Filter convergence and decompositions for vector lattice-valued measures Domenico Candeloro, Anna Rita Sambucini Department of Mathematics and Computer Science - University of Perugia Integration, Vector Measures and Related Topics VI, Be ¸dlewo, June 15-21, 2014 Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
Outline 1 Some background Vector lattices Filter convergence Decompositions 2 Convergence theorems The σ -additive case The finitely additive case 3 The ( SCP ) property Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
Vector lattices Let ( X , + , · , ≤ ) be a real vector space, endowed with a compatible ordering < . If X is stable under finite suprema (and infima) then it is called a vector lattice . Usually we shall assume that: Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
Vector lattices Let ( X , + , · , ≤ ) be a real vector space, endowed with a compatible ordering < . If X is stable under finite suprema (and infima) then it is called a vector lattice . Usually we shall assume that: X is super-Dedekind complete every non-empty upper-bounded subset A ⊂ X ( i.e. has supremum in X and contains a countable subset N such that sup N = sup A ). Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
Vector lattices Let ( X , + , · , ≤ ) be a real vector space, endowed with a compatible ordering < . If X is stable under finite suprema (and infima) then it is called a vector lattice . Usually we shall assume that: X is super-Dedekind complete every non-empty upper-bounded subset A ⊂ X ( i.e. has supremum in X and contains a countable subset N such that sup N = sup A ). X is weakly σ -distributive 0 = � � i a i ,φ ( i ) holds true, for each double (i.e. φ sequence ( a i , j ) such that a i , j ↓ j 0 for every integer i ( regulator), and φ runs among all mappings from N to N ) Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
Vector lattices Let ( X , + , · , ≤ ) be a real vector space, endowed with a compatible ordering < . If X is stable under finite suprema (and infima) then it is called a vector lattice . Usually we shall assume that: X is super-Dedekind complete every non-empty upper-bounded subset A ⊂ X ( i.e. has supremum in X and contains a countable subset N such that sup N = sup A ). X is weakly σ -distributive 0 = � � i a i ,φ ( i ) holds true, for each double (i.e. φ sequence ( a i , j ) such that a i , j ↓ j 0 for every integer i ( regulator), and φ runs among all mappings from N to N ) ( o ) -sequence Any decreasing sequence ( p n ) n in X , such that inf n p n = 0. Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
( o ) -convergence A sequence ( a n ) n in X is said to be ( o ) -convergent to a ∈ X whenever an ( o ) -sequence ( p n ) n exists, such that | a n − a | ≤ p n for all n . If this happens, ( p n ) n will be called a regulating ( o ) -sequence for ( a n ) n . Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
( o ) -convergence A sequence ( a n ) n in X is said to be ( o ) -convergent to a ∈ X whenever an ( o ) -sequence ( p n ) n exists, such that | a n − a | ≤ p n for all n . If this happens, ( p n ) n will be called a regulating ( o ) -sequence for ( a n ) n . Lemma (see a ) Let ( r n ) n be any ( o ) -sequence in a super-Dedekind complete vector lattice X. For every positive element u ∈ X + there exists an increasing mapping ω : N → N such that ∞ � N �→ u ∧ ( r ω ( n ) ) n = N defines an ( o ) -sequence in X. a A. BOCCUTO, D. C., A survey of decomposition and convergence theorems for l-group-valued measures , Atti Sem. Mat. Fis. Univ. Modena e Reggio Emilia, 53 , (2005), 243-260. Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
Filter convergence Let Z be any fixed set. A family F of subsets of Z is called a filter of Z iff ∅ �∈ F A ∩ B ∈ F whenever A , B ∈ F A ∈ F , B ⊃ A ⇒ B ∈ F . Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
Filter convergence Let Z be any fixed set. A family F of subsets of Z is called a filter of Z iff ∅ �∈ F A ∩ B ∈ F whenever A , B ∈ F A ∈ F , B ⊃ A ⇒ B ∈ F . Given any filter F of subsets of Z , the dual ideal of F is I F := { F c : F ∈ F} . If { z } ∈ I F for all z ∈ Z , F is a free filter. Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
