Dominated Convergence and Egorov Theorems for Filter Convergen
We study the filters, such that for convergence with respect to this filters the Lebesgue dominated convergence theorem and the Egorov theorem on almost uniform convergence are valid (the Lebesgue filters and the Egorov filters, respectively). Some characterizations of the Egorov and the Lebesgue fi...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
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irk-123456789-1064452016-09-29T03:02:16Z Dominated Convergence and Egorov Theorems for Filter Convergen Kadets, V. Leonov, A. We study the filters, such that for convergence with respect to this filters the Lebesgue dominated convergence theorem and the Egorov theorem on almost uniform convergence are valid (the Lebesgue filters and the Egorov filters, respectively). Some characterizations of the Egorov and the Lebesgue filters are given. It is shown that the class of Egorov filters is a proper subset of the class of Lebesgue filters, in particular, statistical convergence filter is the Lebesgue but not the Egorov filter. It is also shown that there are no free Lebesgue ultrafilters. Significant attention is paid to the filters generated by a matrix summability method. 2007 Article Dominated Convergence and Egorov Theorems for Filter Convergen / V. Kadets, A. Leonov // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 2. — С. 196-212. — Бібліогр.: 11 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106445 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We study the filters, such that for convergence with respect to this filters the Lebesgue dominated convergence theorem and the Egorov theorem on almost uniform convergence are valid (the Lebesgue filters and the Egorov filters, respectively). Some characterizations of the Egorov and the Lebesgue filters are given. It is shown that the class of Egorov filters is a proper subset of the class of Lebesgue filters, in particular, statistical convergence filter is the Lebesgue but not the Egorov filter. It is also shown that there are no free Lebesgue ultrafilters. Significant attention is paid to the filters generated by a matrix summability method. |
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Article |
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Kadets, V. Leonov, A. |
| spellingShingle |
Kadets, V. Leonov, A. Dominated Convergence and Egorov Theorems for Filter Convergen Журнал математической физики, анализа, геометрии |
| author_facet |
Kadets, V. Leonov, A. |
| author_sort |
Kadets, V. |
| title |
Dominated Convergence and Egorov Theorems for Filter Convergen |
| title_short |
Dominated Convergence and Egorov Theorems for Filter Convergen |
| title_full |
Dominated Convergence and Egorov Theorems for Filter Convergen |
| title_fullStr |
Dominated Convergence and Egorov Theorems for Filter Convergen |
| title_full_unstemmed |
Dominated Convergence and Egorov Theorems for Filter Convergen |
| title_sort |
dominated convergence and egorov theorems for filter convergen |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/106445 |
| citation_txt |
Dominated Convergence and Egorov Theorems for Filter Convergen / V. Kadets, A. Leonov // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 2. — С. 196-212. — Бібліогр.: 11 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT kadetsv dominatedconvergenceandegorovtheoremsforfilterconvergen AT leonova dominatedconvergenceandegorovtheoremsforfilterconvergen |
| first_indexed |
2025-07-07T18:30:07Z |
| last_indexed |
2025-07-07T18:30:07Z |
| _version_ |
1837013936066527232 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2007, vol. 3, No. 2, pp. 196�212
Dominated Convergence and Egorov Theorems
for Filter Convergence
V. Kadets and A. Leonov
Department of Mechanics and Mathematics, V.N. Karazin Kharkiv National University
4 Svobody Sq., Kharkiv, 61077, Ukraine
E-mail:vova1kadets@yahoo.com
leonov_ family@mail.ru
Received May 18, 2006
We study the �lters, such that for convergence with respect to this �lters
the Lebesgue dominated convergence theorem and the Egorov theorem on
almost uniform convergence are valid (the Lebesgue �lters and the Egorov
�lters, respectively). Some characterizations of the Egorov and the Lebesgue
�lters are given. It is shown that the class of Egorov �lters is a proper subset
of the class of Lebesgue �lters, in particular, statistical convergence �lter is
the Lebesgue but not the Egorov �lter. It is also shown that there are no free
Lebesgue ultra�lters. Signi�cant attention is paid to the �lters generated by
a matrix summability method.
Key words: measure theory, dominated convergence theorem, Egorov
theorem, �lter convergence, statistical convergence, matrix summability.
Mathematics Subject Classi�cation 2000: 28A20, 54A20, 40C05.
1. Introduction
The aim of the paper is to study the classical Lebesgue integration theory
results � the Lebesgue dominated convergence theorem and the Egorov theorem
in a general setting when the ordinary convergence of sequences is replaced by
a �lter convergence. We show that for some �lters this theorems are valid and for
some are not, and moreover that the set of �lters on N for which the dominated
The �rst Author would like to express his gratitude to the Department of Mathematics,
University of Murcia (Spain) and especially to Prof. Bernardo Cascales for hospitality and
fruitful discussions as well as to the Seneca Foundation, Murcia whose Grant No. 02122/IV2/05
has made possible this research visit.
The second Author expresses his gratitude to Prof. Je� Connor who generously provided
the Author with his works on statistical convergence.
c
V. Kadets and A. Leonov, 2007
Dominated Convergence and Egorov Theorems for Filters Convergence
convergence theorem takes place is strictly wider than the corresponding set of
�lters for the Egorov theorem.
