real_analysis_I

real_analysis_I - Real Analysis, course outline Denis...

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Unformatted text preview: Real Analysis, course outline Denis Labutin 1 Measure theory I 1. Sigma algebras. Let A be a collection of subsets of some fixed set . It is called a -algebra with the unit element if (a) , A ; (b) E A = c E A ; (c) E j A , j = 1 , 2 ,... = [ j E j A . Prove that a -algebra A is closed under countable number of the set theoretic operations ( , , \ , ( ) c ). 2. For a sequence of sets E j define limsup j E j = { points which belong to infinitely many E j } , liminf j E j = { points which belong to all but finitely many E j } . Thus limsup E j liminf E j . If limsup E j = liminf E j = E , then we write lim j E j = E. (a) Give an example of a sequence { E j } for which the inclusion is strict. (b) Show that limsup j E j = \ j =1 [ k = j E k , liminf j E j = [ j =1 \ k = j E k . (c) Show that if all E j are elements of a -algebra A , then limsup j E j , liminf j E j A . (d) Suppose that E 1 E 2 . Prove that limsup j E j = liminf j E j = \ j E j . 1 (e) Find (and prove) the formula for limsup E j , liminf E j in the case E 1 E 2 . 3. Prove that the following collections of sets are -algebras. (a) Trivial -algebra { , } . (b) Bulean -algebra 2 = { all subsets of } . (c) For any E the collection { E, c E, , } (It is called the - algebra generated by the set E ). (d) Let D = { D 1 ,D 2 ,... } is a countable (disjoint) partition of : = [ j D j , D i D j = if i 6 = j. The family of all at most countable unions of elements of D (includ- ing ) is a -algebra. (e) The family of all E such that either E is at most countable or c E is at most countable. 4. New -algebras from old. Prove the following statements. (a) Give an example showing that the union of two -algebras with the same unit element is not necessarily a -algebra. (b) If ( A 1 , ) and ( A 2 , ) are -algebras then A 1 A 2 is also a - algebra with the same unit element . (c) Similarly, if ( A a , ) is a -algebra for any a A , then \ a A A a is also a -algebra with the same unit element . (d) If ( A , ) is a -algebra, and E , then A E = { A E : A A} is a -algebra with the unit E = E . (e) If ( A , ) is a -algebra, and is any fixed set, then A = { A : A A} is a -algebra with the unit . One should think of as a group of dummy variables. 5. Property (b) is the basis for the following important construction. Theorem 1 Let C be any collection of subsets of . Then there exists one and only one -algebra ( A , ) such that: (i) C A ....
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This note was uploaded on 12/26/2011 for the course MATH 5A taught by Professor Rickrugangye during the Fall '07 term at UCSB.

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real_analysis_I - Real Analysis, course outline Denis...

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