Volume 02 Issue 12-2022
219
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
02
I
SSUE
12
Pages:
219-226
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
METADATA
IF
–
7.356
A
BSTRACT
In this paper, we consider the problem of classifying maximal approximatively finite pairs of discrete
measurable partitions of type II Lebesgue space and construct a complete invariant for such pairs of
partitions.
K
EYWORDS
Lebesgue space, discrete measurable partitions, a one-to-one and mutually measurable mapping.
I
NTRODUCTION
A partition of a Lebesgue space is called discrete
if it is a trajectory partition of some finite or
countable group of automorphisms. By an
automorphism we mean a one-to-one and
mutually measurable mapping of a Lebesgue
space onto itself, taking a zero-measure set to a
zero-measure set. The Lebesgue space in this note
is a space with σ
-finite Lebesgue measure that
does not contain points of positive measure. We
will consider pairs of discrete measurable
Journal
Website:
http://sciencebring.co
m/index.php/ijasr
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Research Article
ON THE CLASSIFICATION OF PAIRS OF DISCRETE
MEASURABLE PARTITIONS OF TYPE II
Submission Date:
December 19, 2022,
Accepted Date:
December 24, 2022,
Published Date:
December 29, 2022
Crossref doi:
https://doi.org/10.37547/ijasr-02-12-31
A.N. Fozilov
Ferghana Polytechnic Institute, Uzbekistan
N. Makhmudova
Ferghana Polytechnic Institute, Uzbekistan
Volume 02 Issue 12-2022
220
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
02
I
SSUE
12
Pages:
219-226
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
METADATA
IF
–
7.356
partitions
1
2
,
,
X
of the space
(
,
X
F, m )
satisfying the condition
1
2
=
. Denote by
1
2
a partition defined as follows: two points
,
x y
belong to the same element
1
2
if and only
if there exists a finite sequence of points
1
2
, ,
,...,
,
n
x x x
x y
for which every two neighboring
points belong to the same element
1
or
2
. Let
,
A
F
0
mA
A
pair
of
partitions
1
2
,
,
A
A A
is called a part of a pair
1
2
,
,
X
if the following conditions are met:
1)
Every element
1
and every element
2
has
a non-empty intersection with
A.
2)
1
2
1
2
(
)
(
) (
)
A
A
A
=
A pair of discrete measurable partitions
1
2
,
,
X
will be called an extension of the pair
1
1
1
2
1
,
,
X
if the pair
1
1
1
2
1
,
,
X
is isomorphic to
some part of the pair
1
2
,
,
X
. A pair of discrete
measurable partitions is said to be maximal if it
has no extensions that are not isomorphic with it.
Partition type
1
2
2
/
is called the type of the
pair
1
2
,
,
X
. A pair
1
2
,
,
X
is called ergodic
if the partitions
1
2
2
/
are ergodic.
The concept of a maximal pair of discrete
measurable partitions was defined in [1]. It was
also shown there that the problem of describing
pairs of discrete measurable partitions is largely
reduced to describing maximal pairs. In [2] , [3-
19] maximal pairs of type II
1
are described
. The
purpose of this paper is to generalize the results
of [4] to the nonergodic case. If
ζ
is some partition
of the space F
(
,
X
, m ) ,
then by [
ζ
] we denote the
group of all automorphisms that leave the
partition
ζ
fixed. If
ζ
is a discrete partition of type
II, then by
m
we denote the invariant measure
with respect to [
ζ
].Let be an ergodic pair of
discrete type
1
2
,
,
X
II measurable partitions
of the space
(
,
X
F , m )
. If A and B are maximal
one-layer sets with respect to
1
( or
2
), then it is
clear that
1
2
1
2
( )
( )
m
A
m
B
=
,
Denote by
λ
1
1
2
m
−
measure of the maximum
one-layer set with respect to
1
, and through
λ
2
1
2
m
−
measure of the maximum one-layer set
with respect to
2
.
Obviously, a pair of
1
2
,
,
X
type II has type II
∞
if and only if
λ
2
=∞ .
For an ergodic pair of discrete measurable
partitions
1
2
,
,
X
of type II
∞
, we denote
=
=
1
1
,
,
1
,
0
2
1
The following theorem was proved in [21-47].