Recall that a �lter F on N is a nonempty collection of subsets of N satisfying
the following axioms: ; =2 F ; if A;B 2 F then A \ B 2 F ; and for every A 2 F
if B � A then B 2 F .
A sequence an 2 R is said to be F-convergent to a (and we write a = F�liman
or an !F a) if for every " > 0 the set fn 2 N : jan � aj < "g belongs to F .
In particular if one takes as F the �lter of those sets whose complement is
�nite (the Fr�echet �lter) then F-convergence coincides with the ordinary one.
The natural ordering on the set of �lters on N is de�ned as follows: F1 � F2 if
F1 � F2. If G is a centered collection of subsets (i.e., all �nite intersections of the
elements of G are nonempty), then there is a �lter containing all the elements of
G. The smallest �lter, containing all the elements of G is called the �lter generated
by G.
Let F be a �lter. A collection of subsets G � F is called the base of F if for
every A 2 F there is a B 2 G such that B � A.
A �lter F on N is said to be free if it dominates the Fr�echet �lter. Below when
we say ��lter" we mean a free �lter on N. In particular every ordinary convergent
sequence will be automatically F-convergent.
A maximal in the natural ordering �lter is called an ultra�lter. The Zorn
lemma implies that every �lter is dominated by an ultra�lter. A �lter F on N is
an ultra�lter if and only if for every A � N either A or N nA belongs to F . More
about �lters, ultra�lters and their applications one can �nd in every advanced
General Topology textbook, for example in [11].
All over the paper (
;�; �) stands for a �nite measure space, and when
we say �a function on
" we mean a real-valued and measurable function.
For a � �
denote �� the collection of intersections of � with elements of �.
Below we several times make use of the following simple remark:
Theorem 1.1. The outer measure �� generated by � is countably additive on
�� even in the case of � 62 �.
De�nition 1.2. A �lter F on N is said to be a Egorov �lter (or is said to
have the Egorov property) if for every measure space (
;�; �), for every point-
wise F-convergent to 0 sequence of functions fn on
and for every " > 0 there
is a subset B 2 � with �(B) < " such that supt2
nB jfn(t)j !F 0.
De�nition 1.3. A �lter F on N is said to be a Lebesgue �lter (or is said to have
the Lebesgue property) if the following statement takes place: for every measure
space (
;�; �), for every point-wise F-convergent to 0 sequence of functions fn
on
if jfnj are dominated by a �xed integrable function g 2 L1(
;�; �) thenR
fnd�!F 0.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 197
V. Kadets and A. Leonov
Below we give some characterizations of Egorov and Lebesgue �lters; we show
that the Egorov property implies the Lebesgue one and that there are no ultra-
�lters with the Lebesgue property. Then we pass to a wide class of the Lebesgue
�lters � to �lters generated by a summability method. Under a natural restric-
tion on the summability method these �lters are the examples of �lters without
the Egorov property.
2. Characterizations of Egorov and Lebesgue Filters
Let us start with some observations to simplify the conditions of Egorov and
Lebesgue theorems. Namely it is su�cient to consider fn in Defs.1.2 and 1.3 being
the functions that take only values 0 and 1, or in other words, it is enough to
consider fn of the form fn = �An
, An 2 �. More precisely:
Theorem 2.1. F is a Egorov �lter if and only if for every measure space
(
;�; �), for every sequence An 2 � such that fn = �An
point-wise F-converge
to 0 and for every " > 0 there is a subset B 2 � with �(B) < " and there is
an I 2 F such that An � B for all n 2 I.
P r o o f. The �only if" part of the theorem is just a particular case of Def. 1.2
when all the fn take only values 0 and 1. So let us prove the �if" part.
Step 1. The functions fn in Def. 1.2 can be taken positive (the corresponding
statements for fn and for jfnj are equivalent) and moreover taking values from
[0; 1], since we may replace fn by minffn; 1g.
Step 2. Each of fn can be uniformly approximated by a simple function
gn :
! [0; 1] taking only irrational values; jfn � gnj < 1=n. If the statement
of Def. 1.2 holds true for these gn, then it holds true for fn as well. So we may
suppose that fn :
! [0; 1] are simple functions taking only irrational values.
Step 3. Now for every irrational x 2 [0; 1] let us write down its binary expan-
sion x =
P1
m=1 2
�mam(x); am(x) 2 f0; 1g. The functions am(x) are continuous
on the set of irrationals, so their compositions with fn are measurable functions
and we have expansions
fn(t) =
1X
m=1
2�mam(fn(t)):
Denote Am;n = supp(am Æ fn), then am Æ fn = �Am;n
. According to our
assumption fn(t)!F 0 for all t 2
, hence for a �xed m 2 N am(fn(t))!F 0 as
n!1. Applying the conditions of the theorem for "=2m, we get for every m 2 N
a subset Bm 2 � with �(Bm) < "=2m and an Im 2 F such that Am;n � Bm for
198 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Dominated Convergence and Egorov Theorems for Filters Convergence
all n 2 Im. Put B =
S
m
Bm. Then �(B) < ". Moreover, for all n 2
T
m
k=1 Ik
sup
t2
nB
fn(t) = sup
t2
nB
1X
k=1
2�kak(fn(t)) � sup
t2
n(
S
m
k=1
Bk)
1X
k=1
2�kak(fn(t))
= sup
t2
n(
S
m
k=1
Bk)
1X
k=m+1
2�kak(fn(t)) �
1X
k=m+1
2�k = 2�m:
This means that supt2
nB fn(t)!F 0, which completes the proof.
