Slides Chapter 1. Measure Theory and Probability

1.4. PROBABILITY MEASURES
Theorem 1.2 (Caratheodory’s Extension Theorem)
Let(Ω,A,P)beaprobabilityspace, whereAisanalgebra. Then,
P can be extended from A to σ(A), and the extension is unique
(i.e., there exists a unique probability measure ˆP over σ(A) with
ˆP(A) = P(A), ∀A ∈ A).
TheextensionofP fromF toσ(F) = B isdonebysteps. First,
P is extended to the collection of the limits of increasing sequences
ofeventsinF,denotedC. ItholdsthatC ⊃ F andC ⊃ σ(F) = B
(Monotone Class Theorem). The probability of each event A from
C is defined as the limit of the probabilities of the sequences of
events from F that converge to A. Afterwards, P is extended
to the σ-algebra of the parts of IR. For each subset A ∈ P(IR),
the probability is defined as the infimum of the probabilities of
the events in C that contain A. This extension is not countably
additive on P(IR), only on a smaller σ-algebra, so P it is not a
probability measure on (IR,P(IR)). Finally, a σ-algebra in which
P is a probability measure is defined as the collection H of subsets
H ⊂ Ω, for which P(H)+P(H c ) = 1. This collection is indeed a
σ-algebra that contains C and P is a probability measure on it. It
holds that σ(F) = B ⊂ H and P restricted to σ(F) = B is also a
probability measure on (IR,B).
Problems
1. ProvethepropertiesoftheprobabilitymeasuresinProposition
1.4.
2. Prove Bool’s inequality in Proposition 1.4.
3. Prove the Sequential Continuity of the probability in Proposition
1.6.
ISABEL MOLINA 18