Summary
Content
STUDY OF FUZZY MEASURE AND SOME
PROPERTIES OF NULL-ADDITIVE FUZZY
MEASURE
A Summary
Submitted to the
KUMAUN UNIVERSITY,
NAINITAL, Uttarakhand, INDIA
FOR THE DEGREE OF
Doctor of Philosophy in Mathematics
SUBMITTED BY
PARUL AGARWAL
M. Sc., M. Phil. (Mathematics)
UNDER THE SUPERVISION OF
DR. H.S. NAYAL
Associate Professor
Department of mathematics
Govt. Postgraduate College, Ranikhet (Almora)
UTTARAKHAND.
2015
SUMMARY OF THE THESIS
STUDY OF FUZZY MEASURE AND SOME
PROPERTIES OF NULL-ADDITIVE FUZZY
MEASURE
BY
PARUL AGARWAL
Submitted to the
KUMAUN UNIVERSITY,
NAINITAL, Uttarakhand, INDIA
FOR THE DEGREE OF
Doctor of Philosophy in Mathematics
UNDER THE SUPERVISION OF
DR. H.S. NAYAL
Associate Professor
Department of mathematics
Govt. Postgraduate College, Ranikhet (Almora)
UTTARAKHAND.
2015
Summary
The thesis entitled,
“STUDY OF FUZZY MEASURE AND SOME
PROPERTIES OF NULL-ADDITIVE FUZZY
MEASURE”
is divided into the following five chapters:
CHAPTER-1
Introduction
CHAPTER-2
Possibility Theory versus Probability Theory in Fuzzy
Measure Theory
CHAPTER-3
Properties of Null-Additive and Absolute Continuity of a
Fuzzy Measure
CHAPTER-4
Properties of Strong Regularity of Fuzzy Measure on
Metric Space
CHAPTER-5
Continuous, auto-continuous and completeness of fuzzy
measure
Fuzzy set theory, describing fuzziness mathematically for the
first time introduced by L. A. Zadeh in 1965, [60]. In 19 th
century, mathematician defined the concepts of sets and functions
to represent a problem. Measure theory was developed in
successive stage during the late 19 th and early 20th centuries by
Emile Borel, Henri Lebesgue, Johann Radon and Maurice
Frechet, among others by [100], [18], [31].
The main applications of measures are in the foundations of the
Lebesgue integral, [64] in Andrey Kolmogorov’s axiomatisation
of probability theory and in ergodic theory [42]. The central
concept of fuzzy measure theory is the fuzzy measure which was
introduced by Choquet in 1953 by [44] and independently
defined by Sugeno in 1974 by [72] in the context of fuzzy
integrals. The name “Theory of possibility was coined by Zadeh
[59], who was inspired by a paper by Gaines and Kohout [8].
Recently, many authors have investigated different type of nonadditive set functions, as sub-measures, k-triangular set
functions, ⊥−¿ decomposable measures [96], null-additive set
functions and others. The range of the null-additive set functions
introduced by Wang [109]. Suzuki [45] introduced and
investigated atoms of fuzzy measures and Pap [26] introduced
and discussed atoms of null-additive set functions.
In the present thesis we deal with the study of fuzzy measure and
null-additive fuzzy measure which have wide applications in
computer, information technology, manufacturing systems,
production, communication, transportation systems etc.
The details of chapter-wise organization of the whole thesis are
as follows:
Chapter-1: Introduction:
In this chapter we discuss some important fundamental concepts
and definitions related to our research work. The concept of
fuzzy measure provide us with a broad framework within which
it is convenient to introduce and examine possibility theory [44],
a theory that is closely connected with fuzzy set theory and pays
an important role in some of its applications [37]. Thus with the
advancement of the research areas the range of mathematical
tools has expanded considerably with the use of fuzzy measure
[26].