Theorem 2.2. F is a Lebesgue �lter if and only if for every measure space
(
;�; �), for every sequence An 2 � if fn = �An
point-wise F-converge to 0, then
�(An)!F 0.
P r o o f. Like the previous theorem the �only if" part is evident. The proof
of the �if" part needs the same three steps as above.
Step 1. The functions fn in the Def. 1.3 can be taken positive (it is su�cient
to prove the statement for positive functions f+n = maxffn; 0g and f
�
n = f+n � fn
and to use the formula fn = f+n � f�n ). Moreover, passing from the measure d�
and functions fn to (g+1)d� and fn=(g+1) respectively, we can reduce the task
to the case of functions taking values from [0; 1].
The Steps 2 and 3 can be taken almost word-to-word from the proof of
Th. 2.2 with the only di�erence that in the Step 3 we do not need Bn and B;
Im 2 F must be selected in such a way that �(Am;n) < " for n 2 Im and in the
�nal part we estimate
R
fnd� for n 2
T
m
k=1 Ik:Z
fnd� =
1X
k=1
2�k�(Ak;n) �
mX
k=1
2�k"+
1X
k=m+1
2�k�(
) � "+ 2�m�(
):
Corollary 2.3. The Egorov property of a �lter implies the Lebesgue property.
A set of naturals is called stationary with respect to a �lter F (or just
F-stationary) if it has nonempty intersection with each member of the �lter.
Denote the collection of all F-stationary sets by F�. For a J 2 F� we call the
collection of sets fI \ J : I 2 Fg the trace of F on J (which is evidently a
�lter on J), and by F(J) we denote the �lter on N generated by the trace of F
on J . Clearly F(J) dominates F . Any subset of naturals is either a member of
F or the complement of a member of F or the set and its complement are both
F-stationary sets. F� is precisely the union of all ultra�lters dominating F . F� is
a �lter base if and only if it is equal to F and F is an ultra�lter.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 199
V. Kadets and A. Leonov
Proposition 2.4. Let N = J0 t J1 be any disjoint partition of naturals into
F-stationary sets. Then F does not have the Egorov property if and only if either
F(J0) or F(J1) is not a Egorov �lter.
P r o o f. First, we show that if F(J0) and F(J1) are both Egorov �lters,
then F is a Egorov �lter. Notice that �An
point-wise F-converges if and only if
it simultaneously converges point-wise in F(J0) and F(J1) senses. For a given
measure � and " > 0 let us choose B0 and B1 with the measures smaller than "=2
and I0, I1 from F(J0) and F(J1), respectively, as in Th. 2.1. Then B = B0 tB1,
I = I0 t I1 2 F are �t to satisfy the conditions of Th. 2.1.
Let now, for instance, F(J0) be not a Egorov �lter. Then for some measure
� and measurable An with F(J0)� lim�An
= 0 there is " > 0 such that for each
I \ J0 2 F(J0), I 2 F , we have �(
S
n2I\J0
An) � ". Now to get an example of
Ân with �
Ân
point-wise F -converging to 0 and with �(
S
n2I Ân) � " we can take
Ân = An when n 2 J0 and to set the rest of Ân to be empty.
De�nition 2.5. We say that a �lter F is nowhere Egorov (nowhere Lebesgue)
if and only if its trace on each F-stationary set generates a non-Egorov (non-
Lebesgue) �lter.
In the sense of this de�nition the preceding proposition says that each Egorov
�lter must be an everywhere Egorov �lter. But it is easy to see that a �lter
without the Egorov property is not necessarily a nowhere Egorov �lter.
Denote by eN the set of all free ultra�lters U on N, equipped with the topology
de�ned by means of its base f ~A : A � Ng, where ~A = fU 2 eN : A 2 Ug. Remark,
that in this topology the basic open sets ~A are at the same time closed. eN can be
identi�ed with �NnN where �N denotes the Stone��Cech compacti�cation of N.
By the de�nition, the support set of a �lter F is the set KF =
T
f ~A : A 2 Fg.
In other words, KF is the set of all ultra�lters dominating F . More about the
support sets in connection with di�erent types of convergence see in [1, 4, 10].
Proposition 2.6. If F is a nowhere Egorov �lter, then KF is nowhere dense
in eN .
P r o o f. Observe that KF is closed in eN . So it is nowhere dense in eN if and
only if for any in�nite A � N there is a U 2 ~A such that U 62 KF . That means
that for each in�nite A � N there is such a U containing A that there is I 2 U
(can be reckon as subset of A) which does not belong to any ultra�lter from KF .
