On ideal equal convergence

<jats:title>Abstract</jats:title><jats:p>We consider ideal equal convergence of a sequence of functions. This is a generalization of equal convergence introduced by Császár and Laczkovich [Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar., 1975, 10(3–4), 463–472]. Our definition of ideal equal convergence encompasses two different kinds of ideal equal convergence introduced in [Das P., Dutta S., Pal S.K., On and *-equal convergence and an Egoroff-type theorem, Mat. Vesnik, 2014, 66(2), 165–177]_and [Filipów R., Szuca P., Three kinds of convergence and the associated I-Baire classes, J. Math. Anal. Appl., 2012, 391(1), 1–9]. We also solve a few problems posed in the paper by Das, Dutta and Pal.</jats:p>

section_content: Let, ∈ N, and be real-valued functions defined on a set X. We say that the sequence ( ) is equally convergent to if there exists a sequence of positive reals (ε ) → 0 such that for every ∈ X there is N with | ( ) -( )| < ε for every > N. The notion of equal convergence was introduced by Császár and Laczkovich in [1]. It is known that equal convergence is weaker than uniform convergence and stronger than pointwise convergence, i. e. if ( ) is uniformly convergent to then ( ) is equally convergent to ; and if ( ) is equally convergent to then ( ) is pointwise convergent to. 

Let I be an ideal on N (i. e. I is a family of subsets of N closed under taking finite unions and subsets of its elements). We say that a sequence of reals ( ) is I-convergent to ∈ R if { ∈ N : | -| ≥ ε} ∈ I for every ε > 0 (see e. g. [6] ). 

We write ( )

The notion of equal convergence was generalized in [3] and [4] with the aid of ideals on N. However, the authors of both papers generalized it in different ways. Let I be an ideal of subsets of N. 

• In [3] the authors say that ( ) is I-equally convergent to if there exists a sequence of positive reals (ε )

• In [4] the authors say that ( ) is I-equally convergent to if there exists a sequence of positive reals (ε ) → 0 such that { ∈ N : | ( ) -( )| ≥ ε } ∈ I for every ∈ X. 

The only difference between these two definitions of ideal equal convergence is the requirement that the sequence (ε ) is either convergent (in the classical meaning) to zero or it is convergent to zero with respect to the ideal I. It is easy to see that if ( ) is I-equally convergent to in the sense of [4], then it is also I-equally convergent to in a sense of [3]. 

In this paper we introduce one more kind of ideal equal convergence (Section 3) which encompasses ideal equal convergence in senses of [3] and [4]. In Section 3 we examine relationships between our ideal equal convergence and ideal equal convergence introduced in [3, 4]. 

In Section 4 we examine relationships between ideal uniform, ideal equal and ideal pointwise convergence. Among other we show (Example 4. 7) that ideal pointwise convergence does not imply ideal equal convergence (this solves a problem posed in [3] ). In Section 5 we examine relationships between ideal equal convergence and ideal σ -uniform convergence. 

For instance we prove (Corollary 5. 4) that ideal equal convergence implies ideal σ -uniform convergence if and only if the ideal is countably generated (this solves a problem posed in [3] ). In Section 6 we consider convergence of big (in a sense of ideals) subsequences (so-called filter convergence). Among other we prove (Corollary 6. 5) that filter equal convergence is equivalent to filter σ -uniform convergence if and only if the ideal is a P-ideal (this solves a problem posed in [3] ). In Section 7 we examine relationships between ideal convergence and filter convergence. For instance we prove (Corollary 7. 16) that ideal equal convergence does not imply filter equal convergence (this solves one more problem posed in [3] ).
section_name: Introduction

section_name: Preliminaries

section_content: An ideal on N is a family of subsets of N closed under taking finite unions and subsets of its elements. In the sequel we assume that ideals contain all finite sets. The ideal of all finite subsets of N is denoted by Fin. We can define the notion of ideal on any set in a similar way. For an ideal I, we write I * = {A : N \ A ∈ I} and called it the filter dual to I. 

Let G be a family of subsets of N. The smallest ideal containing G is called the ideal generated by G. An ideal is countably generated if it is generated by a countable family G. Note that an ideal I is countably generated if and only if there are A 1 A

2 ∈ I such that for every A ∈ I there is ∈ N with A ⊆ A. 

Let I J be ideals on N. An ideal I is a P(J)-ideal if for any sets

for every ∈ N ([7, Definition 3. 10] where the authors call it AP(I J)). Note that P(Fin)-ideals are also called P-ideals. 

