Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
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<strong>1.</strong>4. PROBABILITY MEASURES<br />
Theorem <strong>1.</strong>2 (Caratheodory’s Extension Theorem)<br />
Let(Ω,A,P)beaprobabilityspace, whereAisanalgebra. Then,<br />
P can be extended from A to σ(A), <strong>and</strong> the extension is unique<br />
(i.e., there exists a unique probability measure ˆP over σ(A) with<br />
ˆP(A) = P(A), ∀A ∈ A).<br />
TheextensionofP fromF toσ(F) = B isdonebysteps. First,<br />
P is extended to the collection of the limits of increasing sequences<br />
ofeventsinF,denotedC. ItholdsthatC ⊃ F <strong>and</strong>C ⊃ σ(F) = B<br />
(Monotone Class Theorem). The probability of each event A from<br />
C is defined as the limit of the probabilities of the sequences of<br />
events from F that converge to A. Afterwards, P is extended<br />
to the σ-algebra of the parts of IR. For each subset A ∈ P(IR),<br />
the probability is defined as the infimum of the probabilities of<br />
the events in C that contain A. This extension is not countably<br />
additive on P(IR), only on a smaller σ-algebra, so P it is not a<br />
probability measure on (IR,P(IR)). Finally, a σ-algebra in which<br />
P is a probability measure is defined as the collection H of subsets<br />
H ⊂ Ω, for which P(H)+P(H c ) = <strong>1.</strong> This collection is indeed a<br />
σ-algebra that contains C <strong>and</strong> P is a probability measure on it. It<br />
holds that σ(F) = B ⊂ H <strong>and</strong> P restricted to σ(F) = B is also a<br />
probability measure on (IR,B).<br />
Problems<br />
<strong>1.</strong> ProvethepropertiesoftheprobabilitymeasuresinProposition<br />
<strong>1.</strong>4.<br />
2. Prove Bool’s inequality in Proposition <strong>1.</strong>4.<br />
3. Prove the Sequential Continuity of the probability in Proposition<br />
<strong>1.</strong>6.<br />
ISABEL MOLINA 18