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 Nikodm types properties M. Lpez-Pellicer (DMA, IUMPA) Be dlewo, 1st-7th July, 2018 Pawe Doma nski Memorial
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
Be ¸dlewo, 1st-7th July, 2018 Paweł Doma´ nski Memorial Conference
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
1
Preliminaries.
2
Properties (N), (wN) and (G) in rings.
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Applications and open questions.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
1
Preliminaries.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
R = Ring of subsets of a nonempty set Ω, ℓ∞
0 (R) = span{χA : A ∈ R} with sup norm. The gauge of
Q = acx{χA : 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 ai χAi, with pairwise disjoint
sets A1, . . . , Am ∈ R. The ring R is an algebra (a σ-algebra) of subsets of Ω if Ω ∈ R (resp. if Ω ∈ R and ∪{An : n ∈ N} ∈ R when An ∈ R, n ∈ N).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
∆ ⊂ R is a Nikodým set for ba (R), in brief, ∆ has property(N), if ∆-pointwise bounded ⇒ norm-bounded in ba (R), i. e., sup
α∈Λ
|µα (A)| < ∞, for each A ∈ ∆, ⇒ sup
A∈R
sup
α∈Λ
|µα (A)| < ∞. ∆ is a Nikodým set iff span {χA : 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
web on R such that Rm1 ⊆ Rn1 whenever m1 ≤ n1 and Rn1,n2,...,np,mp+1 ⊆ Rn1,n2,...,np,np+1 whenever mp+1 ≤ np+1 for every ni ∈ N and i p. R is a (wN)-ring if each increasing web on R contains a strand
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
A linear increasing web
subspaces. A locally convex space E is linear-(wN) if each linear increasing web on E contains a strand
verify Banach-Steinhaus theorem. Theorem R is a (wN)-ring if and only the space ℓ∞
0 (R) is linear-(wN).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
Nikodým-Grothendieck theorem states that each σ-algebra Σ has property (N). Nikodým proved sup
α∈Λ
|µα (A)| < ∞, for each A ∈ Σ, ⇒ sup
A∈Σ
sup
α∈Λ
|µα (A)| < ∞, when µα 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) = limn Fn(τn) of Fréchet spaces there exists m such that µ is a bounded vector measure µ : Σ → Fm(τm). 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).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
Proposition Let µ : Σ → F(τ) = limm Fm(τm), (LF-space) be bounded additive measure. ∃q : µ : Σ → Fq(τq) is exhaustive (c. a.). Proposition Let (xn)n is subseries convergent in F(τ) = limm Fm(τm), (LF-space) ∃q : (xn)n is bounded multiplier Fq(τq). Proposition If Σ = ∪mΣm and (µn)n ∈ ba(Σ), ∃p ∈ N : if (µn(A))n is Cauchy, ∀A ∈ Σp, (µn)n converges weakly. Recently, Kakol and LP found that each σ-álgebra has property (wN).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
Proposition Let µ : Σ → F(τ) = limm Fm(τm), (LF-space) be bounded additive measure. ∃q : µ : Σ → Fq(τq) is exhaustive (c. a.). Proposition Let (xn)n is subseries convergent in F(τ) = limm Fm(τm), (LF-space) ∃q : (xn)n is bounded multiplier Fq(τq). Proposition If Σ = ∪mΣm and (µn)n ∈ ba(Σ), ∃p ∈ N : if (µn(A))n is Cauchy, ∀A ∈ Σp, (µn)n converges weakly. Recently, Kakol and LP found that each σ-álgebra has property (wN).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
Proposition Let µ : Σ → F(τ) = limm Fm(τm), (LF-space) be bounded additive measure. ∃q : µ : Σ → Fq(τq) is exhaustive (c. a.). Proposition Let (xn)n is subseries convergent in F(τ) = limm Fm(τm), (LF-space) ∃q : (xn)n is bounded multiplier Fq(τq). Proposition If Σ = ∪mΣm and (µn)n ∈ ba(Σ), ∃p ∈ N : if (µn(A))n is Cauchy, ∀A ∈ Σp, (µn)n converges weakly. Recently, Kakol and LP found that each σ-álgebra has property (wN).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
Proposition Let µ : Σ → F(τ) = limm Fm(τm), (LF-space) be bounded additive measure. ∃q : µ : Σ → Fq(τq) is exhaustive (c. a.). Proposition Let (xn)n is subseries convergent in F(τ) = limm Fm(τm), (LF-space) ∃q : (xn)n is bounded multiplier Fq(τq). Proposition If Σ = ∪mΣm and (µn)n ∈ ba(Σ), ∃p ∈ N : if (µn(A))n is Cauchy, ∀A ∈ Σp, (µn)n converges weakly. Recently, Kakol and LP found that each σ-álgebra has property (wN).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
The N-G fails for algebras: Let R be the algebra of the finite and cofinite subsets of N and ǫn the point mass at {n}. As ǫn+1(R) − ǫn(R) = 0 for n > n(R), for R ∈ R, we get that {µn : n ∈ N}, defined by µn(R) = n(ǫn+1(R) − ǫn(R)), are R pointwise bounded, but no uniformly bounded, because µn({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 Rk the algebra J(K) has property (sN), states the still
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).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
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
n=1 in ba(R) which converges
pointwise on R is uniformly exhaustive, i.e., for each sequence (Ai)i of pairwise disjoints set of R lim
i→∞ sup n∈N
|µn (Ai)| = 0. He proved that J[0, 1] does not have property (G), answering Seever question (N) ⇒ (G)?