It is easy to see that every ideal I is always P(I)-ideal.
section_name: Ideals

section_content: Let I J be ideals on N. We say that a sequence of reals ( ) is

for every ε > 0 (see e. g. [6] ). We write ( )

where the authors call it I J -convergence). We write ( )

---→ in this case. 

Note that (I * Fin)-convergence is also called I * -convergence (see e. g. [6] ).
section_name: Ideal convergence

section_content: Let I J be ideals on N. The following are equivalent. 

(a) For every sequence of reals ( ), if ( )

(b) I is a P(J)-ideal.
section_name: Theorem 2.1 ([7, Theorems 3.11 and 3.12]).

section_content: Let I J be ideals on N. Let, ∈ N, and be real-valued functions defined on a set X. We say that the sequence ( ) is (I J)-equally convergent to if there exists a sequence of positive reals (ε )

for every ∈ X. We write ( )

---→ in this case. For I = J = Fin we obtain the equal convergence introduced by Császár and Laczkovich [1] and write ( ) e -→ instead of ( )

In [3] and [4] the authors also introduced a kind of ideal equal convergence (see Section 1). It is easy to see that ideal equal convergence from [3] is equivalent to (I I)-equal convergence, and ideal equal convergence from [4] is equivalent to (I Fin)-equal convergence. 

The following fact can be easily checked.
section_name: Ideal equal convergence

section_content: Let ( ) be a sequence of real-valued functions defined on a set X. Let I 0 I 1 J 0 J 1 be ideals on N. 

Let X be a nonempty set. Let I J be ideals on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

---→ then ( )

for ∈ X and ∈ N. Let ( ) = 0 for ∈ X. Then ( )

By the assumption we obtain that ( ) Since we assume that ideals contain all finite subsets of N, we obtain the following corollary.
section_name: Fact 3.1.

section_content: Let I be an ideal on N. Let ( ) be a sequence of real-valued functions defined on a set X. If ( )
section_name: Corollary 3.3.

section_content: Let X be a nonempty set. Let I J be ideals on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

Let ( ) = 0 for every ∈ X. It is easy to see that ( )

Thus by the assumption we have ( )

(b) ⇒ (a). Suppose that I is a P(J)-ideal and ( )

every ∈ X. Since I is a P(J)-ideal, so by Theorem 2. 1 we obtain (ε )

Define a sequence (η ) by

It is easy to see that (η ) J -→ 0 and the set { : | ( ) -( )| ≥ η } ∈ I for every ∈ X. Thus ( )

---→.
section_name: Theorem 3.4.

section_content: Let X be a nonempty set. Let I be an ideal on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

(b) I is a P-ideal. 

Thus I-equal convergence in the sense of [3] is equivalent to I-equal convergence in the sense of [4] if and only if I is a P-ideal.
section_name: Corollary 3.5.

section_content: Let I be an ideal on N. A sequence ( ) ∈N of real-valued functions defined on X is

• I-uniformly convergent to if for every ε > 0 the set { : | ( ) -( )| ≥ ε for some } ∈ I. We write ( )

for short. -→ then ( ) -→. Moreover it is known that these implications do not reverse. Below we examine relationships between ideal equal convergence and ideal uniform and ideal pointwise convergence. For instance, we provide necessary and sufficient conditions for I and J so that (I J)-equal convergence is between I-uniform and I-pointwise convergence (Corollary 4. 6). In [3, Theorem 3. 1] the authors proved that if ( )

Let X be a nonempty set. Let I J be ideals on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

It is easy to see that ( )
section_name: Ideal convergence: uniform, equal and pointwise

section_content: Let X be a nonempty set. Let I be an ideal on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

The following example shows that (I J)-equal convergence does not imply I-uniform convergence in general.
section_name: Corollary 4.2.

section_content: Let I J be ideals on N. Let X be an infinite set. Let ∈ X, ∈ N, be distinct elements of X. Let ( ) be a sequence defined by ( ) = χ { } ( ) for ∈ X. Let ( ) = 0 for all ∈ X. Then it is not difficult to see that ( )

---→ and ¬ ( )
section_name: Example 4.3.

section_content: Let X be a nonempty set. Let I J be ideals on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

---→ then ( )

for ∈ X and ∈ N. Let ( ) = 0 for ∈ X. Then ( )

---→ (see the proof of Theorem 3. 2). By the assumption we obtain that ( )

---→. There exists a sequence ε
section_name: Proposition 4.4.

section_content: Let I J be ideals on N. Let ( ) be a sequence of real-valued functions defined on a set X. 

(a) If ( )

---→ then ( )

----→ then ( )

Corollary 4. 6. 

