MI - Measure and Integral 1 Algebras Definition 1.1. An...

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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 two-element 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 pairwise-disjoint 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 ) + ( )....
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MI - Measure and Integral 1 Algebras Definition 1.1. An...

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