Theorem. Let
1
2
,
,
X
,
1.
1
1
1
2
,
,
X
be ergodic
maximal approximatively finite pairs of discrete
measurable partitions of type II
∞
of spaces
(
,
X
F
, m )
,
( X
1
, F
1
, m
1
)
respectively. Pairs of partitions
Volume 02 Issue 12-2022
221
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
02
I
SSUE
12
Pages:
219-226
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
METADATA
IF
–
7.356
1
2
,
,
X
and
1.
1
1
1
2
,
,
X
are isomorphic if and
only if
1
1
1
2
1
2
,
,
=
.[48-54].
Let
1
2
,
,
X
a pair of discrete measurable
partitions of type II
∞
of the space
(
,
X
F , m )
. For
a discrete partition
ζ
, we denote
)
(
the
measurable shell
ζ
, and by
С
(
х
)
the partition
element
)
(
2
1
containing
x
.
Let
0
:
)
(
),
(
)
0
(
,
2
1
2
1
=
=
x
C
x
C
X
x
E
,
1
:
)
(
),
(
)
1
(
,
2
1
2
1
=
=
x
C
x
C
X
x
E
.
It is obvious that
)
0
(
,
2
1
E
,
)
1
(
,
2
1
E
are measurable, Denote
=
=
0
,
1
0
,
0
)
(
mM
mM
M
))
(
),
(
(
)
1
(
,
)
0
(
,
,
2
1
2
1
2
1
E
E
=
.
The following theorem holds:
Theorem. Let
1
2
,
,
X
both
1.
1
1
1
2
,
,
X
maximal approximatively finite pairs of discrete measurable
partitions of type II
∞
spaces
(
,
X
F , m )
,
( X
1
, F
1
, m
1
)
and quotient spaces
)
(
/
2
1
X
,
)
(
/
1
1
1
2
1
X
be continuous Lebesgue spaces
.
Pairs of partitions
1
2
,
,
X
and
1.
1
1
1
2
,
,
X
are isomorphic if and only if
1
2
1
1
2
1
,
,
=
.
R
EFERENCES
1.
Винокуров, В. Г., & Фозилов, А. Н. (1986).
Классификация
пар
дискретных
измеримых
разбиений
пространства
Лебега.
Успехи математических наук
,
41
(2
(248), 185-186.
2.
Винокуров В.Г. , Фозилов А.Н. Об одном
классе пар дискретных дискретных
Volume 02 Issue 12-2022
222
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
02
I
SSUE
12
Pages:
219-226
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
METADATA
IF
–
7.356
измеримых разбиений
.
Изв. АН УзССР, сер.
физ
-
мат. наук, 1982 , № 5,
16-20.
3.
Фозилов
А.Н.
О
неэргодических
максимальных
парах
дискретных
дискретных измеримых разбиений
II
1
.
–
ДАН АН УзССР, 1985, № 6,
8-9.
4.
Винокуров В.Г., Фозилов А.Н. О парах
дискретных
дискретных
измеримых
разбиений
типа
II
∞
.
–
Изв. АН УзССР, сер.
физ
-
мат. наук, 1984, № 1,
23-27.
5.
Акбаров, Д. Е., Кушматов, О. Э., Умаров, Ш.
А., & Фозилов, А. Н. (2021). Исследования
Общих Математических Характеристик
Моделей Булевых Функций Логических
Операций И Табличной Замены В
Криптографических
Преобразованиях.
Central asian journal of
mathematical
theory
and
computer
sciences
,
2
(11), 51-59.
6.
Aybek, T., & Fozilov, A. (2021). Current Issues
of Training Qualified Personnel.
Central Asian
Journal
of
Innovations
on
Tourism
Management And Finance
,
2
(11), 20-24.
7.
Kodirshaevich, S. A., Nabievich, F. A., &
Abdujabbarovna, M. N. (2022). On the
Normalization of Singular Integral Operators
with Carlemann Shift.
Texas Journal of
Multidisciplinary Studies
,
13
, 71-76.
8.
Abdujabbor, A., Nasiba, M., & Nilufar, M.
(2022). Semi-discretization method for
solving boundary value problems for
parabolic
systems.