Or in terms of F-stationary sets: each stationary set has an in�nite nonstationary
subset. In other words, we have that KF is nowhere dense in eN provided the trace
of F on any D 2 F� is not the Fr�echet �lter on D (or as we further say there
is no Fr�echet stationary set with respect to F). Since the Fr�echet �lter has the
Egorov property the claim is proved.
200 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Dominated Convergence and Egorov Theorems for Filters Convergence
In the next section we show that for the �lter generated by a summability
matrix the inverse implication is true as well.
The rest of this section is devoted to one more reformulation of the Egorov
property (Th. 2.8) and the Lebesgue property (Th. 2.14) in a way that reduces
the number of parameters in the de�nitions to minimum.
Lemma 2.7. F does not have the Egorov property if and only if there exists
a measure space (
;�; �) such that:
(1)
� F ;
(2) An = f! 2
: n 62 !g 2 �;
(3) � 6� 0 and for every I 2 F the set
S
n2I An has full measure.
P r o o f. As follows from Th. 2.1, F is not a Egorov �lter if and only if there
are (
1;�1; �), Cn 2 �1 and " > 0 such that for each t 2
1 fn : t 62 Cng 2 F ,
and for all B with �(B) < " and for every I 2 F there is n 2 I such that Cn does
not lie in B. In other words, �(
S
n2I Cn) > " for every I 2 F .
Let I 2 F , consider BI =
S
n2I Cn. The family fBIgI2F has the following
property:
BI1
\BI2
� BI1\I2 : (2.1)
Denoting
� = inf
I2F
�(BI) > 0;
we choose In 2 F such that �(BIn) < �+1=n. Without loss of generality, we can
assume that I1 � I2 � I3 � : : : .
Introducing the notation b
=
T1
n=1BIn , observe that �(BIn) ! �(b
) when
n!1, thus �(b
) = �. Due to (2:1) we have that �(b
\BI) = � for every I 2 F .
From now on we deal with b
instead of
1, bAn = Cn \ b
instead of Cn, andbBI = BI \ b
=
S
n2I
bAn instead of BI . Note that the condition on Cn still holds
for bAn: fn : t 62 bAng 2 F : And for any I 2 F , as we have already mentioned,
�( bBI) = �: (2.2)
Consider the natural map G : b
! F , G(t) = fn : t 62 bAng. De�ne the
measure space (
;�; �) we need as the image of (b
;�1; �) under G, i.e., put
= G(b
), let � be the collection of those D �
that G�1(D) 2 �1 and put
�(D) = �(G�1(D)). De�ne An = G( bAn). Observing that t 2 bAn if and only if
n 62 G(t), we obtain that G�1(An) = bAn, which means that An 2 �. To complete
the proof remark that �(
) = � and for every I 2 F
�
[
n2I
An
!
= �
G�1
[
n2I
An
!!
= �( bBI) = �:
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 201
V. Kadets and A. Leonov
Let us equip 2N (the collection of all subsets of N) with the standard product
topology and denote by B the Borel �-algebra on 2N .
Theorem 2.8. F is not a Egorov �lter if and only if there exists a Borel
measure � on 2N such that:
(1) ��(F) > 0; and
(2) for every I 2 F CI = f! � N : ! � Ig is a �-null set.
P r o o f. First we establish that if such a measure exists, then F is not
a Egorov �lter. Consider
= F , � = BF , and equip BF with the measure
� = �� (Th. 1.1). Now we are in the conditions of the preceding criterion: the
third condition follows from the observation that CI = f! 2 F : ! � Ig is the
complement to
S
n2I An from Lem. 2.7 and the second one from observation that
An are closed in F .
Now suppose that F possesses the Egorov property and so there is a measure
� satisfying conditions (1), (2), (3) of Lem. 2.7. Note that in induced topology
on
the natural neighborhoods base of J 2
is formed by
Un(J) =
�
! 2
: ! \ f1; 2; : : : ; ng = J \ f1; 2; : : : ; ng
=
0@ \
i2f1;2;::: ;ng\J
nAi
1A n
0@ [
i2f1;2;::: ;ngnJ
nAi
1A :
Thus Un(J) 2 � and hence � contains the �-algebra of Borel sets on
. To com-
plete the proof put �(A) = �(A \
) for all A 2 B.
Corollary 2.9. Every �lter F with a countable base possesses the Egorov
property.
P r o o f. Let G = fIng
1
n=1 be a base of F . Suppose that condition (2) of the
theorem holds. Then
0 = �(
1[
n=1
CIn) = �(F);
and hence condition (1) does not hold. This establishes the claim.
Now we can apply Th. 2.8 to show that all ultra�lters do not have the Egorov
property (remind that we consider only free �lters and ultra�lters).
Corollary 2.10. Free ultra�lters do not have the Egorov property.
P r o o f. Consider the standard product measure � on 2N . If U is an ultra-
�lter, then 2N = U t fNnu : u 2 Ug. But, as u 7! Nnu is a preserving measure
bijection of 2N , we have that ��(U) = ��(fNnu : u 2 Ug) and both must be at
least 1/2 (to be precise ��(U) = 1, see [6, Lem. 464Ca]). Since u 2 U is in�nite
all the Cu = f! � N : ! � ug are �-null sets.