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions.
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).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
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Properties (N), (wN) and (G) in rings. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Unless otherwise stated we shall always work over an underlying measure space (Ω, Σ), Σ a σ-algebra. If A ∈ Σ, ΣA := {B ∈ Σ : B ⊆ A} is a σ-algebra A subset H of Σ is Σ-hereditary if H = ∪{ΣA : A ∈ H}. Definition A subset M of a ring R of subsets of Ω is a distinguished subset of R if for each sequence {An : n ∈ N} ⊆ R there is {Mn : n ∈ N} ⊆ M with ∞
n=1 (An \ Mn) ∈ R.
The ring M of finite subsets of N is a distinguished subset of the hereditary ring Z of subsets of density zero of N (Drewnoswki).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Theorem Let M ⊂ R ⊂ Σ, both Σ-hereditary and M distinguished subset of the ring R. If each T ⊂ ba (R) pointwise bounded on R is uniformly bounded on M, then R has property (N). Proof [by contradiction]. If ∃ (An, µn, p) ∈ R × ba (R) × N, n ∈ N, such that {µn : n ∈ N} is pointwise bounded on R, |µn(An)| > n and |µn(M)| p, for each M ∈ M, then let {Mn : n ∈ N} in M satisfying that A := ∞
n=1 (An \ Mn) ∈ R, with Mn ⊂ An for all n ∈ N.
As the σ-algebra ΣA ⊂ R then there exists q > 0 such that |µn(An \ Mn)| q for each n ∈ N. So we get the contradiction p+q |µp+q(Mp+q)|+|µp+q(Ap+q \ Mp+q)| |µp+q(Ap+q)| > p+q.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Theorem Let M ⊂ R ⊂ Σ, both Σ-hereditary and M distinguished subset of the ring R. If each T ⊂ ba (R) pointwise bounded on R is uniformly bounded on M, then R has property (N). Proof [by contradiction]. If ∃ (An, µn, p) ∈ R × ba (R) × N, n ∈ N, such that {µn : n ∈ N} is pointwise bounded on R, |µn(An)| > n and |µn(M)| p, for each M ∈ M, then let {Mn : n ∈ N} in M satisfying that A := ∞
n=1 (An \ Mn) ∈ R, with Mn ⊂ An for all n ∈ N.
