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sigmaalg

# sigmaalg - Chapter 1 Sigma-Algebras 1.1 Definition Consider...

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Unformatted text preview: Chapter 1 Sigma-Algebras 1.1 Definition Consider a set X . A σ –algebra F of subsets of X is a collection F of subsets of X satisfying the following conditions: (a) ∅ ∈ F (b) if B ∈ F then its complement B c is also in F (c) if B 1 , B 2 , ... is a countable collection of sets in F then their union ∪ ∞ n =1 B n Sometimes we will just write “sigma-algebra” instead of “sigma-algebra of subsets of X .” There are two extreme examples of sigma-algebras: • the collection {∅ , X } is a sigma-algebra of subsets of X • the set P ( X ) of all subsets of X is a sigma-algebra Any sigma-algebra F of subsets of X lies between these two extremes: {∅ , X } ⊂ F ⊂ P ( X ) An atom of F is a set A ∈ F such that the only subsets of A which are also in F are the empty set ∅ and A itself. 1 2 CHAPTER 1. SIGMA-ALGEBRAS A partition of X is a collection of disjoint subsets of X whose union is all of X . For simplicity, consider a partition consisting of a finite number of sets A 1 , ..., A N . Thus A i ∩ A j = ∅ and A 1 ∪ ··· ∪ A N = X Then the collect F consisting of all unions of the sets A j forms a σ –algebra. Here are a few simple observations: Proposition 1 Let F be a sigma-algebra of subsets of X . (i) X ∈ F (ii) If A 1 , ..., A n ∈ F then A 1 ∪ ··· ∪ A n ∈ F (iii) If A 1 , ..., A n ∈ F then A 1 ∩ ··· ∩ A n ∈ F (iv) If A 1 , A 2 , ... is a countable collection of sets in F then ∩ ∞ n =1 A n ∈ F (v) If A, B ∈ F then A- B ∈ F . Proof Since ∅ ∈ F and X = ∅ c it follows that X ∈ F . For (ii) we have A 1 ∪ ··· ∪ A n = A 1 ∪ ··· ∪ A n ∪ ∅ ∪ ∅ ∪ ··· ∈ F Then (iii) follows by complementation: A 1 ∩ ··· ∩ A n = ( A c 1 ∪ ··· ∪ A c n ) c which is in F because each A c i ∈ F and, by (i), F is closed under finite unions. Similarly, (iv) follows by taking complements : ∩ ∞ n =1 A n = [ ∪ ∞ n =1 A c n ] c which belongs to F because F is closed under complements and countable unions. Finally, A- B = A ∩ B c is in F , because A, B c ∈ F . QED 1.2. GENERATED SIGMA-ALGEBRA σ ( B ) 3 1.2 Generated Sigma-algebra σ ( B ) Let X be a set and B a non-empty collection of subsets of X . The smallest σ –algebra containing all the sets of B is denoted σ ( B ) and is called the sigma-algebra generated by...
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sigmaalg - Chapter 1 Sigma-Algebras 1.1 Definition Consider...

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