202 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Dominated Convergence and Egorov Theorems for Filters Convergence
As we see, a wide class of �lters that have no countable base are not the
Egorov �lters, among them, as we show in the next section, there is the �lter of
statistical convergence. Before proceeding further it is useful to have an example
of the Egorov �lter that has no countable base.
Example 2.11. A �lter with the Egorov property that has no countable base.
Let fni;jg
1
i;j=1 be an enumeration of N. For each sequence of naturals fkjg
de�ne an element of the �lter base G in the following way: J(kj) = fni;j : i �
kj ; j = 1; 2; : : : g. It is easy to see that the �lter � generated by G has no
countable base. A real-valued sequence fxng �-converge to 0 if and only if for
each j 2 N xni;j ! 0 as i!1.
Let ffng be a point-wise �-convergent to 0 sequence of functions on
. For any
" > 0 put "k = "=2k, k 2 N. Applying the ordinary Egorov theorem for fni;k
for each k, we get Bk with �(Bk) < "k such that the sequence fni;k uniformly
converges to 0 on
nBk as i ! 1. Put B :=
S1
k=1Bk. Then �(B) < " and fn
uniformly �-converge to 0 on
nB.
Let us proceed now with the Lebesgue property. Recall some facts.
Fact 1. Any �-almost everywhere (a.e.) convergent sequence of functions on
is convergent in measure �.
Fact 2. If for a given sequence of measurable functions ffng there are positive
scalars an, "n such that limn an = 0,
P1
n=1 "n < 1 and �(ft : jfnj > ang) < "n
for all n 2 N then fn converge a.e. to 0.
Theorem 2.12. For a �xed �lter F on N the following three properties of
a sequence fn on (
;�; �) are equivalent:
(1) fn is F-convergent to 0 in measure;
(2) every J 2 F� contains an in�nite subset M such that fn converge a.e. to 0
along M ;
(3) for every J 2 F� there is an in�nite subset M � J such that fn converge in
measure to 0 along M .
P r o o f. (2.12)) (2.12). Let J 2 F� and let an,"n be as in Fact 2. From
F-convergence in measure follows that for every n 2 N there is In 2 F such that
for all i 2 In �(ft : jfij > ang < "ng. Let us select an increasing sequence mn
such that mn 2 In\J . Then gn := fmn
satis�es the conditions of Fact 2 and thus
fi converge a.e. to 0 along M := fmng.
The implication (2.12)) (2.12) evidently follows from Fact 1, so let us prove
that (2.12)) (2.12). Suppose fn do not F-converge in measure. Then there are
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 203
V. Kadets and A. Leonov
positive scalars a," such that in each I 2 F there is an j such that �(ft : jfjj >
a) > ". Consequently J = fj 2 N : �(ft : jfjj > a) > "g is stationary and
contains no subset along which fn converge to 0 in measure.
And once more the �Lebesgue" analogues for the properties of the Egorov
(non-Egorov) �lters.
Lemma 2.13. F does not have the Lebesgue property if and only if there exists
a measure space (
;�; �) such that:
(1)
� F ;
(2) An = f! 2
: n 62 !g 2 �;
(3) there is a J 2 F� such that
inf
n2J
�(An) > 0;
or alternatively, for every in�nite subset M � J
�(f! 2
: jMn!j =1g) > 0:
P r o o f. Theorem 2.2 says that F does not have the Lebesgue property
when there are (
1;�1; �), Cn 2 �1 such that for each t 2
1 fn : t 62 Cng 2 F
and the sequence �(Cn) does not F-converge to 0, or in other words, �Cn
do not
F-converge to 0 in measure. Due to the item (2.12) of the previous theorem, there
is a J 2 F� such that the sequence f�(Cn)gn2J does not have converging to 0
subsequences, or in other words infn2J �(Cn) > 0. Alternatively, due to the item
(2.12) of the same theorem, there is a J 2 F� (in fact J can be left the same)
such that �Cn
do not converge a.e. along any subsequence of J . This means that
�(
1\
n=1
[
m>n;m2M
Cm) > 0 (2.3)
for every in�nite subset M � J .
Now, in the same way as we did in Lemma 2.7 we apply map G :
1 ! F ,
G(t) = fn : t 62 Cng to the original measure space in order to get the measure
space (
;�; �) we need. Then An = f! 2
: n 62 !g equals G(Cn) and
f! 2
: jMn!j = 1g =
T1
n=1
S
m>n; m2M Am = G
�T1
n=1
S
m>n; m2M Cm
�
which owing to (2.3) completes the proof.
204 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Dominated Convergence and Egorov Theorems for Filters Convergence
Theorem 2.14. F is not a Lebesgue �lter if and only if there exists a Borel
measure � on 2N such that there is a J 2 F� such that
inf
n2J
��(fI 2 F : n 62 Ig) > 0;
or alternatively, for each in�nite subset M � J
��(fI 2 F : jMnIj =1g) > 0:
P r o o f. The argumentation is the same as in Th. 2.8. Let such a measure
exists. Considering
= F , � = BF and equipping BF with the measure � = ��
we �nd ourselves in the conditions of preceding criterion.