As the σ-algebra ΣA ⊂ R then there exists q > 0 such that |µn(An \ Mn)| q for each n ∈ N. So we get the contradiction p+q |µp+q(Mp+q)|+|µp+q(Ap+q \ Mp+q)| |µp+q(Ap+q)| > p+q.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Definition Let (Ω, Σ) be a measure space, {µn}∞
n=1 a sequence of
[0, 1]-valued finitely additive measures that are countably subadditive and {En : n ∈ N} a pairwise disjoint sequence in Σ such that µn (En) = 1 for each n ∈ N. Then the Σ-hereditary ring R = {A ∈ Σ : µn (A) → 0} is named Σ-subring dominated by the sequence {(µn, En) : n ∈ N}. Clearly, no Σ-subring dominated has property (G), because {µn (A)}∞
n=1 converges for every A ∈ R, but
lim
i→∞ sup n∈N
|µn (Ei)| = 1 = 0
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Theorem If R is a Σ-subring dominated by a sequence {(µn, En) : n ∈ N}, the family M := n
p=1 Ep : n ∈ N
Proof. Let {Ai : i ∈ N} ⊆ R. Select (ns)n ↑ in N with 0 µk(A1) + · · · + µk(As) < s−1, for ns k. As µk(Ep) = δkp then, for ns k < ns+1, we have 0 µk ∞
i=1
p=1 Ep
i=1 µk
p=1 Ep
⇒ lim
k µk
∞
i=1
ni
p=1 Ep
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Theorem If R is a Σ-subring dominated by a sequence {(µn, En) : n ∈ N}, the family M := n
p=1 Ep : n ∈ N
Proof. Let {Ai : i ∈ N} ⊆ R. Select (ns)n ↑ in N with 0 µk(A1) + · · · + µk(As) < s−1, for ns k. As µk(Ep) = δkp then, for ns k < ns+1, we have 0 µk ∞
i=1
p=1 Ep
i=1 µk
p=1 Ep
⇒ lim
k µk
∞
i=1
ni
p=1 Ep
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Theorem If R is a Σ-subring dominated by a sequence {(µn, En) : n ∈ N}, the family M := n
p=1 Ep : n ∈ N
Proof. Let {Ai : i ∈ N} ⊆ R. Select (ns)n ↑ in N with 0 µk(A1) + · · · + µk(As) < s−1, for ns k. As µk(Ep) = δkp then, for ns k < ns+1, we have 0 µk ∞
i=1
p=1 Ep
i=1 µk
p=1 Ep
⇒ lim
k µk
∞
i=1
ni
p=1 Ep
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Example The ring Z of subsets of N of density zero is a 2N-subring
Proof. Let En :=
and µn the [0, 1]-valued measure µn (A) = |A ∩ En| 2n−1 , then µn (En) = 1. If A ∈ Z then lim
n→∞
|A ∩ En| 2n−1 = 2× lim
n→∞
|A ∩ (0, 2n]| 2n − lim
n→∞
= 0. Conversely, if A ⊆ N verifies that µn (A) → 0, then A is a set of density zero as a consequence of the Stolz convergence test. Z is the 2N-subring dominated by (µn, En)n. Then the ring does not have property G (a fact observed
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Example The ring Z of subsets of N of density zero is a 2N-subring
Proof. Let En :=
and µn the [0, 1]-valued measure µn (A) = |A ∩ En| 2n−1 , then µn (En) = 1. If A ∈ Z then lim
n→∞
|A ∩ En| 2n−1 = 2× lim
n→∞
|A ∩ (0, 2n]| 2n − lim
n→∞
= 0. Conversely, if A ⊆ N verifies that µn (A) → 0, then A is a set of density zero as a consequence of the Stolz convergence test. Z is the 2N-subring dominated by (µn, En)n. Then the ring does not have property G (a fact observed
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Theorem Let {µn}∞
n=1 atomless probability measures on Σ and
{En ∈ Σ : n ∈ N} pairwise disjoints with µn (Em) = δn,m for n, m ∈ N. The Σ-subring R dominated by {(µn, En) : n ∈ N} has property (N). Proof. Dn := n
p=1 Ep. Then M = {ΣDn : n ∈ N} is a Σ-hereditary
distinguished subset of R. Let H(⊂ ba(R)), pointwise bounded on R but not uniformly bounded on M. H is not uniformly bounded on Mm := {ΣDn : n > m}.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Proof. Fix n ∈ N and let {Ep,j ∈ Σ : 1 j n} a partition of Ep with µp(Ep,j) = n−1. Let Dnj := n
p=1 Epj and Mmj := {ΣDnj : n > m}. For each n
there exists 1 jn n such that H is not uniformly bounded on Mmjn. Therefore there exists vn ∈ H, mn+1 > mn and An ⊆ {Ep,jn : mn < p mn+1} with |vn(An)| > n.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Proof. A clear induction procedure provides for each n ∈ N a triple (vn, mn, An) ∈ H × N × Σ with {mn}∞
