Generating new ideals using weighted density via ...

GENERATING NEW IDEALS USING WEIGHTED DENSITY VIA MODULUS FUNCTIONS KUMARDIPTA BOSE, PRATULANANDA DAS, AND ADAM KWELA Abstract. In this paper we extend the idea of weighted density of [2] by using a modulus function and introduce the idea f -density of weight g of subsets of ω := {0, 1, . . . } (at the same time extending the notion of f -density [1]), which we name dfg where g : ω → [0, ∞) satisfies g(n) → ∞ and n/g(n) 9 0 and f is a modulus function. The aim of this paper is to show that we can get new ideals Zg (f ) consisting of sets A ⊂ ω for which dfg (A) = 0 different from all the previously constructed ideals Zg of [2] and moreover they retain all the nice properties of the ideals Zg .

1. Introduction Let ω := {0, 1, . . . }. For a, b ∈ ω and a ≤ b, let [a, b] denotes the set {a, a + 1, a + 2, ..., b}. By |A| we denote the cardinality of a set A. The lower and the upper natural densities of A ⊂ ω are defined by d(A) = lim inf n→∞

|A ∩ [0, n − 1]| |A ∩ [0, n − 1]| and d(A) = lim sup . n n n→∞

If d(A) = d(A), we say that the natural density of A exists and it is denoted by d(A). Recall some basic definitions connected with ideals on ω (cf. [7], [11]). A family J ⊂ P(ω) is called an ideal on ω whenever • ω∈ / J, • if A, B ∈ J then A ∪ B ∈ J, • A ⊂ B and B ∈ J then A ∈ J. An ideal J is called proper if J is not 2ω or φ, and is called admissible if {n} ∈ J for all n ∈ ω. If J is an ideal on ω, by J ∗ we denote the dual filter, that is, J ∗ := {ω \ A : A ∈ J}. By F in we denote the ideal of all finite subsets of ω. An ideal J on ω is called a P -ideal if for every sequence (An )n∈ω of sets in J there is a set A ∈ J such that An ⊂∗ A for all n ∈ ω (where An ⊂∗ A means that An \ A ∈ F in). Every ideal J on ω can be treated as a subset of the Cantor space 2ω since P (ω) and 2ω can be identified via the characteristic functions. A submeasure on ω is a function ϕ : P(ω) → [0, ∞] such that: 2010 Mathematics Subject Classification. Primary: 03E15, 11B05, 28A05. Key words and phrases. generalized density, weight function, modulus function, P -ideal, density ideal, Borel complexity. 1

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KUMARDIPTA BOSE, PRATULANANDA DAS, AND ADAM KWELA

• ϕ(∅) = 0; • if A ⊂ B then ϕ(A) ≤ ϕ(B), • ϕ(A ∪ B) ≤ ϕ(A) + ϕ(B), • ϕ({n}) < ∞ for all n ∈ ω. A submeasure ϕ is called a lower semicontinuos submeasure (in short, lscsm) if ϕ(A) = limn→∞ ϕ(A ∩ [0, n − 1]) for all A ⊂ ω (this condition is equivalent to the classical lower semicontinuity of the function ϕ : 2ω → [0, ∞]). For any lscsm ϕ, we consider the exhaustive ideal given by Exh(ϕ) = {A ⊂ ω : lim ϕ(A \ [0, n − 1]) = 0}. n→∞

It follows that for every lscsm ϕ on ω, Exh(ϕ) is an Fσδ P -ideal [7, Lemma 1.2.2]. A highly nontrivial theorem of Solecki [12] states that each analytic P -ideal on ω is of the form Exh(ϕ) for some lscsm ϕ on ω. It is well known that the ideal generated by the natural density Z = {A ⊂ ω : d(A) = 0} is an Fσδ P -ideal on ω which is also an example of a so-called Erd˝os-Ulam ideal (for further information, see [7]). In [3] the authors proposed a modified version of density. Namely, for 0 < α ≤ 1 and A ⊂ ω, they put dα (A) = lim sup n→∞

|A ∩ [0, n − 1]| nα

and dα (A) is defined analogously. Very recently in [2] this idea was extended using a weight function g and density of weight g, namely, the weighted density function dg was defined and the corresponding ideal Zg := {A ⊂ ω : dg (A) = 0} was investigated. One of the most important observation of that paper was that uncountably many different analytic P -ideals can be generated in this process. On the other hand in [1] the idea of natural density was extended in another way by using a modulus function f . The modulus functions are defined as functions f : R+ ∪ {0} → R+ ∪ {0} which satisfy the following properties. (i) f (x) = 0 ⇔ x = 0 (ii) f (x + y) ≤ f (x) + f (y) for all x, y ∈ R+ [Triangle inequality] (iii) f is increasing (iv) f is right continuous at 0. Some examples of such modulus functions [1] are given by 1. f (x) = x, x ∈ R+ ∪ {0}. 2. f (x) =

x 1+x , x

∈ R+ ∪ {0}.