The converse results from Lem. 2.13 by putting �(A) = �(A\
) for all Borel
subsets of 2N .
Corollary 2.15. Let � be the usual product measure on 2N. If F has the Le-
besgue property then ��(F) = 0.
P r o o f. Suppose the contrary, let ��(F) = a > 0. Let us show that, for
instance, an alternative condition of the non-Lebesgue property is valid with N
as a stationary set from this condition (which leads to a contradiction). Denoting
Am = fI 2 F : m 62 Ig, for any in�niteM � N, we have F = (
S
m2M Am)t(fI 2
F : M � Ig). Since M is in�nite ��(fI 2 F : M � Ig) � �(f! � N : M �
!g) = 0. Thus for any in�nite M ��(
S
m2M Am)) � ��(F) = a and hence for
Mn =M \ fn; n+ 1; n+ 2; : : : g as well. Thus applying Th. 1.1, we obtain:
��(fI 2 F : jMnIj =1g) = ��(
1\
n=1
[
m2Mn
Am) � a > 0:
Corollary 2.16. An ultra�lter does not have the Lebesgue property.
For a given non-Lebesgue �lter F Th. 2.14 suggests to consider �lter F(J),
where J is the stationary set from the criterion. It is evident that any �lter
dominating F(J) satis�es the condition in its turn. Consequently we have the
following result.
Corollary 2.17. If F0 is not a Lebesgue �lter, then there is a J 2 F� such
that each F � F0(J) does not have the Lebesgue property.
Now we are going to consider the �lters generated by summability matrices.
We show that all of them possess the Lebesgue property, characterize the nowhere
Egorov ones and give a su�cient condition for a �lter to be non-Egorov.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 205
V. Kadets and A. Leonov
3. Filters Generated by Summability Matrices
In this section we study the matrix generalization of statistical convergence
with respect to the Egorov and the Lebesgue properties. Though matrix
summability methods and statistical convergence were introduced separately and,
until recently, followed independent lines of development, they are closely related.
The de�nition of statistical convergence was introduced by H. Fast [5] with the
natural density of a set in N being used. A real valued sequence xk is statis-
tically convergent to x if for every " > 0 the set fk : jxk � xj > "g has na-
tural density 0 where the natural density of a subset A � N is de�ned to be
Æ(A) := limn n
�1jfk � n : k 2 Agj. Statistical convergence is a generalization of
the usual notion of convergence, and in its turn has been extended in a variety of
ways. A number of authors replaced the natural density with the one generated
by a matrix summability method ([4�9]), or more generally considered statistical
convergence determined by a �nitely additive set function satisfying some ele-
mentary properties ([2]). An overview of the theory of statistical convergence the
reader can �nd in one of the most recent papers [3]. In this section we use an ex-
tension of Fast's de�nition of statistical convergence where the natural density is
replaced by a matrix generated as presented in [4].
An N � N matrix ' = ('i;j) is said to be a summability matrix if:
(1) 'i;j � 0 for all i and j;
(2)
P1
j=1 'n;j � 1 for every n 2 N;
(3) lim supn!1
P1
j=1 'n;j > 0;
(4) limn!1 'n;j = 0 for every n 2 N.
Usually in literature the following regularity condition is also demanded from
a summability matrix: limn!1
P1
j=1 'n;j = 1, but for our purposes it is more
convenient to consider nonregular matrices as well.
For a summability matrix ' and I � N let
d'(I) = lim
i!1
1X
j=1
'i;j�I(j);
when this limit exists. Because d'(I) does not exist for some subsets of N, it is
sometimes convenient to use the upper density d'(I) := lim sup
i!1
P1
j=1 'i;j�I(j).
We say that a set I � N is '-null if d'(I) = 0, and '-nonthin if d'(I) > 0. Having
introduced matrix generated density d', a sequence xk is said to be '-statistically
convergent to x provided for every " > 0, d'(fk : jxk � xj > "g) = 0.
206 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Dominated Convergence and Egorov Theorems for Filters Convergence
For a summability matrix ' denote F' = fI � N : d'(N n I) = 0g and
note that F' is a �lter. As it is easy to see, F'-convergence and '-statistical
convergence coincide, and a set J is F'-stationary when it is '-nonthin.
If ' = C is a C�esaro matrix (i.e., 'i;j = 1=i for j � i and 'i;j = 0 otherwise),
then FC concurs with the usual �lter of statistical convergence and dC is the
usual natural density function. Note that the �lter � from Ex. 2.11 can be also
generated by a summability matrix. To de�ne such a summability matrix put
'i;nl;k = 2�k, where fnl;kg are from the de�nition of �. De�ne all the rest of 'i;j
as zeros.
Each summability matrix ' is equivalent to a summability triangular matrix
T , i.e., for any I � N d'(I) = dT (I). To establish this we note that since seriesP1
j=1 'i;j converge for each i, then there are Ni such that
P1
j=Ni
'i;j � 2�i.