Texas
Journal
of
Multidisciplinary Studies
,
13
, 77-80.
9.
Shaev, A. K., & Makhmudova, N. A. (2021).
Convergence of the method of straight lines
for solving parabolic equations with
applications
of
hydrodynamically
unconnected formations.
Ministry of higher
and secondary special education of the republic
of Uzbekistan national university of Uzbekistan
Uzbekistan academy of sciences vi romanovskiy
institute of mathematics
, 280.
10.
Abdujabbor, A., Nasiba, M., & Nilufar, M.
(2022). The Numerical Solution of Gas
Filtration in Hydrodynamic Interconnected
Two-Layer Reservoirs.
Eurasian Journal of
Physics, Chemistry and Mathematics
,
6
, 18-21.
11.
Шаев, А. К., & Нишонов, Ф. М. (2018).
Сингулярные интегральные уравнения со
сдвигом Карлемана с рациональными
коэффициентами.
Молодой ученый
, (39), 7-
12.
12.
Кравченко, В. Г., & Шаев, А. К. (1991).
Теория
разрешимости
сингулярных
интегральных уравнений с дробно
-
линейным сдвигом Карлемана. In
Доклады
Академии наук
(Vol. 316, No. 2, pp. 288-292).
Российская академия наук.
13.
Абдуразаков, А., Махмудова, Н., &
Мирзамахмудова, Н. (2019). Решения
многоточечной
краевой
задачи
фильтрации газа в многослойных пластах
с
учетом
релаксации.
Universum:
технические науки
, (11-1 (68)), 6-8.
14.
Файзуллаев, Ж. И. (2020). Методика
обучения
ортогональных
проекций
геометрического тела в координатных
Volume 02 Issue 12-2022
223
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
02
I
SSUE
12
Pages:
219-226
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
METADATA
IF
–
7.356
плоскостях
на
основе
развития
математической
компетентности.
Вестник
Ошского
государственного университета
, (1-4),
285-289.
15.
Расулов, Р., Сатторов, А., & Махкамова, Д.
(2022). Вычисленние Квадрат Нормы
Функционала Погрошности Улучшенных
Квадратурных
Формул
В
Пространстве.
Central asian journal of
mathematical
theory
and
computer
sciences
,
3
(4), 114-122.
16.
Rashidjon, R., & Sattorov, A. (2021). Optimal
Quadrature Formulas with Derivatives in the
Space.
Middle European Scientific Bulletin
,
18
,
233-241.
17.
Абдуразаков, А., Махмудова, Н. А., &
Мирзамахмудова, Н. Т. (2022). Об одном
численном решении краевых задач для
вырождающихся
параболических
уравнений имеющие приложении в
теории
фильтрации.
Universum:
технические науки
, (5-1 (98)), 41-45.
18.
Абдуразаков, А., Махмудова, Н. А., &
Мирзамахмудова, Н. Т. (2021). Численное
решение
краевых
задач
для
вырождающихся
уравнений
параболического
типа,
имеющих
приложения в фильтрации газа в
гидродинамических невзаимосвязанных
пластах
.
Universum: технические науки
, (10-
1 (91)), 14-17.
19.
Далиев, Б. С. (2021). Оптимальный
алгоритм решения линейных обобщенных
интегральных
уравнений
Абеля.
Проблемы
вычислительной
и
прикладной математики
,
5
(35), 120-129.
20.
Акбаров, Д. Е., Абдуразоков, А., & Далиев, Б.
С.
(2021).
О
Функционально
Аналитической
Формулировке
И
Существования
Решений
Системы
Эволюционных Операторных Уравнений С
Краевыми
И
Начальными
Условиями.
Central
asian
journal
of
mathematical
theory
and
computer
sciences
,
2
(11), 14-24.
21.
Kosimova,
М
. Y., Yusupova, N. X., & Kosimova,
S. T. (2021).
Бернулли тенгламасига
келтирилиб
ечиладиган
иккинчи
тартибли оддий дифференциал тенглама
учун учинчи чегаравий масала
.
Oriental
renaissance: Innovative, educational, natural
and social sciences
,
1
(10), 406-415.
22.
Yusupova, N. K., & Abduolimova, M. Q. (2022).
Use fun games to teach geometry.