3. For any α ∈ (0, 1), f (x) = xα for x ∈ R+ ∪ {0}. 4. f (x) = log(1 + x), x ∈ R+ ∪ {0}.

GENERATING NEW IDEALS USING WEIGHTED DENSITY VIA MODULUS FUNCTIONS

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Specifically the upper f density function was defined in the following way: df (A) = lim sup n→∞

f (|A ∩ [0, n − 1]|) . f (n)

Similarly the lower f density function df is defined. In [1] and very recently in [10] the corresponding notion of convergence, namely f statistical convergence, was studied and its implications were investigated in Banach spaces, however the underlying ideal generated by an f density function, i.e. the ideal {A ⊂ ω : df (A) = 0} was not investigated at all. As a natural consequence, in this paper, we use the modulus function f to obtain the new notion of density function, called f density of weight g, dfg on ω, which naturally extends all the existing notions mentioned above. We primarily investigate the generated ideal Zg (f ) showing that it is an Fσδ P -ideal on ω generated by a lower semicontinuous submeasure on ω. Further we show that it is also a density ideal in the sense of Farah but it is never an Fσ ideal. Finally we present some comparative studies of these newly introduced ideals Zg (f ) and the existing ideals Zg and most importantly an example is given to show that we can indeed get a new ideal different from all Zg ’s. 2. The ideals Zg (f ) and their comparison with ideals of the form Zg In [2] the authors defined a new class of densities using weight functions. Let g : ω → [0, ∞) be a function with lim g (n) = ∞. The upper density of weight g was defined in [2] by the formula n→∞

dg (A) = lim sup n→∞

|A ∩ [0, n − 1]| g (n)

for A ⊂ ω. The lower density of weight g, dg (A) is defined in a similar way. Then the family Zg = {A ⊂ ω : dg (A) = 0} forms an ideal. It has been observed in [2] that ω ∈ Zg iff

n g(n)

→ 0. So we additionally assume that

n/g (n) 9 0 so that ω ∈ / Ig and Ig becomes a proper admissible P -ideal of ω ([2]). In addition here we also assume the weight function g to be such that dg (ω) exists (even though infinitely). Throughout the paper G will stand for the set of all weight functions having all the above mentioned properties. Now we recall a basic result from [2]. Proposition 2.1. [2] Let g1 , g2 ∈ G be such that there exist M > 0 and k ∈ ω such that g1 (n)/g2 (n) ≤ M for all n ≥ k. Then Zg1 ⊂ Zg2 . Consequently, if there exist 0 < m < M and k ∈ ω such that m ≤ g1 (n)/g2 (n) ≤ M for all n ≥ k, then Zg1 = Zg2 . In view of the above result, from now on, we will assume without loss of generality, g to be such that g(n) ≤ n for all n. Moreover from [2] we can see that

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Proposition 2.2. [2] For each function g ∈ G there exists a nondecreasing function g 0 ∈ G such that Zg0 = Zg . Moreover, g 0 (n) ≤ g(n) for all n ∈ ω. Therefore we can in addition assume g to be non-decreasing throughout the paper. Let us denote the set of all such weight functions g by G. We now introduce our main definition. For a modulus function f and g ∈ G we define f (|A ∩ [0, n − 1]|) . n→∞ f (g(n))

dfg (A) = lim

It is easy to verify that all the axiomatic conditions to be a density function, as given in [9], are satisfied by f (n) n→∞ f (g(n))

this new density function dfg except for the fact that dfg (ω) = lim consider only those density functions

dfg

for which

dfg (ω)

may not be equal to 1. We also

> 0 for obvious reasons. One should also note

that the modulus function f applied to get the generalized density function as above must be unbounded. Otherwise let for all x, |f (x)| ≤ M for some M > 0. Then for any A ⊂ ω

|f (A∩[0,n−1])| f (g(n))

≤

M f (g(n))

0, k ∈ ω for which c1 ≤ f1 (x)/f2 (x) ≤ c2 for all x and c3 ≤ f1 (g1 (n))/f2 (g2 (n)) ≤ c4 for all n ≥ k, then Zg1 (f1 ) = Zg2 (f2 ). However, the condition for equality is not necessary, as we can see in the following example. Example 2.1. Let f1 (x) = x1/2 , and f2 (x) = x whereas g1 (n) = g2 (n) = n1/6 . Then for any A ⊂ ω, f1 (|A∩[0,n−1]|) f1 (g1 (n)) |A∩[0,n−1]| → n1/6 −1/2 x0 ≤ c.