n=1 ↑⊆ N such that
An ⊆ {Ep,jn : mn < p mn+1} for 1 jn n, µp(Ep,jn) = n−1 if mn < p mn+1, and |vn(An)| > n, for each n ∈ N. Then A := {An : n ∈ N} ∈ Σ and, by construction, µp(A) = µp(An) µp(Ep,jn) = n−1 if mn < p mn+1, hence limp µp(A) = 0 and consequently A ∈ R. Since the σ-algebra ΣA is contained in the Σ-hereditary ring R, it turns out that H must be uniformly bounded in ΣA, which contradicts the inequalities |vn(An)| > n.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Proof. A clear induction procedure provides for each n ∈ N a triple (vn, mn, An) ∈ H × N × Σ with {mn}∞
n=1 ↑⊆ N such that
An ⊆ {Ep,jn : mn < p mn+1} for 1 jn n, µp(Ep,jn) = n−1 if mn < p mn+1, and |vn(An)| > n, for each n ∈ N. Then A := {An : n ∈ N} ∈ Σ and, by construction, µp(A) = µp(An) µp(Ep,jn) = n−1 if mn < p mn+1, hence limp µp(A) = 0 and consequently A ∈ R. Since the σ-algebra ΣA is contained in the Σ-hereditary ring R, it turns out that H must be uniformly bounded in ΣA, which contradicts the inequalities |vn(An)| > n.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Proof. A clear induction procedure provides for each n ∈ N a triple (vn, mn, An) ∈ H × N × Σ with {mn}∞
n=1 ↑⊆ N such that
An ⊆ {Ep,jn : mn < p mn+1} for 1 jn n, µp(Ep,jn) = n−1 if mn < p mn+1, and |vn(An)| > n, for each n ∈ N. Then A := {An : n ∈ N} ∈ Σ and, by construction, µp(A) = µp(An) µp(Ep,jn) = n−1 if mn < p mn+1, hence limp µp(A) = 0 and consequently A ∈ R. Since the σ-algebra ΣA is contained in the Σ-hereditary ring R, it turns out that H must be uniformly bounded in ΣA, which contradicts the inequalities |vn(An)| > n.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Example Let Ω = [0, 1] and Σ be the σ-algebra of Lebesgue measurable subsets of the interval [0, 1]. Define the atomless measures µn (A) =
fn (t) dλ (t)
and λ is the Lebesgue probability measure of [0, 1]. The ring R of subsets of [0, 1] dominated by {(µn, En) : n ∈ N} has property (N) and, as each dominated Σ-subring, does not have property (G).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
We exhibit a class of rings for which properties (N) and (wN) are equivalent. We need the following two lemmas. Lemma Let R be a ring and W= ∪nWn ↑ ⊂ R. If W is a Nikodým set for ba(R), ∃m ∈ N such that span{χA : A ∈ Wm} = ℓ∞
0 (R).
If W is not a Nikodým set for ba(R), span{χA : A ∈ W} = ℓ∞
0 (R) and M(⊂ R) is countable then
Wn M is not a Nikodým set for ba(R).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
The lemma follows from the fact that Banach-Steinhaus theorem is hereditary in countable codimensional subspaces. In fact: If does not exist m, Hahn-Banach theorem provides {µn : n ∈ N} ⊆ ba(R), {mn}∞
n=1 ↑⊆ N : µn = n and
µn(Wmn) = {0}. Contradiction (W has (N)). When W has not (N) and span{χA : A ∈ W} = ℓ∞
0 (R), then
the subspace span{χA : A ∈ W} does not verifies Banach-Steinhaus theorem. Then span{χA : A ∈ W M} does not verifies Banach-Steinhaus theorem. Hence W M is not a Nikodým set for ba(R).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Lemma Let R be a ring with property (N) which fails to have property (wN). Then there exists an increasing web {Rt : t ∈
s∈N Ns}
in R such that for each countable subset M of R the increasing web {Rt M : t ∈
s∈N Ns} does not contain any strand
consisting entirely of Nikodým sets for ba(R). Proof. Let {R′
t : t ∈ s∈N Ns} be an increasing web in R without any
strand consisting of Nikodým sets. By property (N) ∃m1 : span{χA : A ∈ R′
m1+t1} = ℓ∞ 0 (R) and it may happen:
∀ R′
m1+t1 non (N). Rt1 := R′ m1+t1 and Rt := R′ m1+t1,t2.