Central
asian journal of mathematical theory and
computer sciences
,
3
(7), 58-60.
23.
Yusupova, N. X., & Nomoanjonova, D. B.
(2022). Innovative technologies and their
significance.
Central
asian
journal
of
mathematical
theory
and
computer
sciences
,
3
(7), 11-16.
24.
Shakhnoza, S. (2022). Application of Topology
in Variety Fields.
Eurasian Journal of Physics,
Chemistry and Mathematics
,
11
, 63-71.
25.
Bozarov, B. I. (2019). An optimal quadrature
formula with sinx weight function in the
Volume 02 Issue 12-2022
224
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
02
I
SSUE
12
Pages:
219-226
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
METADATA
IF
–
7.356
Sobolev space.
Uzbekistan academy of sciences
vi romanovskiy institute of mathematics
,
47
.
26.
Akbarov, D. E., Kushmatov, O. E., Umarov, S. A.,
Bozarov, B. I., & Abduolimova, M. Q. (2021).
Research
on
General
Mathematical
Characteristics of Boolean Functions’ Models
and their Logical Operations and Table
Replacement
in
Cryptographic
Transformations.
Central asian journal of
mathematical
theory
and
computer
sciences
,
2
(11), 36-43.
27.
Shavkatjon o‘g‘li, T. B. (2022). Proving The
Inequalities Using a Definite Integral and
Series.
Texas Journal of Engineering and
Technology
,
13
, 64-68.
28.
Shavkatjon o’g’li, T. B. (2022).
Some integral
equations for a multivariable function.
Web of
Scientist: International Scientific Research
Journal
,
3
(4), 160-163.
29.
Alimjonova, G. (2021). Modern competencies
in the techno-culture of future technical
specialists.
Current research journal of
pedagogics
,
2
(06), 78-84.
30.
Kupaysinova, Z. S. (2022). Attempts of Central
Asian Scholars to Prove Euclid's Fifth
Postulate.
Eurasian
Journal
of
Physics,
Chemistry and Mathematics
,
12
, 71-75.
31.
Yakubjanovna,
Q.
M.
(2022).
Some
Methodological Features of Teaching the
Subject «Higher Mathematics» in Higher
Educational Institutions.
Eurasian Journal of
Physics, Chemistry and Mathematics
,
4
, 62-65.
32.
Abdurahmonovna, N. G. (2022). Factors for
the Development of Creativity and Critical
Thinking in Future Economists Based on
Analytical Thinking.
Journal of Ethics and
Diversity
in
International
Communication
,
2
(5), 70-74.
33.
Abdurahmonovna, N. G. (2022). Will Be on the
Basis of Modern Economic Education
Principles of Pedagogical Development of
Analytical Thinking in Economists.
European
Multidisciplinary Journal of Modern Science
,
6
,
627-632.
34.
Nazarova, G. A. (2022). Will be on the basis of
modern economic education Principles of
pedagogical development of analytical
thinking in economists.
Journal of Positive
School Psychology
, 9579-9585.
35.
Назарова,
Г.
А.
(2022).
Аналитик
тафаккурни
ривожлантиришнинг
педагогик зарурати.
Integration of science,
education and practice. Scientific-methodical
journal
,
3
(3), 309-314.
36.
Kosimova, M. Y. (2022).
Talabalarni ta’lim
sifatini oshirishda fanlararo uzviyligidan
foydalanish.
Nazariy va amaliy tadqiqotlar
xalqaro jurnali
,
2
(2), 57-64.
37.
Файзуллаев, Д. И. (2022). Развитие
профессиональной
компетентности
студентов технических высших учебных
заведений на основе деятельностного
подхода.
Central
asian
journal
of
mathematical
theory
and
computer
sciences
,
3
(10), 102-107.
Volume 02 Issue 12-2022
225
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
02
I
SSUE
12
Pages:
219-226
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
METADATA
IF
–
7.356
38.
Файзуллаев, Ж. И. (2022). Техника олий
таълим муассасалари талабаларининг
математик
компетенциясини
ривожлантириш
математик
таълим
сифатини
оширишнинг
асоси
сифатида.
Pedagogs jurnali
,
9
(2), 248-256.
39.
Abdujabbor, A., & Nabiyevich, F. A. (2022).
