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 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 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 ( B ) 3 1.2 Generated Sigma-algebra ( B ) Let X be a set and B a non-empty collection of subsets of X . The smallest algebra containing all the sets of B is denoted ( B ) and is called the sigma-algebra generated by...
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This note was uploaded on 01/27/2010 for the course MATH 7312 taught by Professor Sengupta during the Spring '02 term at LSU.

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

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