Each R′
m1+t1 is (N). We continue in and obvious way.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Theorem Let M ⊂ R ⊂ Σ, both Σ-hereditary and M countable R
Proof. If R ∈ (N) \(wN), ∃{Rt : t ∈
s∈N Ns} ↑ in R and
{Rt M : t ∈
s∈N Ns} /
∈ (wN). Let J := {t ∈
If t ∈ J, ∃Tt ⊂ ba(R) pointwise bounded on Rt M and not uniformly bounded on R. As R is Nikodým, ∃At ∈ R with Tt unbounded in At, t ∈ J.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Theorem Let M ⊂ R ⊂ Σ, both Σ-hereditary and M countable R
Proof. If R ∈ (N) \(wN), ∃{Rt : t ∈
s∈N Ns} ↑ in R and
{Rt M : t ∈
s∈N Ns} /
∈ (wN). Let J := {t ∈
If t ∈ J, ∃Tt ⊂ ba(R) pointwise bounded on Rt M and not uniformly bounded on R. As R is Nikodým, ∃At ∈ R with Tt unbounded in At, t ∈ J.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Proof. Let Mt ∈ M such that A := {At \ Mt : t ∈ J} ∈ R. ΣA ∈ (wN), hence ∃ {mp}∞
p=1 ⊆ N:
{Rm1m2···mp ∪ M} ∩ ΣA is a Nikodým set for ba(ΣA), p ∈ N. Let t = (m1, m2, · · · , mq) ∈ J. As Tt ⊂ ba(R) is pointwise bounded on Rt M we get that Tt is pointwise bounded on Mt and Tt is pointwise bounded on {Rm1m2···mp ∪ M} ∩ ΣA, hence is uniformly bounded in ΣA, and in particular in At \ Mt. We have the contradiction that Tt is bounded in At. As Z has property (N) (D) then it has (wN).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Proof. Let Mt ∈ M such that A := {At \ Mt : t ∈ J} ∈ R. ΣA ∈ (wN), hence ∃ {mp}∞
p=1 ⊆ N:
{Rm1m2···mp ∪ M} ∩ ΣA is a Nikodým set for ba(ΣA), p ∈ N. Let t = (m1, m2, · · · , mq) ∈ J. As Tt ⊂ ba(R) is pointwise bounded on Rt M we get that Tt is pointwise bounded on Mt and Tt is pointwise bounded on {Rm1m2···mp ∪ M} ∩ ΣA, hence is uniformly bounded in ΣA, and in particular in At \ Mt. We have the contradiction that Tt is bounded in At. As Z has property (N) (D) then it has (wN).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Proof. Let Mt ∈ M such that A := {At \ Mt : t ∈ J} ∈ R. ΣA ∈ (wN), hence ∃ {mp}∞
p=1 ⊆ N:
{Rm1m2···mp ∪ M} ∩ ΣA is a Nikodým set for ba(ΣA), p ∈ N. Let t = (m1, m2, · · · , mq) ∈ J. As Tt ⊂ ba(R) is pointwise bounded on Rt M we get that Tt is pointwise bounded on Mt and Tt is pointwise bounded on {Rm1m2···mp ∪ M} ∩ ΣA, hence is uniformly bounded in ΣA, and in particular in At \ Mt. We have the contradiction that Tt is bounded in At. As Z has property (N) (D) then it has (wN).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Proof. Let Mt ∈ M such that A := {At \ Mt : t ∈ J} ∈ R. ΣA ∈ (wN), hence ∃ {mp}∞
p=1 ⊆ N:
{Rm1m2···mp ∪ M} ∩ ΣA is a Nikodým set for ba(ΣA), p ∈ N. Let t = (m1, m2, · · · , mq) ∈ J. As Tt ⊂ ba(R) is pointwise bounded on Rt M we get that Tt is pointwise bounded on Mt and Tt is pointwise bounded on {Rm1m2···mp ∪ M} ∩ ΣA, hence is uniformly bounded in ΣA, and in particular in At \ Mt. We have the contradiction that Tt is bounded in At. As Z has property (N) (D) then it has (wN).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Definition A subset X of the dual closed unit ball BE∗ of a Banach space E is a Rainwater set for E if every bounded sequence {xn}∞
n=1
for each x∗ ∈ X, converges weakly in E. By Corollary 11 in Simons’ paper A convergence theorem with boundary, each James boundary J for BE∗ is a Rainwater set for E (the converse is not true). Hence, in particular, ExtBE∗ is a Rainwater set for E (appears first time in Rainwater’s paper Weak convergence
integral representation theorem).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Proposition Let R be a ring of subsets of Ω. The following are equivalent
1
R has property (G).
2
R is a Rainwater set for ba (R). Proof. Assume 1. Each bounded sequence H = {µn}∞
n=1 ⊂ ba (R)
that converges pointwise on R is uniformly exhaustive. Then H weakly rel. compact subset and (Eberlein th.) H is weakly rel.
n=1 converges weakly.