Let X be a nonempty set. Let I J be ideals on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

---→ and if ( ) Proof. Let ( α ) α<c be an enumeration of all sequences of positive reals (ε ) such that (ε )

Let : X → R be given by ( ) = 0 for every ∈ X. It is easy to see that ( )

Suppose that ( )

---→. Let (ε ) be a sequence of positive reals such that (ε )
section_name: Corollary 4.5.

section_content: Let I be an ideal on N. A sequence ( ) of real-valued functions defined on X is σ -I-uniformly convergent to : X → R if there are sets X ⊆ X, ∈ N, such that X = ∈N X and ( X ) I---→ X for every ∈ N. We write ( )

---→ in this case. For I = Fin we obtain the σ -uniform convergence introduced in [2] and write ( )

It is easy to see that for every ideal I, if ( )

---→ ; and if ( )

In [2] the authors proved that the equal convergence and σ -uniform convergence are the same, i. e. ( ) e -→ iff ( )

we examine relationships between ideal equal convergence and ideal σ -uniform convergence. In [3, Theorem 3. 2] the authors proved that for every ideal I, if ( )

---→. 

Proposition 5. 1. 

Let X be a nonempty set. Let I J be ideals on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

Let ( ) = 0 for every ∈ X. In the proof of Proposition 4. 1 we showed that ( )
section_name: Ideal σ -uniform convergence

section_content: Let X be a nonempty set. Let I be an ideal on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

(b) I is a P-ideal. 

In [3, Theorem 3. 2] the authors proved that if I is a countably generated ideal then ( )

---→ implies ( )

They also stated a problem if the later implication holds for every ideal. Below we answer the problem in the negative (Corollary 5. 4).
section_name: Corollary 5.2.

section_content: Let |X | ≥ c. Let I J be ideals on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

(b) I is countably generated and J ⊆ I.
section_name: Theorem 5.3.

section_content: Let : X → R be given by ( ) = 0 for every ∈ X. It is not difficult to check that ( )

---→. So, by the assumption we have ( )

---→. Hence, there are sets X ⊆ X, ∈ N, such that X = ∈N X and ( X )

-→ X for every ∈ N. For every ∈ N we define

for some ∈ X

It is easy to see that C ∈ I for all ∈ N. We will show that I is generated by the family {C : ∈ N} (hence I is countably generated). Let A ∈ I. Let α < c be such that A = A α. Let ∈ N be such that α ∈ X. Then It is easy to see that X = ∈N X. Indeed, let ∈ X. Then there is ∈ N with A ⊆ C, so ∈ X. Now we will show that ( X )
section_name: Proof. (a) ⇒ (b). Let

section_content: Let |X | ≥ c. Let I be an ideal on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

(b) For every sequence ( ) of real-valued functions defined on a set X, if ( )
section_name: Corollary 5.4.

section_content: Let X be a countable set. Let I J be ideals on N. The following are equivalent.
section_name: Proposition 5.5.

section_content: In connection with Theorem 5. 3 and Proposition 5. 5, the question arises what happens when X is uncountable and ¬CH is assumed.
section_name: Remark 5.6.

section_content: Let I J be ideals on N. A sequence ( ) of real-valued functions defined on X is • I * -uniformly convergent to : X → R if there exists a set F ∈ I * with ( ) ∈F u -→. We write ( )

--→ in this case. 

• I * -pointwise convergent to : X → R if there exists a set F ∈ I * with ( ) ∈F -→. We write ( )

-→ in this case. 

• (I * J)-equally convergent to : X → R if there exists a set F ∈ I * with ( ) ∈F (Fin J)-e ----→. We write ( )

----→ in this case. 

• σ -I * -uniformly convergent to : X → R if there exist X, ∈ N, with X = ∈N X and ( X )

--→ X for every ∈ N. We write ( )

The notions of I * -uniform, I * -pointwise and σ -I * -uniform convergence were introduced in [3]. In [3] the authors introduced also I * -equal convergence, which is equivalent to (I * Fin)-equal convergence in our notation.
section_name: Filter convergence: uniform, equal, σ -uniform and pointwise

section_content: Let X be a nonempty set. Let I J be ideals on N. Then for every sequence ( ) of real-valued functions defined on a set X, if ( )

Proof. Let ( ) Example 6. 2. 

