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Unformatted text preview: Measure and Integral 1 Algebras Definition 1.1. An algebra on the set Ω is a collection F of subsets of Ω with the properties: (i) Ω ∈ F (ii) C ∈ F ⇒ C c = Ω C ∈ F (iii) A,B ∈ F ⇒ A ∪ B ∈ F . Note that if F is an algebra on Ω then ∅ = Ω c ∈ F and if A,B ∈ F then A ∩ B = ( A c ∪ B c ) c ∈ F . Note also that an algebra on Ω is a true algebra over the twoelement field, with intersection as product and symmetric difference as sum. Example 1.1. Each set Ω has two extreme algebras: {∅ , Ω } and the power set P (Ω) comprising all subsets of Ω . Example 1.2. The intersection of any (nonempty) collection of algebras on Ω is an algebra on Ω . Theorem 1.1. If C is any class of subsets of Ω then there exists on Ω a unique smallest algebra containing C . Proof. The intersection of all algebras on Ω containing C does what is re quired. Definition 1.2. A σalgebra on the set Ω is an algebra F on Ω with the property that S n ∈ N C n ∈ F whenever C n ∈ F for each n ∈ N . Note that a σalgebra 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 Ω has two extreme σalgebras: {∅ , Ω } and the power set P (Ω) . Example 1.4. The intersection of any (nonempty) collection of σalgebras on Ω is a σalgebra on Ω . Theorem 1.2. If C is any class of subsets of Ω then there exists on Ω a unique smallest σalgebra containing C . Proof. The intersection of all σalgebras on Ω containing C does the trick. We call the smallest σalgebra on Ω containing C the σalgebra σ ( C ) generated by C . As a special case, if Ω is a topological space then its Borel σalgebra B (Ω) is the σalgebra generated by its class of open (equivalently, closed) subsets. Definition 1.3. A measurable space is a pair (Ω , F ) with Ω a set in which F is a σalgebra of subsets. When (Ω , F ) is a measurable space, we may refer to elements of F as measurable subsets of Ω. 2 Measures Definition 2.1. Let C be a class of subsets of the set Ω . The function μ : C → [0 , ∞ ] is said to be: additive iff μ ( ∅ ) = 0 and μ ( A ∪ B ) = μ ( A ) + μ ( B ) whenever A and B are disjoint elements of C with union in C ; σadditive (or countably additive) iff μ ( ∅ ) = 0 and μ ( [ n ∈ N C n ) = X n ∈ N μ ( 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 ( σ)algebra, for then its closure under (countable) unions is automatic. Note also that the definitions simplify when μ takes any finite value: the hypothesis μ ( ∅ ) = 0 is then redundant, for if C ∈ C and μ ( C ) < ∞ then μ ( ∅ ) = 0 follows from μ ( C ) = μ ( C ∪ ∅ ) = μ ( C ) + μ ( ∅ )....
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 Fall '09
 Robinson
 measure, µs, Lebesgue integration

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