Assume 2. If H = {µn}∞
n=1 is a bounded sequence in ba (R)
and converges pointwise on R, then {µn}∞
n=1 converges weakly.
Then H is weakly relatively compact and then {µn}∞
n=1 is
uniformly exhaustive. Hence R has property (G).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Proposition Let R be a ring of subsets of Ω. The following are equivalent
1
R has property (G).
2
R is a Rainwater set for ba (R). Proof. Assume 1. Each bounded sequence H = {µn}∞
n=1 ⊂ ba (R)
that converges pointwise on R is uniformly exhaustive. Then H weakly rel. compact subset and (Eberlein th.) H is weakly rel.
n=1 converges weakly.
Assume 2. If H = {µn}∞
n=1 is a bounded sequence in ba (R)
and converges pointwise on R, then {µn}∞
n=1 converges weakly.
Then H is weakly relatively compact and then {µn}∞
n=1 is
uniformly exhaustive. Hence R has property (G).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Corollary The ring Z has (wN) and is not Rainwater set for ba (Z). Corollary If a subset N of R is (N) and Rainwater then each sequence {µn}∞
n=1 in ba (R) pointwise convergent on N imply weakly
convergence. In particular, if a ring R has both properties (N) and (G), i. e., R is a so-called a ring with property (VHS), each sequence in ba (R) pointwise convergent on R is weakly convergent in ba (R).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Corollary The ring Z has (wN) and is not Rainwater set for ba (Z). Corollary If a subset N of R is (N) and Rainwater then each sequence {µn}∞
n=1 in ba (R) pointwise convergent on N imply weakly
convergence. In particular, if a ring R has both properties (N) and (G), i. e., R is a so-called a ring with property (VHS), each sequence in ba (R) pointwise convergent on R is weakly convergent in ba (R).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. A class of rings with property (N) that fail property (G) Rings for which (N) ⇔ (wN) . Rainwater sets for ba (R) and property (G).
Corollary The ring Z has (wN) and is not Rainwater set for ba (Z). Corollary If a subset N of R is (N) and Rainwater then each sequence {µn}∞
n=1 in ba (R) pointwise convergent on N imply weakly
convergence. In particular, if a ring R has both properties (N) and (G), i. e., R is a so-called a ring with property (VHS), each sequence in ba (R) pointwise convergent on R is weakly convergent in ba (R).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
3
Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
ℓ∞ (Ω) is Banach space of all scalar bounded functions defined
R is a hereditary ring of subsets of Ω. R (ℓ∞ (Ω)) := {f ∈ ℓ∞ (Ω) : supp f ∈ R}. R (ℓ∞ (Ω)) is a linear space just because the ring R is hereditary. Theorem If R is a ring of subsets of Ω with property (wN) then R (ℓ∞ (Ω)) is linear-(wN).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Lemma R (ℓ∞ (Ω)) is isomorphically isometric to a quotient of ℓ∞
0 (R) ⊗π ℓ∞ (Ω).
Proof. The linearization of the map T : ℓ∞
0 (R) ⊗π ℓ∞ (Ω) → R (ℓ∞ (Ω))
defined by T (χA ⊗ g) = χA · g, with A ∈ R and g ∈ ℓ∞ (Ω), verifies that Tv∞ ≤ n
i=1 fi∞ gi∞, for v = n i=1 fi ⊗ gi,
hence Tv∞ ≤ vπ. T is onto, because if g ∈ R (ℓ∞ (Ω)) and supp g = E ∈ R then T (χE ⊗ g) = g, and therefore T = 1.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Lemma R (ℓ∞ (Ω)) is isomorphically isometric to a quotient of ℓ∞
0 (R) ⊗π ℓ∞ (Ω).
