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Measure Algebras Measure Algebras Thomas Jech Warsaw, October 2013 Thomas Jech Measure Algebras Boolean σ-algebras Boolean algebra: a set B, operations a ∨ b, a ∧ b and −a; a partial ordering a ≤ b and the smallest and the greatest element, 0 and 1. If every countable subset A of B has a least upper bound W A (and the greatest lower bound V A) then B is a Boolean σ-algebra. Thomas Jech Measure Algebras Boolean σ-algebras Boolean algebra: a set B, operations a ∨ b, a ∧ b and −a; a partial ordering a ≤ b and the smallest and the greatest element, 0 and 1. If every countable subset A of B has a least upper bound W A (and the greatest lower bound V A) then B is a Boolean σ-algebra. B+ is the set of all nonzero elements of B. An antichain in B is a set A ⊂ B+ whose any two elements a, b are disjoint, i.e. a ∧ b = 0. A is maximal if W A = 1. Thomas Jech Measure Algebras Boolean σ-algebras Boolean algebra: a set B, operations a ∨ b, a ∧ b and −a; a partial ordering a ≤ b and the smallest and the greatest element, 0 and 1. If every countable subset A of B has a least upper bound W A (and the greatest lower bound V A) then B is a Boolean σ-algebra. B+ is the set of all nonzero elements of B. An antichain in B is a set A ⊂ B+ whose any two elements a, b are disjoint, i.e. a ∧ b = 0. A is maximal if W A = 1. B satisfies the countable chain condition (ccc) if it has no uncountable antichains. Thomas Jech Measure Algebras Measure algebras A measure on a Boolean σ-algebra B is a real-valued function m with the following properties: m(0) = 0, m(a) > 0 for a = 0, and m(1) = 1, m(a) ≤ m(b) if a ≤ b, m(a ∨ b)= m(a)+ m(b) if a ∧ b = 0, Thomas Jech Measure Algebras Measure algebras A measure on a Boolean σ-algebra B is a real-valued function m with the following properties: m(0) = 0, m(a) > 0 for a = 0, and m(1) = 1, m(a) ≤ m(b) if a ≤ b, m(a ∨ b)= m(a)+ m(b) if a ∧ b = 0, if a1 > a2 > a3 > ... and Vn an = 0, then lim m(an) = 0. Thomas Jech Measure Algebras Measure algebras A measure on a Boolean σ-algebra B is a real-valued function m with the following properties: m(0) = 0, m(a) > 0 for a = 0, and m(1) = 1, m(a) ≤ m(b) if a ≤ b, m(a ∨ b)= m(a)+ m(b) if a ∧ b = 0, if a1 > a2 > a3 > ... and Vn an = 0, then lim m(an) = 0. (It follows that m(Wn∈ω an)= Pn∈ω m(an) whenever the an are pairwise disjoint.) A Boolean σ-algebra that carries a measure is a measure algebra. Thomas Jech Measure Algebras Measure algebras For every n let Cn = {a ∈ B : m(a) ≥ 1/n}. We have + C1 ⊂ C2 ⊂ C3 ⊂ ... and S Cn = B . If A ⊂ Cn is an antichain then it has size at most n. Hence every antichain is at most countable, and so every measure algebra satisfies ccc. Thomas Jech Measure Algebras Measure algebras For every n let Cn = {a ∈ B : m(a) ≥ 1/n}. We have + C1 ⊂ C2 ⊂ C3 ⊂ ... and S Cn = B . If A ⊂ Cn is an antichain then it has size at most n. Hence every antichain is at most countable, and so every measure algebra satisfies ccc. Definition. A Boolean σ-algebra B is weakly distributive if for any sequence {An}n of maximal antichains there exist finite sets En ⊂ An such that Wn Vk≥n(W Ek )= 1. Thomas Jech Measure Algebras Measure algebras For every n let Cn = {a ∈ B : m(a) ≥ 1/n}. We have + C1 ⊂ C2 ⊂ C3 ⊂ ... and S Cn = B . If A ⊂ Cn is an antichain then it has size at most n. Hence every antichain is at most countable, and so every measure algebra satisfies ccc. Definition. A Boolean σ-algebra B is weakly distributive if for any sequence {An}n of maximal antichains there exist finite sets En ⊂ An such that Wn Vk≥n(W Ek )= 1. Every measure algebra is weakly distributive: Given {An}n, we let n+1 En ⊂ An be such that m(W En) ≥ 1 − 1/2 . Thomas Jech Measure Algebras Algebraic characterizations of measure algebras The problem of von Neumann. John von Neumann asked in 1937 if ccc and weak distributivity are sufficient for a Boolean σ-algebra to carry a measure. Thomas Jech Measure Algebras Algebraic characterizations of measure algebras The problem of von Neumann. John von Neumann asked in 1937 if ccc and weak distributivity are sufficient for a Boolean σ-algebra to carry a