Thus for any I � N
d'(I) = lim
i!1
NiX
j=1
'i;j�I(j):
Consequently, if we erase all the elements 'i;j with j > Ni (writing zeros instead
of them) we pass to an equivalent matrix. Adding to this new matrix �rst N1� 1
zero rows and for every i repeating the i-th row of f'i;jg Ni+1 � Ni � 1 times,
we reduce our matrix to an equivalent triangular matrix. So for the remainder of
the note all the summability matrices are triangular.
Recall that for a summability matrix ', a scalar valued sequence xk is said to
be strongly '-summable if there is a scalar x such that limi
P
j
'i;j jx � xij = 0.
It is known that a bounded sequence is '-statistically convergent if and only if it
is strongly '-summable [2, Th. 8]. Let us apply this fact.
Proposition 3.1. If F' is a �lter generated by a summability matrix ', then
F' is a Lebesgue �lter.
P r o o f. To establish this let us use the very �rst reformulation of the
Lebesgue property (Th. 2.2). Let a measure space (
;�; �) and An 2 � such
that �An
point-wise F'-converge to 0 be given. In terms of strong '-summability
this means that Si =
P
j
'i;j�Aj
point-wise converge to 0. Note that Si � 1 for
all i and are integrable. We can apply classical dominated convergence theorem
to get
0 = lim
i
Z
Si d� = lim
i
iX
j=1
'i;j�(Aj);
so, once more using the connection of strong '-summability with '-statistical
convergence, �(An)!F' 0.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 207
V. Kadets and A. Leonov
Corollary 3.2. ��(F') = 0, where � is the usual product measure on 2N and
' is an arbitrary summability matrix.
Thus for a �lter generated by a summability matrix it is the Egorov property to
be studied. Before we proceed, let us introduce some more notations: a lacunary
sequence is an increasing sequence of integers fnig such that n0 = 0 and ni �
ni�1 !1 as i!1. We set (ni�1; ni] = fn : ni�1 < n � nig. For two sequences
fn1
i
g and fn2
i
g we write fn1
i
g � fn2
i
g if every n1
i
� n2
i
.
From now on � is the Lebesgue measure on [0,1] and F is a �lter on N.
De�nition 3.3. A lacunary sequence fnig is called F-special if Nnfaig 2 F
for every faig � N such that ai 2 (ni�1; ni] for all i.
Lemma 3.4. If there is an F-special lacunary sequence fnig and there are
�n � 0; � > 0 such that: X
k2(ni�1;ni]
�k � 1 (3.1)
and for every I 2 F
sup
i2N
X
k2(ni�1;ni]
�k�I(k) � �; (3.2)
then F is not a Egorov �lter.
P r o o f. Using the condition (3.1) one can easily construct such a sequence
Ak of the Lebesgue measurable subsets of [0,1] that
(1) Aj for j 2 (ni�1; ni] are disjoint;
(2) �(Aj) = �j and
F
j2(ni�1;ni]
Aj � [0; 1].
The condition () guaranties that for each t 2 [0; 1] there is no more than one ai 2
(ni�1; ni] such that t 2 Aai
. Because of this the de�nition of F-special lacunary
sequence ensures that �Ak
point-wise F-converge to 0. As it was observed in the
proof of Lem. 2.7, Th. 2.1 guarantees that the �lter does not have the Egorov
property when there is such an " > 0 that �(
S
n2I An) > " for every I 2 F .
The obvious inequality
�(
[
n2I
An) � sup
i2N
X
k2(ni�1;ni]
�(Ak)�I(k) = sup
i2N
X
k2(ni�1;ni]
�k�I(k) � �
establishes the claim.
208 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Dominated Convergence and Egorov Theorems for Filters Convergence
Note that for a summability matrix ' Def. 3.3 of F -special sequence fnig can
be rewritten as follows: for any faig � N with ai 2 (ni�1; ni], i = 1; 2; : : :
lim
i!1
iX
j=1
'i;j�fakg(j) = 0: (3.3)
Lemma 3.5. For a summability matrix ' and the corresponding �lter F = F'
the following assertions are equivalent:
(1) There is an F-special lacunary sequence fnig.
(2) The matrix ' satis�es
lim
i!1
max
j2N
'i;j = 0: (3.4)
Moreover, under the second condition fnig can be selected in such a way that every
lacunary sequence fmig � fnig is also F-special.
P r o o f. (1) ) (2). If fnig is F -special but (2) is not true, then there is
a " > 0 and there is an increasing sequence of naturals ik with the corresponding
bk � ik, such that 'ik;bk � ". For every k let m(k) be the index for which
bk 2 (nm(k)�1; nm(k)]. Since each column of ' tends to zero, only �nite number
of m(k) for di�erent k can coincide. So passing if necessary to a subsequence of
indices k we may assume that m(1) < m(2) < : : : . Now selecting an arbitrary
sequence ai 2 (ni�1; ni] of naturals in such a way that am(k) = bk, we get
ikX
j=1
'ik;j�fakg(j) � 'ik;bk � ":
This contradicts the property (3.3) of F-special sequence.