Proof. The linearization of the map T : ℓ∞
0 (R) ⊗π ℓ∞ (Ω) → R (ℓ∞ (Ω))
defined by T (χA ⊗ g) = χA · g, with A ∈ R and g ∈ ℓ∞ (Ω), verifies that Tv∞ ≤ n
i=1 fi∞ gi∞, for v = n i=1 fi ⊗ gi,
hence Tv∞ ≤ vπ. T is onto, because if g ∈ R (ℓ∞ (Ω)) and supp g = E ∈ R then T (χE ⊗ g) = g, and therefore T = 1.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
R is a hereditary ring of subsets of N. R (ℓp) with 1 ≤ p ≤ ∞ the linear subspace of ℓp consisting of those sequences ξ such that supp ξ ∈ R. Theorem If Z is the ring of subsets of N of density zero, the normed space Z (ℓp) is linear-(wN).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Lemma R (ℓp) is a quotient of ℓ∞
0 (R) ⊗π ℓp for 1 ≤ p ≤ ∞ by means of
a norm-one linear quotient map. Proof. As p = ∞ is done with Ω = N, we assume 1 ≤ p < ∞. The linearization of the map T : ℓ∞
0 (R) ⊗π ℓp → R (ℓp)
given by T (χA ⊗ ξ) = ζA, where ζA
n = ξn if n ∈ A and ζn = 0
i=1 fi∞ ξip,
for v = n
i=1 fi ⊗ gi, hence Tvp ≤ vπ.
T is onto, because if ζ ∈ R (ℓp) and supp ζ = E ∈ R then ζ ∈ ℓp and T (χE ⊗ ζ) = ζ, and then T = 1.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Lemma R (ℓp) is a quotient of ℓ∞
0 (R) ⊗π ℓp for 1 ≤ p ≤ ∞ by means of
a norm-one linear quotient map. Proof. As p = ∞ is done with Ω = N, we assume 1 ≤ p < ∞. The linearization of the map T : ℓ∞
0 (R) ⊗π ℓp → R (ℓp)
given by T (χA ⊗ ξ) = ζA, where ζA
n = ξn if n ∈ A and ζn = 0
i=1 fi∞ ξip,
for v = n
i=1 fi ⊗ gi, hence Tvp ≤ vπ.
T is onto, because if ζ ∈ R (ℓp) and supp ζ = E ∈ R then ζ ∈ ℓp and T (χE ⊗ ζ) = ζ, and then T = 1.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
The properties wN, s(wN), w(sN) and w(wN) ae equivalents. For a ring R there are Baire’s type properties of ℓ∞
0 (R) stronger
than linear-(wN). Problem Characterize those rings R for which ℓ∞
0 (R) is Baire like,
unordered Baire like or Baire. Problem Characterize those rings R for which (N) ⇔ (sN) (Valdivia 2013 for algebras). And the same for (N) ⇔ (wN). Problem To find a theorem for the (N) ⇒ (wN) in J(K).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
The properties wN, s(wN), w(sN) and w(wN) ae equivalents. For a ring R there are Baire’s type properties of ℓ∞
0 (R) stronger
than linear-(wN). Problem Characterize those rings R for which ℓ∞
0 (R) is Baire like,
unordered Baire like or Baire. Problem Characterize those rings R for which (N) ⇔ (sN) (Valdivia 2013 for algebras). And the same for (N) ⇔ (wN). Problem To find a theorem for the (N) ⇒ (wN) in J(K).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
The properties wN, s(wN), w(sN) and w(wN) ae equivalents. For a ring R there are Baire’s type properties of ℓ∞
0 (R) stronger
than linear-(wN). Problem Characterize those rings R for which ℓ∞
0 (R) is Baire like,
unordered Baire like or Baire. Problem Characterize those rings R for which (N) ⇔ (sN) (Valdivia 2013 for algebras). And the same for (N) ⇔ (wN). Problem To find a theorem for the (N) ⇒ (wN) in J(K).
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Number 92 in Graduate Texts in Mathematics. Springer, New York, Berlin, Heidelberg, 1984.
Mathematical Surveys and Monographs. American Mathematical Society, Providence, 1977.
General Theory. Wiley Classics Library, New York, 1988.
.J. Paúl. Barrelled subspaces of spaces with subseries decompositions or Boolean rings of projections, Glasgow Math. J., 36:57-69, 1994.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
.J. Paúl. Some new classes of rings of sets with the Nikodým property. In Functional analysis: proceedings of the first international workshop held at Trier University, Germany, September 26-October 1, 1994 (Editors, S. Dierolf, S. Dineen and P . Doma´ nski), pp. 143–152. Walter de Gruyter & Co., Berlin, New York, 1996.
bounded function space. J. Math. Anal. Appl., 184:437–440, 1994.
41:270–274, 2002.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Metrizable Barrelled Spaces. Number 332 in Pitman Research Notes in Mathematics. Longman, copublished with John Wiley & Sons Inc., New York, 1995.
spaces with strong barrelledness conditions, Proc. R. Ir. Acad., 92A:69–75, 1992.
advances on the Nikodým boundedness theorem and spaces of simple functions, Rocky Mount. J. Math., 34:139–172, 2004.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
¸kol and M. López-Pellicer. On Valdivia strong version
446:1–17, 2017.
in algebras of Jordan measurable sets, RACSAM, Rev. R.
