This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Measure and Integral 1 Algebras Definition 1.1. An algebra on the set is a collection F of subsets of with the properties: (i) F (ii) C F C c =  C F (iii) A,B F A B F . Note that if F is an algebra on then = c F and if A,B F then A B = ( A c B c ) c F . Note also that an algebra on is a true algebra over the twoelement field, with intersection as product and symmetric difference as sum. Example 1.1. Each set has two extreme algebras: { , } and the power set P () comprising all subsets of . Example 1.2. The intersection of any (nonempty) collection of algebras on is an algebra on . Theorem 1.1. If C is any class of subsets of then there exists on a unique smallest algebra containing C . Proof. The intersection of all algebras on containing C does what is re quired. Definition 1.2. A algebra on the set is an algebra F on with the property that S n N C n F whenever C n F for each n N . Note that a algebra F is also closed under countable intersections, for if { C n : n N } F then n N C n = ( [ n N C c n ) c . 1 Example 1.3. Each set has two extreme algebras: { , } and the power set P () . Example 1.4. The intersection of any (nonempty) collection of algebras on is a algebra on . Theorem 1.2. If C is any class of subsets of then there exists on a unique smallest algebra containing C . Proof. The intersection of all algebras on containing C does the trick. We call the smallest algebra on containing C the algebra ( C ) generated by C . As a special case, if is a topological space then its Borel algebra B () is the algebra generated by its class of open (equivalently, closed) subsets. Definition 1.3. A measurable space is a pair ( , F ) with a set in which F is a algebra of subsets. When ( , F ) is a measurable space, we may refer to elements of F as measurable subsets of . 2 Measures Definition 2.1. Let C be a class of subsets of the set . The function : C [0 , ] is said to be: additive iff ( ) = 0 and ( A B ) = ( A ) + ( B ) whenever A and B are disjoint elements of C with union in C ; additive (or countably additive) iff ( ) = 0 and ( [ n N C n ) = X n N ( C n ) whenever { C n : n N } is a countable family of pairwisedisjoint elements of C with union in C . 2 Note that these definitions simplify when the class C is a ( )algebra, for then its closure under (countable) unions is automatic. Note also that the definitions simplify when takes any finite value: the hypothesis ( ) = 0 is then redundant, for if C C and ( C ) < then ( ) = 0 follows from ( C ) = ( C ) = ( C ) + ( )....
View
Full
Document
 Fall '09
 Robinson

Click to edit the document details