Let X be a nonempty set, I J be ideals on N such that I ⊆ J and I = J. Then there exists a sequence ( ) of real-valued functions defined on a set X such that ( )

----→ and ¬ ( )

Let : X → R be given by ( ) = 0 for every ∈ X. Then ( )

We have (ε )

Fin for every ∈ X. We will show that ¬ ( )

Let G ∈ I * and ε = 1/2. Then the set G ∩ A is infinite, so for every N ∈ N there exists ≥ N, ∈ G ∩ A, such that | ( ) -( )| ≥ ε for every ∈ X. 

In [3, Theorem 3. 3] the authors proved that if I is a P-ideal then for every sequence ( ), ( )

They also posed a problem if one can remove the assumption that I is a P-ideal in their theorem. Below we answer the problem in the negative (Corollary 6. 5).
section_name: Proposition 6.1.

section_content: Let X be a nonempty set. Let I be an ideal on N and J be a P-ideal. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

----→ then ( )

(b) For every sequence ( ) of real-valued functions defined on a set X, if ( )

(c) J ⊆ I.
section_name: Proposition 6.3.

section_content: for ∈ X and ∈ N. Let ( ) = 0 for ∈ X. It is easy to see that ( )

In the first case, from (a) we obtain that ( )

In the second case, from (b) we obtain that ( )

---→. There exist X, ∈ N, with X = ∈N X and ( X )

for every ∈ N. Let ε = 1/2 and ∈ N. Then there exists for every ∈ X. Since J is a P-ideal and J ⊆ I, we have ε J * -→ 0, so there exists G ∈ I * such that (ε ) ∈G → 0. 

Clearly, ( ) ∈F ∩G →, so ( )

for every ≥ max{ N}, ∈ F ∩ G and ∈ X. We obtain ( )

---→, so (b) holds.
section_name: Proof. (a) ⇒ (c) and (b) ⇒ (c). Let

section_content: Let X be a nonempty set. Let I be an ideal on N, J be a P-ideal and J ⊆ I. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

Let : X → R be given by ( ) = 0 for every ∈ X. It is not difficult to check that ( )

---→. So, by the assumption we have ( )
section_name: Theorem 6.4.

section_content: Let X be a nonempty set. Let I be an ideal on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X we have ( )

(b) I is a P-ideal. 

Proposition 6. 6. 

Let X be a nonempty set. Let I be an ideal on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

Proof. (a) ⇒ (b). Let A X, ∈ N, and, ∈ N, be as in the proof of Theorem 6. 4. Then ( )

by the assumption we have ( )

---→. There exist X, ∈ N, with X = ∈N X and F ∈ I * such that ( ) ∈F X u -→ X for every ∈ N. I is a P-ideal, so there exists F ∈ I * such that F \ F ∈ Fin for every ∈ N. Clearly, ( ) ∈F →, so ( )
section_name: Corollary 6.5.

section_content: Let I be an ideal on N and ( ) be a sequence of real-valued functions defined on a set X. It is easy to see that if ( )

--→ then ( )

-→ and if ( )

Let X be a nonempty set. Let I be an ideal on N. The following are equivalent. 

(a) For every sequence ( ) of real-valued functions defined on a set X, if ( )

-→ then ( )

(b) For every sequence ( ) of real-valued functions defined on a set X, if ( )

(c) I is a P-ideal. 

Proof. ---→, resp. ). So, by the assumption ( )

---→, resp. ), hence ( )

-→. By Theorem 2. 1 we obtain that I is a P-ideal. 

(c) ⇒ (a). Let ( )

-→ and ∈ N. There exists A ∈ I such that | ( ) -( )| < 1/( + 1) for every ∈ N \ A and ∈ X. Since I is a P-ideal, there exists A ∈ I such that A \ A ∈ Fin. Let F = N \ A and fix ε > 0. There exists such that 1/( + 1) < ε. We obtain { ∈ F : | ( ) -( )| ≥ ε for every ∈ X } ⊆ A ∩ F ∈ Fin, so ( )

---→. There are sets X ⊆ X, ∈ N, such that X = ∈N X and ( X )

By the implication (c) ⇒ (a), we obtain ( X )

--→ X for every ∈ N, so ( )

It is easy to see that for every sequence ( ) of real-valued functions defined on a set X, if ( )
section_name: Ideal convergence versus filter convergence

section_content: Let X be a countable set. Let I be an ideal on N. The following are equivalent. Proof. (a) ⇒ (b). Let I = {A α : α < c}. Let α ∈ X, α < c, be distinct points of X. We define functions :

Let : X → R be given by ( ) = 0 for every ∈ X. It is easy to see that ( ) I -→. By (a), ( )

Clearly, ( ) ∈F →, so ( )

In [3, Theorem 3. 4] it was proved that for every sequence ( ) of real-valued functions defined on a set X if ( )

---→.
section_name: Proposition 7.2.

section_content: Let I J be ideals on N. Let ( ) be a sequence of real-valued functions defined on a set X. If ( )

Proof. Let ( )

---→.
section_name: Proposition 7.5.

section_content: Let I be an ideal on N and let ( ) be a sequence of real-valued functions defined on a set X. Then ( )

----→ (so ( )

---→ as well). 