Econometric Assessment of the Perspective of
Business Entities.
Eurasian Journal of Physics,
Chemistry and Mathematics
,
8
, 25-29.
40.
Абдураззаков, А., Фозилов, А. Н., &
Ташпулатов, А. (2021). Некоторые вопросы
оптимизации рынка труда.
The Scientific
Heritage
, (76-2), 29-32.
41.
Qosimova, M. Y., & Yusupova, N. X. (2020). On
a
property
of
fractional
integro-
differentiation operators in the kernel of
which the meyer function.
Scientific-technical
journal
,
24
(4), 48-50.
42.
Jalilov, I. (2019). To the problems of
innovation
into
the
educational
process.
Scientific Bulletin of Namangan State
University
,
1
(3), 344-347.
43.
Акбаров, Д. Е., Кушматов, О. Э., Умаров, Ш.
А., & Расулов, Р. Г. (2021). Исследования
Вопросов Необходимых Условий Крипто
Стойкости
Алгоритмов
Блочного
Шифрования
С
Симметричным
Ключом.
Central
asian
journal
of
mathematical
theory
and
computer
sciences
,
2
(11), 71-79.
44.
Mirzakarimov, E. M., & Fayzullaev, J. S. (2020).
Improving the quality and efficiency of
teaching
by
developing
students*
mathematical
competence
using
the
animation method of adding vectors to the
plane using the maple system.
scientific
bulletin of namangan state university
,
2
(9),
336-342.
45.
Mirzakarimov, E. M., & Faizullaev, J. I. (2019).
Method of teaching the integration of
information and educational technologies in a
heterogeneous parabolic equation.
scientific
bulletin of namangan state university
,
1
(5), 13-
17.
46.
Mirzaboevich, M. E. (2021). Using Maple
Programs in Higher Mathematics. Triangle
Problem Constructed on Vectors in
Space.
Central asian journal of mathematical
theory and computer sciences
,
2
(11), 44-50.
47.
Мирзакаримов, Э. М., & Файзуллаев, Д. И.
(2021). Выполнять Линейные Операции
Над Векторами В Пространстве В Системе
Maple.
Central asian journal of mathematical
theory and computer sciences
,
2
(12), 10-16.
48.
Мирзакаримов, Э. М. (2022). Использовать
Систему
Maple
Для
Определения
Свободных Колебаний Прямоугольной
Мембраны
При
Начальных
Условиях.
Central
Asian
Journal
Of
Mathematical
Theory
And
Computer
Sciences
,
3
(1), 9-18.
49.
Мамаюсупов, Ж. Ш. (2022). Интегральное
преобразование Меллина для оператора
интегродифференцирования
дробного
порядка.
Periodica
Journal
of
Modern
Philosophy, Social Sciences and Humanities
,
11
,
186-188.
Volume 02 Issue 12-2022
226
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
02
I
SSUE
12
Pages:
219-226
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
METADATA
IF
–
7.356
50.
Mamayusupov, J. S. O. (2022).
“Iqtisod”
yo’nalishi
mutaxassislarini tayyorlashda
matematika
fanini
o’qitish
uslubiyoti.
Academic research in educational
sciences
,
3
(3), 720-728.
51.
Qo‘Ziyev, S. S., & Mamayusupov
, J. S. (2021).
Umumiy o ‘rta ta’lim maktablari uchun
elektron darslik yaratishning pedagogik
shartlari.
Oriental renaissance: Innovative,
educational, natural and social sciences
,
1
(10),
447-453.
52.
Kosimov, K., & Mamayusupov, J. (2019).
Transitions melline integral of fractional
integrodifferential
operators.
Scientific
Bulletin of Namangan State University
,
1
(1),
12-15.
53.
Qosimova, S. T. (2021). Two-point second
boundary value problem for a quadratic
simple second-order differential equation
solved by the bernoulli equation.
Innovative
Technologica:
Methodical
Research
Journal
,
2
(11), 14-19.
54.
Jalilov, I. I. U. (2022).
К актуальным
проблемам становления педагогического
мастерства
преподавателя
.
Nazariy
va
amaliy tadqiqotlar xalqaro jurnali
,
2
(9), 81-
89.
