introduction-probability.pdf

Proof we let j c is a σ algebra on ω such that g c

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Proof . We let J := {C is a σ –algebra on Ω such that G ⊆ C} . According to Example 1.1.2 one has J = , because G ⊆ 2 Ω and 2 Ω is a σ –algebra. Hence σ ( G ) := C∈ J C yields to a σ -algebra according to Proposition 1.1.5 such that (by construc- tion) G ⊆ σ ( G ). It remains to show that σ ( G ) is the smallest σ -algebra containing G . Assume another σ -algebra F with G ⊆ F . By definition of J we have that F ∈ J so that σ ( G ) = C∈ J C ⊆ F . The construction is very elegant but has, as already mentioned, the slight disadvantage that one cannot construct all elements of σ ( G ) explicitly. Let us now turn to one of the most important examples, the Borel σ -algebra on R . To do this we need the notion of open and closed sets. Definition 1.1.7 [open and closed sets] (1) A subset A R is called open , if for each x A there is an ε > 0 such that ( x - ε, x + ε ) A . (2) A subset B R is called closed , if A := R \ B is open. Given -∞ ≤ a b ≤ ∞ , the interval ( a, b ) is open and the interval [ a, b ] is closed. Moreover, by definition the empty set is open and closed. Proposition 1.1.8 [Generation of the Borel σ -algebra on R ] We let G 0 be the system of all open subsets of R , G 1 be the system of all closed subsets of R , G 2 be the system of all intervals ( -∞ , b ] , b R , G 3 be the system of all intervals ( -∞ , b ) , b R , G 4 be the system of all intervals ( a, b ] , -∞ < a < b < , G 5 be the system of all intervals ( a, b ) , -∞ < a < b < . Then σ ( G 0 ) = σ ( G 1 ) = σ ( G 2 ) = σ ( G 3 ) = σ ( G 4 ) = σ ( G 5 ) .
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1.1. DEFINITION OF σ -ALGEBRAS 13 Definition 1.1.9 [Borel σ -algebra on R ] 1 The σ -algebra constructed in Proposition 1.1.8 is called Borel σ -algebra and denoted by B ( R ). In the same way one can introduce Borel σ -algebras on metric spaces: Given a metric space M with metric d a set A M is open provided that for all x A there is a ε > 0 such that { y M : d ( x, y ) < ε } ⊆ A . A set B M is closed if the complement M \ B is open. The Borel σ -algebra B ( M ) is the smallest σ -algebra that contains all open (closed) subsets of M . Proof of Proposition 1.1.8. We only show that σ ( G 0 ) = σ ( G 1 ) = σ ( G 3 ) = σ ( G 5 ) . Because of G 3 ⊆ G 0 one has σ ( G 3 ) σ ( G 0 ) . Moreover, for -∞ < a < b < one has that ( a, b ) = n =1 ( -∞ , b ) \ ( -∞ , a + 1 n ) σ ( G 3 ) so that G 5 σ ( G 3 ) and σ ( G 5 ) σ ( G 3 ) . Now let us assume a bounded non-empty open set A R . For all x A there is a maximal ε x > 0 such that ( x - ε x , x + ε x ) A. Hence A = x A Q ( x - ε x , x + ε x ) , which proves G 0 σ ( G 5 ) and σ ( G 0 ) σ ( G 5 ) . Finally, A ∈ G 0 implies A c ∈ G 1 σ ( G 1 ) and A σ ( G 1 ). Hence G 0 σ ( G 1 ) and σ ( G 0 ) σ ( G 1 ) . The remaining inclusion σ ( G 1 ) σ ( G 0 ) can be shown in the same way. 1 elix Edouard Justin ´ Emile Borel, 07/01/1871-03/02/1956, French mathematician.
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14 CHAPTER 1. PROBABILITY SPACES 1.2 Probability measures Now we introduce the measures we are going to use: Definition 1.2.1 [probability measure, probability space] Let , F ) be a measurable space.
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