
suffices to define the probabilities of the elementary events {a i } as
P({a i }) = p i , ∀a i ∈ Ω with the condition that ∑ i p i = 1, p i ≥ 0,
∀i. Then, ∀A ⊂ Ω,
P(A) = ∑ a i ∈A
P({a i }) = ∑ a i ∈Ap i .
Example 1.14 (Construction of a probability measure
in (IR,B))
How can we construct a probability measure in (IR,B)? In general,
it is not possible to define a probability measure by assigning
directly a numerical value to each A ∈ B, since then probably the
axioms of Kolmogoroff will not be satisfied.
ISABEL MOLINA 16