Sets and rings with Nikodm types properties M. Lpez-Pellicer (DMA, - PowerPoint PPT Presentation

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. Sets and rings with Nikodým type’s properties M. López-Pellicer (DMA, IUMPA) Be ¸dlewo, 1st-7th July, 2018 Paweł Doma´ nski Memorial Conference M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. Outline Preliminaries. 1 Properties ( N ) , ( wN ) and ( G ) in rings. 2 Applications and open questions. 3 M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. Outline Preliminaries. 1 M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. Basic definitions. R = Ring of subsets of a nonempty set Ω , ℓ ∞ 0 ( R ) = span { χ A : A ∈ R} with sup norm. The gauge of Q = acx { χ A : A ∈ R} is an equivalent norm. Its dual is the Banach space ba ( R ) of bounded finitely additive measures defined on R . The polar norms are the variation and the supremum. The completion of ℓ ∞ 0 ( R ) is the space ℓ ∞ ( R ) of all bounded R -measurable functions. As A ∩ B ∈ R and A ∆ B ∈ R , if A , B ∈ R , then f ∈ ℓ ∞ 0 ( R ) admits representation f = � m i = 1 a i χ A i , with pairwise disjoint sets A 1 , . . . , A m ∈ R . The ring R is an algebra (a σ -algebra) of subsets of Ω if Ω ∈ R (resp. if Ω ∈ R and ∪{ A n : n ∈ N } ∈ R when A n ∈ R , n ∈ N ). M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. Nikodým set (Schachermayer). ∆ ⊂ R is a Nikodým set for ba ( R ) , in brief, ∆ has property ( N ) , if ∆ -pointwise bounded ⇒ norm-bounded in ba ( R ) , i. e., sup | µ α ( A ) | < ∞ , for each A ∈ ∆ , ⇒ sup sup | µ α ( A ) | < ∞ . α ∈ Λ A ∈R α ∈ Λ ∆ is a Nikodým set iff span { χ A : A ∈ ∆ } verifies Banach-Steinhaus theorem and is dense in ℓ ∞ 0 ( R ) , iff ∆ is strong norming, i.e., if ∆ = ∪ n ∆ n ↑ there exists ∆ m norming. � � An increasing web R n 1 , n 2 ,..., n p : p , n 1 , n 2 , . . . , n p ∈ N on R is a web on R such that R m 1 ⊆ R n 1 whenever m 1 ≤ n 1 and R n 1 , n 2 ,..., n p , m p + 1 ⊆ R n 1 , n 2 ,..., n p , n p + 1 whenever m p + 1 ≤ n p + 1 for every n i ∈ N and i � p . R is a ( wN ) -ring if each increasing web on R contains a strand � � R m 1 , m 2 ,..., m p : p ∈ N formed by Nikodým sets. M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. A characterization of ( wN ) property. � � A linear increasing web E n 1 , n 2 ,..., n p : p , n 1 , n 2 , . . . , n p ∈ N on a lcs E is an increasing web on E formed by linear subspaces. A locally convex space E is linear-( wN ) if each linear increasing web on E contains a strand � E m 1 , m 2 ,..., m p : p ∈ N � formed by dense subspaces that verify Banach-Steinhaus theorem. Theorem R is a ( wN ) -ring if and only the space ℓ ∞ 0 ( R ) is linear- ( wN ) . M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. A bit of history for σ -algebras. Nikodým-Grothendieck theorem states that each σ -algebra Σ has property ( N ) . Nikodým proved sup | µ α ( A ) | < ∞ , for each A ∈ Σ , ⇒ sup sup | µ α ( A ) | < ∞ , α ∈ Λ A ∈ Σ α ∈ Λ when µ α is countably additive. Valdivia had the conjecture that for each bounded additive vector measure µ defined in Σ and with values in a inductive limit F ( τ ) = lim n F n ( τ n ) of Fréchet spaces there exists m such that µ is a bounded vector measure µ : Σ → F m ( τ m ) . To obtain this localization theorem Valdivia proved that each σ -algebra Σ has property ( sN ) , i.e., each increasing covering of Σ contains a Nikodým set, improving Nikodým-Grothendieck theorem. Recently, Kakol and LP found that each σ -álgebra has property ( wN ) . M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. Consequences of ( sN ) property of a σ -algebra Σ . Proposition Let µ : Σ → F ( τ ) = lim m F m ( τ m ) , ( LF-space ) be bounded additive measure. ∃ q : µ : Σ → F q ( τ q ) is exhaustive ( c. a. ) . Proposition Let ( x n ) n is subseries convergent in F ( τ ) = lim m F m ( τ m ) , ( LF-space ) ∃ q : ( x n ) n is bounded multiplier F q ( τ q ) . Proposition If Σ = ∪ m Σ m and ( µ n ) n ∈ ba (Σ) , ∃ p ∈ N : if ( µ n ( A )) n is Cauchy, ∀ A ∈ Σ p , ( µ n ) n converges weakly. Recently, Kakol and LP found that each σ -álgebra has property ( wN ) . M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. Consequences of ( sN ) property of a σ -algebra Σ . Proposition Let µ : Σ → F ( τ ) = lim m F m ( τ m ) , ( LF-space ) be bounded additive measure. ∃ q : µ : Σ → F q ( τ q ) is exhaustive ( c. a. ) . Proposition Let ( x n ) n is subseries convergent in F ( τ ) = lim m F m ( τ m ) , ( LF-space ) ∃ q : ( x n ) n is bounded multiplier F q ( τ q ) . Proposition If Σ = ∪ m Σ m and ( µ n ) n ∈ ba (Σ) , ∃ p ∈ N : if ( µ n ( A )) n is Cauchy, ∀ A ∈ Σ p , ( µ n ) n converges weakly. Recently, Kakol and LP found that each σ -álgebra has property ( wN ) . M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. Consequences of ( sN ) property of a σ -algebra Σ . Proposition Let µ : Σ → F ( τ ) = lim m F m ( τ m ) , ( LF-space ) be bounded additive measure. ∃ q : µ : Σ → F q ( τ q ) is exhaustive ( c. a. ) . Proposition Let ( x n ) n is subseries convergent in F ( τ ) = lim m F m ( τ m ) , ( LF-space ) ∃ q : ( x n ) n is bounded multiplier F q ( τ q ) . Proposition If Σ = ∪ m Σ m and ( µ n ) n ∈ ba (Σ) , ∃ p ∈ N : if ( µ n ( A )) n is Cauchy, ∀ A ∈ Σ p , ( µ n ) n converges weakly. Recently, Kakol and LP found that each σ -álgebra has property ( wN ) . M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. Consequences of ( sN ) property of a σ -algebra Σ . Proposition Let µ : Σ → F ( τ ) = lim m F m ( τ m ) , ( LF-space ) be bounded additive measure. ∃ q : µ : Σ → F q ( τ q ) is exhaustive ( c. a. ) . Proposition Let ( x n ) n is subseries convergent in F ( τ ) = lim m F m ( τ m ) , ( LF-space ) ∃ q : ( x n ) n is bounded multiplier F q ( τ q ) . Proposition If Σ = ∪ m Σ m and ( µ n ) n ∈ ba (Σ) , ∃ p ∈ N : if ( µ n ( A )) n is Cauchy, ∀ A ∈ Σ p , ( µ n ) n converges weakly. Recently, Kakol and LP found that each σ -álgebra has property ( wN ) . M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. A bit of history for algebras. The N-G fails for algebras: Let R be the algebra of the finite and cofinite subsets of N and ǫ n the point mass at { n } . As ǫ n + 1 ( R ) − ǫ n ( R ) = 0 for n > n ( R ) , for R ∈ R , we get that { µ n : n ∈ N } , defined by µ n ( R ) = n ( ǫ n + 1 ( R ) − ǫ n ( R )) , are R pointwise bounded, but no uniformly bounded, because µ n ( { n } ) = − n . By Schachermayer the algebra J ([ 0 , 1 ]) has Nikodym property and, in 2013, Valdivia, after his proof that for a compact interval K of R k the algebra J ( K ) has property ( sN ) , states the still open problem whether ( N ) ⇔ ( sN ) holds in an algebra. Recently, our group found that J ( K ) hs property ( wN ) . Drewnowski - F . and P . - proves that the ring Z of subsets of density zero of N has property ( N ) and Ferrando get that Z has property ( wN ) . M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. Property ( G ) and Seever question. A ring R has property ( G ) if ℓ ∞ ( R ) is a Grothendieck space, i.e., each weak* convergent sequence in ba ( R ) is weak convergent. Schachermayer proved that A ring R has property ( G ) if and only if a bounded sequence { µ n } ∞ n = 1 in ba ( R ) which converges pointwise on R is uniformly exhaustive, i.e., for each sequence ( A i ) i of pairwise disjoints set of R i →∞ sup lim | µ n ( A i ) | = 0. n ∈ N He proved that J [ 0 , 1 ] does not have property ( G ) , answering Seever question ( N ) ⇒ ( G ) ? M. López-Pellicer Sets and rings with Nikodým type’s properties

Preliminaries. Properties ( N ) , ( wN ) and ( G ) in rings. Applications and open questions. Some obtained results. We will present the following results: Concerning the Valdivia open problem ( N ) ⇒ ( sN ) in an algebra of subsets of Ω? , we present a class of rings without property ( G ) for which the equivalence ( N ) ⇔ ( sN ) ⇔ ( wN ) holds. We characterize that a ring R has property ( G ) if and only if R is a Rainwater set for ba ( R ) . M. López-Pellicer Sets and rings with Nikodým type’s properties