2016.
boundedness for webs in σ-algebras, RACSAM, Rev. R.
2016.
measures, J. Math. Anal. Appl., 210:257–267, 1997.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Inform., 32:125–127, 2005. P . Pérez Carreras and J. Bonet. Barrelled locally convex
North-Holland, Amsterdam, 1987.
Soc., 48:516–541, 1940. S- A. Saxon. Nuclear and product spaces, Baire-like spaces and the strongest locally convex topology, Math. Ann., 197:87–106 (1972), .
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
theorems for non-sigma-complete Boolean algebras. Dissertationes Math. (Rozprawy Mat.), 214:1–33, 1982. G.L. Seever. Measures on F-spaces, Trans. Amer. Math. Soc., 133: 267–280, 1968.
Fourier (Grenoble), 29:39–56, 1979.
107:355–372, 2013.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
The first example of a non-complete closed and non-separable subspace of a separable tvs was given by Lohman and Stiles. Drenowski and Lohman prove that a subspace of finite codimension of a separable lcs is separable and give an example of a separable lcs E containing a non-separable closed ℵ0-codimensional subspace of a separable lcs E. Obviously E is not barrelled.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
d(X) : minimal of cardinals of dense subspaces of X. dt(X) : minimal of cardinal of its subsets A such that A + U = X for every 0-neighborhood U. If dt(X) = ℵ0 then X is called trans-separable. χ(X) : minimal of cardinals of 0-neighborhoods bases of a lcs X. Theorem (Doma´ nski) A tvs X is isomorphic to a subspace of a tvs Z with d(Z) = m iff X ֒ →
i∈I Xi, |I| 2m, of metrizable tvs with d(Xi) m, iff
χ(X) 2m and dt(X) m. Hence a tvs X may be embedded into a separable tvs if and
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
For |A| = n, c0(n) := {f : A → K : |{a ∈ A : |f(a)| > ǫ}| < ℵ0,∀ǫ > 0}. c0(n, m) = c0(n) with the topology of uniform convergence on subsets of cardinality m verifies:
1
χ(c0(2m, m)) = 2m and
2
dt(c0(2m, m)) = m, hence embeds in a lcs of density m.
3
d(c0(2m, m)) = 2m, if B ⊂ c0(2m, m), |B| < 2m, ∃xB ∈ A : f(xB) = 0, ∀f ∈ B. Hence g / ∈ B if g(xB) = 1. The non-separable complete c0(2ℵ0, ℵ0) embeds in a separable lcs.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Definition Let (xn)n be a sequence in a tvs X.
1
(xn)n is semibasic if each xn / ∈ span{xk : k = n, k ∈ N}, or, equivalently, there exists a biortogonal sequence (x∗
n)n to
(xn)n, i.e., x∗
n(xk) = δnk, ∀n, k ∈ N.
2
A semibasic sequence (xn)n is strongly regular if there exists a biortogonal sequence (x∗
n)n to (xn)n such that
lim
n→∞ x∗ n(x) = 0, for ∀x ∈ span{xk : k ∈ N}.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Theorem (Doma´ nski [1]) Let Xt be a tvs with a strongly regular semibasic sequence (xt,n)n. t ∈ I, |I| = 2ℵ0. X := {Xt : t ∈ I} contains a closed nonseparable subspace. A tvs is minimal if it admits no strictly weaker linear Hausdorff topology, and it is q-minimal if all its Hausdorff quotients are
Drewnowski, Doma´ nski got the following characterization: Theorem (Doma´ nski) Let X be a tvs. The f.a.e.: 1) X contains a strongly regular semibasic sequence. 2) X is non-q-minimal. 3)The completion
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Corollary (KLM Proposition 3) If |I| c and each Ei does not have the weak topology, then
P . Doma´
products of topological vector spaces, and q-minimality,
P . Doma´
closed vector subspaces of separable topological vector spaces, Monatsh. Math. 182 (2017) 39-47.
Sets and rings with Nikodým type’s properties
Preliminaries. Properties (N), (wN) and (G) in rings. Applications and open questions. On R-supported functions of ℓ∞ (Ω) and ℓp. Open problems.
Sets and rings with Nikodým type’s properties