=

(|A∩[0,n−1]|)1/2 n1/12

whereas

f2 (|A∩[0,n−1]|) f2 (g2 (n))

=

|A∩[0,n−1]| . n1/6

Observe that

(|A∩[0,n−1]|)1/2 n1/12

→ 0 ⇔

0. So Zg1 (f1 ) = Zg2 (f2 ). But for any c > 0 there is x0 > 0 such that f1 (x0 )/f2 (x0 ) =

Note that, in particular, for two modulus functions f1 , f2 and for g ∈ G, if the function f1 /f2 is bounded then Zg (f1 ) = Zg (f2 ). In particular, taking f1 = f , any modulus function and f2 as the identity function we obtain sufficient conditions under which the ideal Zg (f ) coincides with the ideal Zg . In particular, we have the following concrete observation.


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Proposition 2.4. For any g ∈ G there are uncountably many modulus functions f such that Zg (f ) = Zg . Proof. For α ∈ (0, 1) let fα (x) = xα . Then for any A ⊂ ω fα (|A ∩ [0, n − 1]|) (|A ∩ [0, n − 1]|)α |A ∩ [0, n − 1]| = →0⇔ → 0. α fα (g(n)) (g(n)) g(n) Hence Zg (fα ) = Zg for all α ∈ (0, 1).

Now we show that there is at least one modulus function f and g ∈ G for which Zg (f ) 6= Zg . Example 2.2. Let f (x) = log(1 + x) and g(n) = n. Now let A = {n2 : n ∈ ω}. Then lim sup n→∞

But lim sup n→∞

|A ∩ [0, n − 1]|) n = lim 2 = 0. n→∞ g(n) n

f (|A ∩ [0, n − 1]|) log(1 + n) 1 = lim = . 2 n→∞ log(1 + n ) f (g(n)) 2

So A ∈ Zg but A ∈ / Zg (f ). Moving one step further, we show that for the above modulus function f , the generated ideal Zg (f ) is distinct from uncountably many Zg ’s. Proposition 2.5. There is a modulus function f for which Z(f ) is not equal to Znα for any α ∈ (0, 1). Proof. Let f (x) = log(1 + x). Let us take any α ∈ (0, 1). Then by Archimedean property,there is k ∈ ω such that kα > 1. Let us take A = {nk : n ∈ ω}. Then lim sup n→∞

But

|A ∩ [0, n − 1]|) n = lim kα = 0 α n→∞ n n

log(1 + n) 1 = >0 k n→∞ log(1 + n ) k lim

So A ∈ Znα \ Z(f ).

The following simple result shows that using the modulus function actually decreases the size of the ideal Zg . Proposition 2.6. For any modulus function f and g ∈ G, Zg (f ) ⊂ Zg . Proof. Let A ∈ Zg (f ) and let ε > 0 be given. We take M ∈ ω such that 1/M < ε. Now lim f (|A ∩ n→∞

[0, n − 1]|)/f (g(n)) = 0. So there exists k ∈ ω such that f (|A ∩ [0, n − 1]|) < f (g(n))/M for all n ≥ k and consequently f (M · |A ∩ [0, n − 1]|) ≤ M · f (|A ∩ [0, n − 1]|) < f (g(n)) for all n ≥ k. Now as f is increasing we infer that M · |A ∩ [0, n − 1]| < g(n) for all n ≥ k. Hence |A ∩ [0, n − 1]|/g(n) < 1/M < ε for all n ≥ k which implies that A ∈ Zg .

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Remark 2.7. The above inclusion is strict for at least one modulus function, namely, f (x) = log(1 + x). Let g ∈ G and as before. As n/g(n) 9 0 there are M > 0 and a sequence nk such that nk /g(nk ) > 1/M for all k. We have Zg = Zg/M as |B ∩ [0, n − 1]| |B ∩ [0, n − 1]| = 0 ⇔ lim = 0. n→∞ n→∞ g(n) g(n)/M q for all B ⊂ ω. We construct A ⊂ ω by taking at most g(n) elements from the set {0.1, 2, ..., n − 1} and M q q g(n) g(n) k) exactly [ g(n M ] elements from the set {0, 1, 2, ..., nk − 1}. As M < M < n, it is possible to construct lim

such a set. Now it is easy to show that A ∈ Zg/M \ Zg/M (f ) as p g(n)/M |A ∩ [0, n − 1]| ≤ lim =0 lim sup n→∞ g(n)/M g(n)/M n→∞ and lim sup n→∞

log(1 + |A ∩ [0, n − 1]|) log([g(nk )/M ]) ≥ lim > 1/2. k→∞ log(1 + g(nk )/M ) log(1 + g(n)/M )