In [3, Theorem 3. 7] it was proved that if for every sequence ( ) of real-valued functions defined on a set X if ( )

-----→, then I is a P-ideal.
section_name: Corollary 7.6 ([3, Theorem 3.4]).

section_content: Let X be an infinite set. Let I J be ideals on N. If for every sequence ( ) of real-valued functions defined on a set X, ( )

----→, then I is a P(J)-ideal. 

Proof. Let A ∈ I, ∈ N. We will show that there is A ∈ I such that A \ A ∈ J for every ∈ N. Let ∈ X, ∈ N, be distinct elements of X. We define : X → R by

Let ( ) = 0 for every ∈ X. Then ( )

By the assumption we obtain that ( ) -----→, then I is a P-ideal.
section_name: Proposition 7.7.

section_content: Let X be an infinite set. Let I be an ideal on N. If for every sequence ( ) of real-valued functions defined on a set X, ( )

---→ implies ( )

-----→, then I is a P-ideal. 

In [3, Theorem 3. 6 ] the authors proved that if X is a countable set and I is a P-ideal then ( ) Remark 7. 13. 

In [8] (see also [5] ) the author proved that it is consistent with ZFC that ω 1 < add * (I) = c for analytic P-ideals. Hence we obtain that it is consistent with ZFC that if I is an analytic P-ideal, |X | = ω 1 < c and ( ) is a sequence of real-valued functions defined on a set X such that ( )
section_name: Corollary 7.9 ([3, Theorem 3.7]).

section_name: Acknowledgements

section_content: The authors would like to thank Tomasz Natkaniec and Piotr Szuca for fruitful discussions and careful reading early version of this article. 

The authors are also greatly indebted to the referee for his helpful suggestions concerning presentation of results and especially for drawing our attention to the notion add * (I) (for instance Remark 7. 3 was provided by the referee).
section_name: Acknowledgements

section_name: Then (ε )

section_content: Let : X → R be given by ( ) = 0 for every ∈ X. It is easy to see that ( ) (I J)-e ---→. Indeed, let ε = 1/( + 2).

section_content: J -→ 0 and if = α for every α < c then { : | ( ) -( )| ≥ ε } = ∅ ∈ I; and if = α for some α < c then

By (a), ( )

. Fix A ∈ I. We will show that (F ∩G)∩A is finite. Since (F ∩G)∩A ∈ I, so there is α < c with A α = (F ∩G)∩A. (a) For every sequence ( ) of real-valued functions defined on a set X, if ( )
section_name: Then (ε )

section_content: (b) There exists F ∈ I * such that if A ∈ I then F ∩ A is finite. 

Remark 7. 17. 

An ideal I on N is dense if for every infinite A ⊆ N there is an infinite B ∈ I with B ⊆ A. It is easy to see that if I is a dense ideal then for every F ∈ I * there is an infinite A ∈ I with A ⊆ F. Hence for every dense P-ideal there exists a sequence ( ) of functions defined on a set X with |X | ≥ c such that ( )

-----→. 

We will characterize ideals that satisfy condition (b) in Theorem 7. 14. We say that ideals I and J are isomorphic (I J) if there exists a bijection : N → N such that A ∈ I iff [A] ∈ J for every A ⊆ N. By Fin ⊕ P(N) we mean the ideal on N × {0 1} given by A ∈ Fin ⊕ P(N) iff { ∈ N : ( 0) ∈ A} ∈ Fin.
section_name: -----→ .

section_content: Let I be an ideal on N. The following are equivalent. 

(a) There exists F ∈ I * such that if A ∈ I then F ∩ A is finite. 

(b) I is isomorphic to Fin or Fin ⊕ P(N). 

Proof. (a) ⇒ (b). Let F ∈ I * be such that every set A ⊆ F if A ∈ I then A is finite. If N \ F is finite then clearly I Fin. Suppose that N \ F is infinite. We will show that I is isomorphic to Fin ⊕ P(N). Let : N → F and : N → N \ F be bijections. We define a function :

Now it is easy to see that is an isomorphism of ideals Fin ⊕ P(N) and I.
section_name: Proposition 7.18.