aa r X i v : . [ m a t h . GN ] J u l A GENERALIZATION OF THE DENSITY ZERO IDEAL
SUMIT SOM
Abstract.
Let F = ( F n ) be a sequence of nonempty finite subsets of ω suchthat lim n | F n | = ∞ and define the ideal I ( F ) := { A ⊆ ω : | A ∩ F n | / | F n | → n → ∞} . The case F n = { , . . . , n } corresponds to the classical case of density zero ideal.We show that I ( F ) is an analytic P-ideal but not F σ . As a consequence, weshow that the set of real bounded sequences which are I ( F )-convergent to 0 isnot complemented in ℓ ∞ . Introduction
Let I be an ideal on the nonnegative integers ω , that is, a collection of subsetsof ω closed under subsets and finite unions. It is also assume, unless otherwisestated, that I is proper (i.e., ω / ∈ I ) and admissible (i.e., I contains that idealFin of finite sets). I is said to be a P-ideal if it is σ -directed modulo finitesets. Moreover, I is said to be a density ideal if there exists a sequence ( µ n )of finitely additive measures P ( ω ) → R supported on disjoint finite sets suchthat I = { A ⊆ ω : lim n µ n ( A ) = 0 } , cf. [3]. Lastly, we endow P ( ω ) with theCantor-space-topology, hence we may speak about analytic ideals, F σ -ideals, etc.At this point, let F = ( F n ) be a sequence of nonempty finite subsets of ω suchthat lim n | F n | = ∞ and define the ideal I ( F ) := { A ⊆ ω : | A ∩ F n | / | F n | → n → ∞} . (1.1)This extends the classical density zero ideal Z , which corresponds to the sequence( F n ) defined by F n = { , . . . , n } for all n ∈ ω . Similar ideals were considered inthe literature, see e.g. [4, 5].It is easy to see that the function d ⋆ F : P ( ω ) → R : A lim sup n →∞ | A ∩ F n | / | F n | . is a monotone subadditive function, cf. also [7, Example 4] and the notion ofabstract upper density given in [2]. It is not difficult to show that there exists asequence F such that I ( F ) = Z : let F n := [ n ! , n !+ n ] ∩ ω for all n and A := S n F n .Then A ∈ Z \ I ( F ). Our main result follows.2. Main Results
Theorem 2.1. I ( F ) is a density ideal. Mathematics Subject Classification.
Key words and phrases.
Ideal convergence, density ideal.
Proof.
It follows by (1.1) that the ideal I ( F ) corresponds to { A ⊆ ω : lim n →∞ µ n ( A ) = 0 } , where, for each n ∈ ω , µ n : P ( ω ) → R is the finitely additive probability measuredefined by ∀ A ⊆ ω, µ n ( A ) = | A ∩ F n | / | F n | . This concludes the proof. (cid:3)
It is worth noticing that every density ideal is an analytic P-ideal, cf. [3]. Itis known that every density ideal is also meager. Hence Theorem 2.1 implies,thanks to [6, Corollary 1.3], the following consequence:
Corollary 2.2.
The set of bounded real sequences which are I ( F ) -convergent to is not complemented in ℓ ∞ .Remark . By a classical result of Solecki, an (not necessarily proper or admis-sible) ideal I is an analytic P-ideal if and only if I = Exh( ϕ ) := { A ⊆ ω : lim n →∞ ϕ ( A \ [0 , n ]) = 0 } , for some lower semicontinuous submeasure ϕ : P ( ω ) → [0 , ∞ ] (that is, ϕ ismonotone, subadditive, ϕ ( ∅ ) = 0, and ϕ ( A ) = lim n ϕ ( A ∩ [0 , n ]) for all A ⊆ ω ),cf. [3]. Accordingly, it is not difficult to see that, in our case, I ( F ) = Exh( ϕ ),where ϕ is the lower semicontinuous submeasure defined by ∀ A ⊆ ω, ϕ ( A ) := sup n ∈ ω µ n ( A ) . (2.1)The proof is straightforward and left to the reader.We conclude with another property of all ideals I ( F ). Theorem 2.4. I ( F ) is not an F σ -ideal.Proof. Let ϕ be the lower semicontinuous submeasure defined in (2.1). By Re-mark 2.3, we have that I ( F ) = Exh( ϕ ), hence I ( F ) = (cid:26) A ⊆ ω : lim n →∞ sup k ∈ ω µ k ( A \ [0 , n ]) = 0 (cid:27) = { A ⊆ ω : k A k ϕ = 0 } , where k A k ϕ := lim sup n →∞ µ n ( A ). At this point, define recursively, for each n ∈ ω , the following sets: P n := (cid:16)S k ∈ ω h max F k − | F k | n , max F k − | F k | n +1 i ∩ ω (cid:17) \ P n − . where by convention P − := ∅ . Lastly, set G := P ∪ ( ω \ S n ∈ ω P n ) and G n := P n for all nonzero n ∈ ω .By construction we have that { G n : n ∈ ω } is a partition of ω such that: k G n k ϕ > n and lim n k S k>n G k k ϕ = 0.The existence of such partition implies, thanks to [1, Theorem 2.5], that I ( F )is not an F σ -ideal. (cid:3) Acknowledgments.
The author is greateful to Paolo Leonetti (Bocconi Uni-versity, Italy) for useful discussions.
GENERALIZATION OF THE DENSITY ZERO IDEAL 3
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Sumit Som, Research Associate, Department of Mathematics, National Insti-tute of Technology Durgapur, India.
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