Finally we proceed to the last step and present an example of an ideal of the form Zg (f ) which is not of the form Zg . Example 2.3. Let f : R+ ∪ {0} → R+ ∪ {0} be given by f (x) = log(1 + x) for each x ∈ R+ ∪ {0}. Then f is a modulus function. Indeed, it is increasing, continuous and f (x) = 0 only for x = 0. What is more, for all x, y ∈ R+ ∪ {0} we have 1 + x + y ≤ (1 + x)(1 + y). Hence, f (x + y) = log(1 + x + y) ≤ log((1 + x)(1 + y)) = log(1 + x) + log(1 + y) = f (x) + f (y). Moreover, f is unbounded. Consider the ideal Zidω (f ), where idω is the identity mapping on ω. Fix any g ∈ H. We will show that Zidω (f ) is not equal to Zg . There are two possible cases: Case 1.: There are α > 0 and k0 ∈ ω such that 1 + g(n) > nα for each n ≥ k0 . Without loss of generality we may assume that α < 1 and k0 > 1. Let A ⊆ ω be such that |A∩[0, n−1]| = bnα/2 c (note that bnα/2 c ≤ nα/2 ≤ n since α < 1). Then A ∈ Zg as |A ∩ [0, n − 1]| nα/2 1 ≤ lim α = lim α/2 = 0. n→∞ n→∞ n − 1 n→∞ n g(n) lim

On the other hand, A ∈ / Zidω (f ) as log(1 + |A ∩ [0, n − 1]|) log(nα/2 ) α ≥ lim = > 0. n→∞ n→∞ log(1 + n) log(1 + n) 2 lim

Hence, Zidω (f ) 6= Zg in this case.


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Case 2.: For each α > 0 and k ∈ ω one can find n > k with 1 + g(n) ≤ nα . 1/(k+1)

Then we can inductively pick an increasing sequence (nk ) of elements of ω such that 1 + g(nk ) ≤ nk 1/(k+1)

and nk

1/(k+2)

< nk+1

for each k ∈ ω (this is possible as limn→∞ n1/(k+1) = ∞ for each k ∈ ω). Without 1/(k+1)

loss of generality we may assume that g(n0 ) > 1. Let A ⊆ ω be such that |A ∩ [0, nk − 1]| = bnk

− 1c

and |A ∩ [0, n − 1]| ≤ n1/(k+1) − 1 for all nk ≤ n < nk+1 (note that such set exists by our assumptions on the sequence (nk ) and the fact that n1/(k+1) > n1/(k+2) for all n, k ∈ ω). Then A ∈ / Zg as 1/(k+1)

n |A ∩ [0, nk − 1]| ≥ k g(nk )

1/(k+1)

−1−1 n −1 1 1 ≥ k − ≥1− >0 g(nk ) g(nk ) g(n0 ) g(n0 )

for all k ∈ ω. On the other hand, A ∈ Zidω (f ) as log(1 + |A ∩ [0, n − 1]|) log(n1/(k+1) ) 1 ≤ ≤ , log(1 + n) log(1 + n) k+1 where nk ≤ n < nk+1 . Hence, Zidω (f ) 6= Zg also in this case. 3. Set theoretic properties of the ideal Zg (f ) First note that in view of the last example of Section 2, the properties of the ideals of the form Zg (f ) do not necessarily follow from the properties of the ideals of the form Zg already established in [2] as the the latter class is strictly smaller. In this section we will show that with some modifications we can get all the set theoretic properties of Zg for the larger class of ideals Zg (f ). We start with showing that the ideal Zg (f ) is also a Fσδ P -ideal. Proposition 3.1. For a modulus function f and g ∈ G, the ideal Zg (f ) is a P -ideal. In fact Zg (f ) is equal to Exh(ϕ), where ϕ is a lower semicontinuos submeasure on ω given by ϕ(A) = sup n∈ω

f (|A ∩ [0, n − 1]) f (g(n))

for A ⊂ ω. Proof. First we check that ϕ is a submeasure on ω. It is clear that

f (|A∩[0,n−1]|) f (g(n))

≤

f (|B∩[0,n−1]|) f (g(n))

whenever

A ⊂ B, because f is non-decreasing. Now for A, B ∈ ω |A ∩ [0, n − 1]| + |B ∩ [0, n − 1]| ≥ |(A ∪ B) ∩ [0, n − 1]| ⇒ f (|A ∩ [0, n − 1]| + |B ∩ [0, n − 1]|) ≥ f (|(A ∪ B) ∩ [0, n − 1]|) ⇒ f (|A ∩ [0, n − 1]|) + f (|B ∩ [0, n − 1]|) ≥ f (|A ∩ [0, n − 1]| + |B ∩ [0, n − 1]|) ≥ f (|(A ∪ B) ∩ [0, n − 1]|) f (|A ∩ [0, n − 1]|) f (|B ∩ [0, n − 1]|) f (|(A ∪ B) ∩ [0, n − 1]|) ⇒ sup + sup ≥ sup f (g(n)) f (g(n)) f (g(n)) n∈ω n∈ω n∈ω Or ϕ(A) + ϕ(B) ≥ ϕ(A ∪ B) which shows that ϕ is a submeasure.