Filter convergence Let Z be any fixed set. A family F of subsets of Z is called a filter of Z iff ∅ �∈ F A ∩ B ∈ F whenever A , B ∈ F A ∈ F , B ⊃ A ⇒ B ∈ F . Given any filter F of subsets of Z , the dual ideal of F is I F := { F c : F ∈ F} . If { z } ∈ I F for all z ∈ Z , F is a free filter. Examples: Z = N | J ∩ [ 0 , n ] | Statistical filter: F := { J ⊂ N : lim n = 1 } n Countably generated filters: I F is generated by a countable partition of N . (free) Ultrafilters Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
Definition o F A sequence ( x k ) k ∈ N in X ( o F ) -converges to x ∈ X ( x k → x ) iff there exists an ( o ) -sequence ( σ p ) p in X such that the set { k ∈ N : | x k − x | ≤ σ p } is an element of F for each p ∈ N . If this is the case, then ( σ p ) p is said to be a regulator for ( o F ) -convergence of ( x k ) k . Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
Definition o F A sequence ( x k ) k ∈ N in X ( o F ) -converges to x ∈ X ( x k → x ) iff there exists an ( o ) -sequence ( σ p ) p in X such that the set { k ∈ N : | x k − x | ≤ σ p } is an element of F for each p ∈ N . If this is the case, then ( σ p ) p is said to be a regulator for ( o F ) -convergence of ( x k ) k . Lemma (see a ) Let ( σ j p ) p be an ( o ) -sequence for all j ∈ N , and assume that the set { σ j p : p ∈ N , j ∈ N } is bounded in X . Then there exists an ( o ) -sequence ( r n ) n such that, for every j and every n there exists p satisfying σ j p ≤ r n . a B. Riecan-T.Neubrunn Integral, Measure and Ordering , Kluwer, Ister Science, Dordrecht/Bratislava (1997). Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
∀ filter F in Z , a subset H ⊂ Z is stationary if Z / ∈ I F , i.e. if and only if H ∩ F � = ∅ for all F ∈ F . 1A. AVILES LOPEZ, B. CASCALES SALINAS, V. KADETS, A. LEONOV, The Schur l 1 -theorem for filters , J. Math. Phys. Anal. Geom. 3 (2009), 383-398. Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
∀ filter F in Z , a subset H ⊂ Z is stationary if Z / ∈ I F , i.e. if and only if H ∩ F � = ∅ for all F ∈ F . The filter F is block-respecting if, for every stationary set H and every block { D k : k ∈ N } of H there exists a stationary set J ⊂ H such that card ( J ∩ D k ) ≤ 1 for all k . ( ∀ infinite I ⊂ Z a block of I is any partition { D k , k ∈ N } of I , obtained with finite sets D k in Z ). 1A. AVILES LOPEZ, B. CASCALES SALINAS, V. KADETS, A. LEONOV, The Schur l 1 -theorem for filters , J. Math. Phys. Anal. Geom. 3 (2009), 383-398. Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
∀ filter F in Z , a subset H ⊂ Z is stationary if Z / ∈ I F , i.e. if and only if H ∩ F � = ∅ for all F ∈ F . The filter F is block-respecting if, for every stationary set H and every block { D k : k ∈ N } of H there exists a stationary set J ⊂ H such that card ( J ∩ D k ) ≤ 1 for all k . ( ∀ infinite I ⊂ Z a block of I is any partition { D k , k ∈ N } of I , obtained with finite sets D k in Z ). The filter F is said to be diagonal if for every sequence ( A n ) n in I F and every stationary set I ⊂ Z , there exists a stationary set J ⊂ I such that J ∩ A n is finite for all n ∈ N . 1A. AVILES LOPEZ, B. CASCALES SALINAS, V. KADETS, A. LEONOV, The Schur l 1 -theorem for filters , J. Math. Phys. Anal. Geom. 3 (2009), 383-398. Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
∀ filter F in Z , a subset H ⊂ Z is stationary if Z / ∈ I F , i.e. if and only if H ∩ F � = ∅ for all F ∈ F . The filter F is block-respecting if, for every stationary set H and every block { D k : k ∈ N } of H there exists a stationary set J ⊂ H such that card ( J ∩ D k ) ≤ 1 for all k . ( ∀ infinite I ⊂ Z a block of I is any partition { D k , k ∈ N } of I , obtained with finite sets D k in Z ). The filter F is said to be diagonal if for every sequence ( A n ) n in I F and every stationary set I ⊂ Z , there exists a stationary set J ⊂ I such that J ∩ A n is finite for all n ∈ N . In the paper 1 the simplified Schur property has been proved to be equivalent to the block-respecting property. 1A. AVILES LOPEZ, B. CASCALES SALINAS, V. KADETS, A. LEONOV, The Schur l 1 -theorem for filters , J. Math. Phys. Anal. Geom. 3 (2009), 383-398. Integration, Vector Measures and Related Topics Candeloro-Sambucini (D.M.I.) Filter convergence ... / 16
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