(2)) (1). If (2) holds, then there are nk such that for all i � nk�1 maxj2N 'i;j
< (k)�2. Now for any fakg � fnkg and any i 2 (nk�1; nk]
iX
j=1
'i;j�fakg(j) =
nkX
j=1
'i;j�fakg(j) =
X
am: am�nk
'i;am � kmax
j2N
'i;j < k�1;
and thus converge to 0 when i ! 1. So condition (3.3) holds and lemma is
proved.
Theorem 3.6. Under the condition (3.4) F = F' is not a Egorov �lter. More-
over, for every sequence fmig there are an F-special lacunary sequence fnig �
fmig and corresponding �k and � such that the conditions of Lem. 3.4 are ful�lled.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 209
V. Kadets and A. Leonov
P r o o f. Fix a sequence fmig. Due to Lem. 3.5 there is an F-special
lacunary sequence fn0
i
g � fmig, such that all lacunary sequences fnig � fn0
i
g
are F-special.
By the de�nition of summability matrix there are fimg and � > 0 such that
for all m
imX
k=1
'im;k � 3�:
Since the columns of ' converge to 0 we can choose a lacunary sequence fnig �
fn0
i
g such that fnig
1
i=1 � fimg, and for all i 2 NX
k2(ni�1;ni]
'ni;k > 2�:
For every k 2 (ni�1; ni] put �k = 'ni;k. Let us prove that this sequence of �k
satis�es the conditions of Lem. 3.4 for fnig.
Let I 2 F'. Then NnI is a '-null set and so for i su�ciently largeP
k2(ni�1;ni]
'ni;k�NnI (k) < �, and thus for such iX
k2(ni�1;ni]
�k�I(k) =
X
k2(ni�1;ni]
'ni;k�I(k) > �
and we are in the conditions of Lem. 3.4.
Theorem 3.7. Let F' be a �lter generated by a summability matrix '. The fol-
lowing assertions are equivalent:
(1) F' is nowhere Egorov;
(2) KF' is nowhere dense in eN ;
(3) there is no Fr�echet stationary set with respect to F';
(4) the condition (3.4) holds true.
P r o o f. (1) ) (2). As we have already shown (Prop. 2.6) it is true for
an arbitrary nowhere Egorov �lter.
(2)) (3). Once again see the proof of Prop. 2.6.
(3) ) (4). This result for regular summability matrices is shown in [10].
For the general case it is true as well. Namely, if (4) does not hold, then there is
an " > 0 and there are increasing sequences of naturals fimg and fjmg such that
'im;jm � " for all m. Under these conditions J = fjmg is an F'-stationary set,
such that the trace of F' on J coincides with the Fr�echet �lter on J .
(4) ) (1). Remark that for a J 2 F�' the �lter F'(J) is generated by the
following summability matrix : i;j = 'i;j�J(j). If the condition (3.4) holds,
210 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
Dominated Convergence and Egorov Theorems for Filters Convergence
then limi!1maxj2N i;j = limi!1maxj2J 'i;j = 0. So Theorem 3.6 ensures us
that F'(J) does not have the Egorov property, and hence F' is nowhere Egorov.
It is easy to show that there are non-Egorov �lters that are not nowhere Egorov
ones. As a consequence we get an example of the Egorov �lter dominating a non-
Egorov �lter. Moreover:
Example 3.8. There are some �lters F4 � F3 � F2 � F1, such that F1, F3
are the Egorov �lters and F2, F4 are not.
For F1 we take a Fr�echet �lter on N. Then let N = fnig t fmig be a disjoint
partition of naturals. For F2 we take the �lter whose restriction on fnig is the
image bF of the statistical convergence �lter under map i! ni and the restriction
of F2 on fmig is F(fmig) � the Fr�echet �lter on fmig, i.e., ! 2 F2 if and only if
! \ fnig 2 bF and ! \ fmig 2 F(fmig). F2 dominates F1, but F2 does not have
the Egorov property because the trace of F2 on fnig is a non-Egorov �lter bF .
Then take F(fmig) for the base of F3. Then F3 is a Egorov �lter that dominates
F2. And �nally, for F4 one can take, say, an ultra�lter dominating F3.
The next theorem shows that for the �lters generated by summability matrices
there is a connection between domination and the Egorov property.
Theorem 3.9. Let '1 and '2 be summability matrices, F'2 � F'1 such that
limi!1maxj '
1
i;j
= 0. Then any �lter F such that F'2 � F � F'1 is not a Egorov
�lter.
P r o o f. Since F'2 � F'1 it follows that KF
'2
� KF
'1
. By Theorem 3.7
KF
'1
and hence KF
'2
are nowhere dense in eN , thus limi!1maxj2N '
2
i;j
= 0
too. Applying Lemma 3.5 (the �moreover" part), we can �nd a lacunary sequence
fmig such that every lacunary sequence fnig � fmig is at the same time F'1
and F'2 -special. By Theorem 3.6 there are fnig � fmig and corresponding �k
and � such that the conditions of Lem. 3.4 are ful�lled for the �lter F'2 . Now
the inequality F � F'1 ensures that the F'1-special sequence fnig is at the same
time F-special; and due to the inequality F'2 � F , the conditions of Lem. 3.4
are ful�lled for the �lter F and for the sequences ni; �k and �.
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