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To prove that ϕ is lower semicontinuos, let A ⊂ ω. By the monotonicity of ϕ, we infer that ϕ(A ∩ [0, n − 1]) ≤ ϕ(A ∩ [0, n]) ≤ ϕ(A) for all n ∈ ω. Hence lim ϕ(A ∩ [0, n − 1]) exists and is equal to sup ϕ(A ∩ [0, n − 1]) ≤ ϕ(A). On the n→∞

n∈ω

other hand, from the definition of ϕ(A) it follows that, either there exists n0 ∈ ω such that ϕ(A) = f (|A ∩ [0, n0 −1]|)/f (g(n0 )) or we can find a sequence nk → ∞ such that ϕ(A) = lim f (|A∩[0, nk −1]|)/f (g(nk )). k→∞

In the first case, we obtain ϕ(A) = ϕ(A ∩ [0, n0 − 1]) and the remaining part of the proof is obvious. In the second case, for each k ∈ ω we have f (|A ∩ [0, nk − 1]|) f (|A ∩ [0, nk − 1] ∩ [0, n − 1]|) ≤ sup = ϕ(A ∩ [0, nk − 1]) ≤ sup ϕ(A ∩ [0, n − 1]). f (g(nk )) f (g(n)) n∈ω n∈ω Letting k → ∞, we have ϕ(A) ≤ sup ϕ(A∩[0, n−1]) = lim ϕ(A∩[0, n−1]). Hence lim ϕ(A∩[0, n−1]) = n→∞

n∈ω

n→∞

ϕ(A). Now, let us first show that Zg (f ) ⊂ Exh(ϕ). Let A ∈ Zg (f ). For each m ∈ ω, using the increasing property of f we get f (|A ∩ [0, n − 1]|) f (|A ∩ [0, n − 1] \ [0, m − 1])| ≤ sup . f (g(n)) f (g(n)) n≥m n≥m

ϕ(A \ [0, m − 1]) = sup Letting m → ∞, we obtain

lim ϕ(A \ [0, m − 1]) ≤ lim sup

m→∞

m→∞ n≥m

f (|A ∩ [0, n − 1]) = dfg (A) = 0. f (g(n))

Hence lim ϕ(A \ [0, m − 1]) = 0 and consequently, A ∈ Exh(ϕ). m→∞

Finally to show that Exh(ϕ) ⊂ Zg (f ), take A ∈ Exh(ϕ), that is, lim ϕ(A \ [0, m − 1]) = 0. Fix ε > 0 m→∞

and pick m0 ∈ ω such that ϕ(A \ [0, m0 − 1]) < ε/2. Using the assumption lim f (g(n)) = ∞, choose n→∞

n0 ≥ m0 such that f (|A ∩ [0, m0 − 1]) ε < f (g(n)) 2

for all n ≥ n0 .

Then for all n ≥ n0 , using the triangle inequality property of f we get f (|A ∩ [0, n − 1]|) f (|A ∩ [0, m0 − 1]|) f ((|A ∩ [0, n − 1]) \ [0, m0 − 1]|) ε ε ε ≤ + < +ϕ(A\[0, m0 −1]) < + = ε. f (g(n)) f (g(n)) f (g(n)) 2 2 2 f

Hence dg (A) = 0 which implies that A ∈ Zg (f ).

Before we establish some more set theoretic properties of the ideal Zg (f ), we need to do some comparative investigations of the ideal Zg (f ) in line of [2]. Note that all these results extend the corresponding results of [2]. We start with the following observation.


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Proposition 3.2. Let f be an unbounded modulus function as specified. For each function g ∈ G there exists a nondecreasing function g 0 ∈ G such that Zg0 (f ) = Zg (f ). Moreover, f (g 0 (n)) ≤ f (g(n)) for all n ∈ ω. Proof. If g is nondecreasing, we just take g 0 := g. Otherwise, define the function g 0 as follows. Let a1 = min{f (g(n)) : n ∈ ω}, i1 = max{i ∈ ω : f (g(i)) = a1 } and g 0 (i) be such that f (g 0 )(i) = a1 for 0 ≤ i ≤ i1 . Next, let a2 = min{f (g(n)) : n > i1 }, i2 = max{i ∈ ω : f (g(i)) = a2 } and g 0 (i) be such that f (g 0 )(i) = a2 for i1 < i ≤ i2 . The rest of the construction goes inductively. g 0 is well-defined as f is increasing. Clearly, the function g 0 ◦ f is nondecreasing and f (g 0 (n)) → ∞. As f is increasing and unbounded we conclude that g 0 is nondecreasing and g 0 (n) → ∞, for otherwise if g 0 (n) ≤ M < ∞ for all n ∈ ω, then f (g 0 (n)) ≤ f (M ) < ∞ for all n ∈ ω. Now note that f (g 0 (n)) ≤ f (g(n)) for all n ∈ ω and as f is increasing, we must have g 0 (n) ≤ g(n) for all n ∈ ω. Therefore n/g(n) ≤ n/g 0 (n) for all n which implies that n/g 0 (n) 9 0. Hence g 0 ∈ G. Moreover, g 0 ◦ f is non-decreasing implies that g 0 is also non-decreasing. Let A ⊂ ω. Then