measure. Convergence For any sequence {an}n in B let limsup an = Vn Wk≥n ak . A sequence {an}n converges to 0, lim an = 0, if limsup an = 0. Thomas Jech Measure Algebras Algebraic characterizations of measure algebras The problem of von Neumann. John von Neumann asked in 1937 if ccc and weak distributivity are sufficient for a Boolean σ-algebra to carry a measure. Convergence For any sequence {an}n in B let limsup an = Vn Wk≥n ak . A sequence {an}n converges to 0, lim an = 0, if limsup an = 0. If a1 > a2 > a3 > ... and Vn an = 0 then lim an = 0. Thomas Jech Measure Algebras Algebraic characterizations of measure algebras The problem of von Neumann. John von Neumann asked in 1937 if ccc and weak distributivity are sufficient for a Boolean σ-algebra to carry a measure. Convergence For any sequence {an}n in B let limsup an = Vn Wk≥n ak . A sequence {an}n converges to 0, lim an = 0, if limsup an = 0. If a1 > a2 > a3 > ... and Vn an = 0 then lim an = 0. If {an}n is an antichain then lim an = 0. Thomas Jech Measure Algebras Algebraic characterizations of measure algebras The problem of von Neumann. John von Neumann asked in 1937 if ccc and weak distributivity are sufficient for a Boolean σ-algebra to carry a measure. Convergence For any sequence {an}n in B let limsup an = Vn Wk≥n ak . A sequence {an}n converges to 0, lim an = 0, if limsup an = 0. If a1 > a2 > a3 > ... and Vn an = 0 then lim an = 0. If {an}n is an antichain then lim an = 0. If m is a measure and lim an = 0, then lim m(an)=0. Thomas Jech Measure Algebras Algebraic characterizations of measure algebras The problem of von Neumann. John von Neumann asked in 1937 if ccc and weak distributivity are sufficient for a Boolean σ-algebra to carry a measure. Convergence For any sequence {an}n in B let limsup an = Vn Wk≥n ak . A sequence {an}n converges to 0, lim an = 0, if limsup an = 0. If a1 > a2 > a3 > ... and Vn an = 0 then lim an = 0. If {an}n is an antichain then lim an = 0. If m is a measure and lim an = 0, then lim m(an)=0. n If Pn m(an) < ∞ (e.g., if m(an) ≤ 1/2 ) then lim an = 0. Thomas Jech Measure Algebras Algebraic characterizations of measure algebras Definition. A Boolean σ-algebra B is concentrated if whenever {An : n = 1, 2, 3,...} is a sequence of finite antichains with n |An|≥ 2 then we can choose an ∈ An such that lim an = 0. Thomas Jech Measure Algebras Algebraic characterizations of measure algebras Definition. A Boolean σ-algebra B is concentrated if whenever {An : n = 1, 2, 3,...} is a sequence of finite antichains with n |An|≥ 2 then we can choose an ∈ An such that lim an = 0. B is uniformly concentrated if there is a choice function F acting on finite antichains so that whenever {An : n = 1, 2, 3,...} is a n sequence of finite antichains with |An|≥ 2 then, letting an = F (An), we have lim an = 0. Thomas Jech Measure Algebras Algebraic characterizations of measure algebras Definition. A Boolean σ-algebra B is concentrated if whenever {An : n = 1, 2, 3,...} is a sequence of finite antichains with n |An|≥ 2 then we can choose an ∈ An such that lim an = 0. B is uniformly concentrated if there is a choice function F acting on finite antichains so that whenever {An : n = 1, 2, 3,...} is a n sequence of finite antichains with |An|≥ 2 then, letting an = F (An), we have lim an = 0. Every measure algebra is uniformly concentrated: Let m be a measure on B. If A is a finite antichain, let F (A) be an element of A of least possible measure. If |A|≥ 2n then m(F (A)) ≤ 1/2n. Thomas Jech Measure Algebras Algebraic characterizations of measure algebras Definition. A Boolean σ-algebra B is uniformly weakly distributive if for every n there is a function Fn acting on maximal antichains such that Fn(A) is a finite subset of A, Thomas Jech Measure Algebras Algebraic characterizations of measure algebras Definition. A Boolean σ-algebra B is uniformly weakly distributive if for every n there is a function Fn acting on maximal antichains such that Fn(A) is a finite subset of A, and for any sequence {An}n of maximal antichains, letting En = Fn(An), we have Wn Vk≥n(W Ek )= 1.
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