This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full
Document
This note was uploaded on 12/26/2011 for the course MATH 5A taught by Professor Rickrugangye during the Fall '07 term at UCSB.
 Fall '07
 RickRugangYe
 Algebra, Sets

Click to edit the document details