f (|A ∩ [0, n − 1]|) f (|A ∩ [0, n − 1]|) ≤ for all n ∈ ω f (g(n)) f (g 0 (n)) which shows that if A ∈ Zg0 (f ) then A ∈ Zg (f ). Now by construction, for each n ∈ ω there exists m ≥ n such that f (g 0 (n)) = f (g 0 (m)) = f (g(m)). Suppose that A ∈ / Zg0 (f ). Then there exist a real α > 0 and an increasing sequence (ni ) of indices such that

f (|A ∩ [0, ni − 1]|) ≥ α for all i ∈ ω. f (g 0 (ni )) For each i ∈ ω we can find mi ≥ ni such that f (g 0 (ni )) = f (g 0 (mi )) = f (g(mi )). As f is increasing, we get f (|A ∩ [0, mi − 1]|) f (|A ∩ [0, ni − 1]|) ≥ for all i ∈ ω f (g(mi )) f (g 0 (ni )) which implies that A ∈ / Zg (f ). This completes the proof of the result.

Lemma 3.3. Let f be an unbounded modulus function and let g ∈ G be such that f (n)/f (g(n)) → ∞. Then there exists a set A ⊂ ω such that the sequence (f (|A ∩ [0, n − 1]|)/f (g(n))) is bounded but not convergent to 0. Proof. As f is increasing, pick the smallest k ∈ ω such that f (g(n)) > 2 for all n ≥ k. We will define a set A ⊂ ω \ [0, k − 1] inductively, deciding whether n ≥ k should belong to A or not. First let n ∈ / A for all n < k. Assume that n ≥ k and that we have defined A ∩ [0, n − 1]. If f (|A ∩ [0, n − 1]|)/f (g(n + 1)) < 1 then let n ∈ A. Otherwise, n ∈ / A. The set A is constructed proceeding in this way and further note that the condition f (n) → ∞ implies that A is infinite.

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We claim that ω \ A is also infinite. Indeed, if it is finite then pick n0 ∈ ω such that n ∈ A for all n ≥ n0 . Again from the fact that f is increasing it follows that f (n − n0 ) f (|A ∩ [0, n − 1]|) ≤ 0, k ∈ ω for which f1 (x)/f2 (x) ≥ c1 for all x and f1 (g1 (n))/f2 (g2 (n)) ≤ c2 for all n ≥ k, then Zg1 (f1 ) ⊂ Zg2 (f2 ). Proof. Observe that for any A ⊂ ω and n ≥ k f2 (|A ∩ [0, n − 1]|) f1 (|A ∩ [0, n − 1]|) f1 (g1 (n)) f2 (|A ∩ [0, n − 1]|) = . . f2 (g2 (n)) f1 (|A ∩ [0, n − 1]|) f1 (g1 (n)) f2 (g2 (n)) ≤ This shows that

f1 (|A∩[0,n−1]|) f1 (g1 (n))

→0⇒

c2 f1 (|A ∩ [0, n − 1]|) · . c1 f1 (g1 (n))

f2 (|A∩[0,n−1]|) f2 (g2 (n))

→ 0 and so Zg1 (f1 ) ⊂ Zg2 (f2 ).

Theorem 3.5. Let f be an unbounded modulus function. If g1 , g2 ∈ G are such that f (n)/f (g2 (n)) → ∞ and f (g2 (n))/f (g1 (n)) → ∞ then Zg1 (f ) Zg (f )

Z(f ).

Zg2 (f ). Further if g ∈ G and f (n)/f (g(n)) → ∞ then


11

Proof. Observe that the inclusion Zg1 (f ) ⊂ Zg2 (f ) follows from Lemma 3.4. Let us choose g3 : ω → [0, ∞] p in such a manner that f (g3 (n)) := f (g1 (n)) · f (g2 (n)) holds for all n ∈ ω. The function g3 so constructed is well-defined as f is increasing. Now (1)

lim

n→∞

f (g3 (n)) f (g2 (n)) = lim = ∞. n→∞ f (g1 (n)) f (g3 (n))

Also we have f (n) f (g2 (n)) f (n) = · → ∞, f (g1 (n)) f (g2 (n)) f (g1 (n))

so

f (n) = f (g3 (n))

s

f (n)2 → ∞. f (g1 (n))f (g2 (n))

Hence f satisfies the assumptions of Lemma 3.3. Take the set A as constructed in Lemma 3.3. Then / Zg1 (f ). Indeed, using (1) we have A ∈ Zg2 (f ) but A ∈ f (|A ∩ [0, n − 1]|) f (|A ∩ [0, n − 1]|) f (g3 (n)) = · →0 f (g2 (n)) f (g3 (n)) f (g2 (n))

since

f (|A ∩ [0, n − 1]|) f (g3 (n))

is bounded. n∈ω

This shows that A ∈ Zg2 (f ). To prove that A ∈ / Zg1 (f ) observe that f (|A ∩ [0, n − 1]|) f (g3 (n)) f (|A ∩ [0, n − 1]|) = · . f (g1 (n)) f (g3 (n)) f (g1 (n)) Therefore A ∈ / Zg1 (f ) since f (|A ∩ [0, n − 1]|)/f (g3 (n)) 9 0 and f (g3 (n))/f (g1 (n)) → ∞ (by (1)). The proof of the second assertion is similar to the above proof. Before establishing more set theoretic properties of the ideal Zg (f ), we present the following interesting result which we will need in our endeavor. Lemma 3.6. Let f be an unbounded modulus function and let g ∈ G be nondecreasing. Further let us define a sequence (nk ) inductively by n0 := 1 and nk+1 := min{n ∈ ω : f (g(n)) ≥ 2f (g(nk ))}. Then f (|A ∩ [nk , nk+1 )|) = 0}. k→∞ f (g(nk ))

Zg (f ) = {A ⊂ ω : lim

Proof. Let A ∈ Zg (f ) and let ε > 0 be given. Choose N0 ∈ ω such that

f (|A∩[0,n−1]|) f (g(n))

N0 .

Then for all k ∈ ω such that nk > N0 we have f (|A ∩ [nk , nk+1 )|) f (|A ∩ [0, nk+1 )|) f (|A ∩ [0, nk+1 )|) ε ≤ ≤ ≤ 2. = ε. f (g(nk )) f (g(nk )) f (g(nk+1 − 1))/2 2 f (|A∩[nk ,nk+1 )|) f (g(nk )) k→∞

So lim

= 0. f (|A∩[nk ,nk+1 )|) = 0. f (g(nk )) k→∞ f (|A∩[nk ,nk+1 )|) < 4ε for all f (g(nk ))

Conversely, let A ⊂ ω be such that lim any ε > 0 and take k0 ∈ ω such that

We will show that A ∈ Zg (f ). Fix k ≥ k0 . Now choose k1 > k0 so that

12


f (|A∩[0,nk0 )|) f (g(nk1 ))

< 2ε . Fix n > nk1 and pick a unique l ∈ ω such that n ∈ [nl , nl+1 ). Then observe that

f (|A ∩ [0, n − 1]|) f (|A ∩ [0, n − 1]|) ≤ f (g(n)) f (g(nl ))

= ≤ ≤

Consequently lim

n→∞

f (|A∩[0,n−1]|) f (g(n))

f (|A ∩ [0, nk0 − 1]|) f (|A ∩ [nk0 , nk0 +1 )|) + f (g(nl )) f (g(nl )) f (|A ∩ [0, nk0 − 1]|) f (|A ∩ [nk0 , nk0 +1 )|) + f (g(nk1 )) 2l−k0 f (g(nk0 )) 1 ε ε 1 + ( l−k0 + l−k0 −1 + ... + 1/2 + 1) · < 2 4 2 2

f (|A ∩ [nl , nl+1 )|) f (g(nl )) f (|A ∩ [nl , nl+1 )|) + ... + f (g(nl )) ε ε + 2 · = ε. 2 4 + ... +

= 0, i.e.,A ∈ Zg (f ).

Let us recall the following definition (cf. [7], Section 1.3). For a measure µ defined on subsets of ω, the support of µ is the set {n ∈ ω : µ({n}) > 0}. We say that an ideal I on ω is a density ideal if I = Exh(ϕ) where ϕ := supi∈ω µi and µi are measures with pairwise disjoint supports being finite subsets of ω. We then say that I is a density ideal generated by the sequence (µi )i∈ω . Solecki [12] had observed that every density ideal is an Fσδ P -ideal. We are now in a position to show that Zg (f ) is a density ideal. Theorem 3.7. Let f be an unbounded modulus function. Then Zg (f ) is a density ideal for every g ∈ G. Moreover, it is generated by the sequence of measures (µk ) given by µk (A) =

f (|A ∩ [nk , nk+1 )|) f (g1 (nk ))

where g1 ∈ G is a nondecreasing function such that Zg (f ) = Zg1 (f ) as in Proposition 3.2 and the sequence (nk ) is defined as in Lemma 3.6. Proof. We take the nondecreasing function g1 ∈ G such that Zg (f ) = Zg1 (f ) as obtained in Proposition 3.2. Let ϕ := sup µk . In order to prove the result we have to show that for any B ⊂ ω k∈ω

f (|B ∩ [nk , nk+1 )|) = 0. k→∞ f (g1 (nk ))

B ∈ Exh(ϕ) if and only if lim

First let B ∈ Exh(ϕ). Then lim ϕ(B \ [0, n − 1]) = 0. Subsequently n→∞

lim sup

n→∞ k

f (|(B \ [0, n − 1]) ∩ [nk , nk+1 )|) = 0. f (g1 (nk ))

So for any ε > 0 we can choose n0 ∈ ω such that for all k ∈ ω and n ≥ n0 f (|B ∩ [n, ∞) ∩ [nk , nk+1 )|) < ε. f (g1 (nk ))


13

Now f (|B ∩ [nk , nk+1 )|) ≤ f (|B ∩ [n0 , ∞) ∩ [nk , nk+1 )| + |[0, n0 − 1] ∩ [nk , nk+1 )|) ≤ f (|B ∩ [n0 , ∞) ∩ [nk , nk+1 )|) + f ([0, n0 − 1] ∩ |[nk , nk+1 )|) ≤ f (|B ∩ [n0 , ∞) ∩ [nk , nk+1 )|) + f (n0 ). So f (|B ∩ [nk , nk+1 )|) f (n0 ) < + ε. f (g1 (nk )) f (g1 (nk )) We know that f (g1 (nk )) → 0 when k → ∞. Hence the right hand side tends to ε as k → ∞ and eventually we obtain lim

k→∞ f (|B∩[nk ,nk+1 )|) f (g1 (nk )) k→∞

Conversely, if lim

f (|B ∩ [nk , nk+1 )|) = 0. f (g1 (nk ))

= 0, then for any ε > 0 we can find k0 ∈ ω such that for k ≥ k0 f (|B ∩ [nk , nk+1 )|) 0. k→∞

we have Φ × Fin ≤RB I. Fix Zg (f ). Now from Proposition 3.2 it follows that there is nondecreasing h ∈ G such that Zg (f ) = Zh (f ). Since Theorem 3.7 says that Zh (f ) is a density ideal so the sequence (µk ) in the canonical representation of Zh (f ) satisfies the conditions (D2) and (D3). Indeed µk ({i}) = sup µk ({i}) = i∈ω

f (1) f (h(nk ))

So

and hence lim sup µk ({i}) = 0 as lim f (h(nk )) = ∞, and k→∞ i∈ω

lim sup µk (ω) = lim sup k→∞

f (|{i}∩[nk ,nk+1 )|) . f (h(nk ))

k→∞

= lim sup k→∞

≥ lim sup k→∞

f (|ω ∩ [nk , nk+1 )|) f (h(nk )) f (nk+1 − nk ) f (h(nk )) f (nk+1 ) − f (nk ) f (h(nk ))

≥ lim sup[2 k→∞

= lim sup k→∞

k→∞

f (nk+1 ) f (nk ) − ] [from the construction of (nk ) in Lemma3.6] f (h(nk+1 )) f (h(nk ))

f (nk ) > 0 [ or else dfh (ω) = 0]. f (h(nk ))

So we have Φ × Fin ≤RB Zh (f ) = Zg (f ). Now it is known [12] that an analytic P -ideal I is not Fσ if and only if Φ × Fin ≤RB I. This proves that Zg (f ) is not Fσ .

Acknowledgement: The authors are thankful to the Learned Referee for valuable comments and suggestions which improved the presentation of the paper considerably.

References [1] A. Aizpuru, M.C. Listan-Garcia, F. Rambla-Barreno, Density by moduli and statistical convergence. Quaestiones Math., (2014), 525-530. [2] M. Balcerzak, P. Das, M. Filipczak, J. Swaczyna, Generalized kinds of density and the associated ideals, Acta Math. Hungar., 147 (1) (2015), 97 - 115. [3] S. Bhunia, P. Das, S.K. Pal, Restricting statistical convergence, Acta Math. Hungar. 134 (2012), 153 - 161. [4] L. Bukovsk´ y, The structure of the real line, Monografie Matematyczne, vol. 71 (new series), Springer Basel AG, 2011.


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[5] H. Cartan, Th´eorie des filtres, C. R. Acad. Sci. Paris, 205 (1937), 595 - 598. [6] H. Cartan, Filtres et ultrafiltres, C. R. Acad. Sci. Paris, 205 (1937), 777 - 779. [7] I. Farah, Analytic quotients: Theory of lifting for quotients over analytic ideals on integers, Mem. Amer. Math. Soc. 148 (2000). [8] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241 - 244. [9] A.R. Freedman and J.J. Sember, Densities and summability, Pacific J. Math., 95 (1981), 293 - 305. [10] M.C. List´ an-Garc´ıa, f -statistical convergence, completeness and f -cluster points, Bull. Belg. Math. Soc. Simon Stevin, 23 (2016), 235 - 245. [11] D. Meza Alc´ antara, Ideals and filters on countable sets, PhD thesis, Univ. Nacional Auton´ oma de M´exico, June 2009. [12] S. Solecki, Analytic ideals, Bull. Symbolic Logic, 2 (1996), 339 - 348. [13] S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic, 99 (1999), 51 - 71. Department of Mathematics, Jadavpur University, Kolkata-700032, India E-mail address: [email protected] Department of Mathematics, Jadavpur University, Kolkata-700032, India E-mail address: [email protected] ´ sk, ul. Institute of Mathematics, Faculty of Mathematics, Physics and Informatics, University of Gdan ´ sk, Poland Wita Stwosza 57, 80 - 308 Gdan E-mail address: [email protected]