CHAPTER-2: Possibility Theory versus Probability Theory
in Fuzzy Measure Theory:
The aim of this chapter is to discuss the differences and
similarities between probability measures and possibilities
measures [21]. This chapter thus establishes a new link between
probability and possibility theories. We discuss some
mathematical properties of probability theory and possibility
theory. These basic properties help us to compare probability
theory and possibility theory [30]. Some of properties define the
similarity between probability and possibility [17].
Secondly we discuss some variants of fuzzy subsets entropies
and the linking relations of these quantities with indicate a
similarity of many important characteristics of fuzzy subsets
entropies and Shannon’s entropy [10]. We analyze some basic
properties of entropy of fuzzy measure [61].
CHAPTER-3: Properties of Null-Additive and Absolute
Continuity of a Fuzzy Measure:
In this chapter we discussed the connection between two types of
absolute continuity of a fuzzy measure with respect to a given
null additive fuzzy measure [107]. The notion of absolute
continuity allows one to obtain generalizations of the relationship
between the two central operations of calculus, differentiation
and integration, expressed by the fundamental theorem of
calculus in the framework of Riemann integration [78]. Such
generalizations are often formulated in terms of Lebesgue
decomposition type for null additive fuzzy measure [65]. We
proposed generalization of the universal set. At last we discussed
some properties of fuzzy measure [96].
CHAPTER- 4: Properties of Strong Regularity of Fuzzy
Measure on Metric Space:
We discussed the regularity of a null- additive fuzzy measure and
proved Egoroff’s theorem and Lusin’s theorem for fuzzy
measures on a metric space [110]. The Egoroff’s theorem and
Lusin’s theorem in the classical measure theory are important and
useful for discussion of convergence and continuity of
measurable functions [106]. We discussed further regularity
properties of null-additive fuzzy measure on metric spaces. We
studied some properties of inner/ outer regularity and the
regularity of fuzzy measure under null-additive condition [56].
Also the strong regularity of fuzzy measure discussed on
complete separable metric spaces.
In this chapter, we shall investigate strong regularity of fuzzy
measure on metric spaces. Under the null-additivity, weekly nulladditivity and converse null-additivity condition, we shall discuss
the relation among the inner regularity, the outer regularity and
the strong regularity of fuzzy measure [82].
CHAPTER-5: Continuous, Auto-Continuous and Completeness
of Fuzzy Measure:
We show that any uniformly auto-continuous finite fuzzy
measure is equivalent to a sub-additive finite fuzzy measure in
the sense of absolute continuity to each other [109]. We discuss
σ −¿
finite and purely atomic fuzzy measures [83]. In this case
the uniform auto-continuity can be replaced with the nulladditive [106]. The concepts of two structured characteristics that
are the pseudo-metric generating property and the auto-continuity
play important roles in the fuzzy measure theory [81].
By mean of the asymptotic structural characteristics of fuzzy
measure we discuss the convergence in fuzzy measure space and
some results [105], such as Lebesgue’s theorem, Riesz’s theorem,
Egoroff’s theorem and their generalizations, of fuzzy measure
theory [106]. In this chapter we studied some theorems on the
completeness of fuzzy measure space and some properties of
fuzzy measure and convergence in measure [108].
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PROPERTIES OF NULL-ADDITIVE FUZZY
MEASURE
A Summary
Submitted to the
KUMAUN UNIVERSITY,
NAINITAL, Uttarakhand, INDIA
FOR THE DEGREE OF
Doctor of Philosophy in Mathematics
SUBMITTED BY
PARUL AGARWAL
M. Sc., M. Phil. (Mathematics)
UNDER THE SUPERVISION OF
DR. H.S. NAYAL
Associate Professor
Department of mathematics
Govt. Postgraduate College, Ranikhet (Almora)
UTTARAKHAND.
2015
SUMMARY OF THE THESIS
STUDY OF FUZZY MEASURE AND SOME
PROPERTIES OF NULL-ADDITIVE FUZZY
MEASURE
BY
PARUL AGARWAL
Submitted to the
KUMAUN UNIVERSITY,
NAINITAL, Uttarakhand, INDIA
FOR THE DEGREE OF
Doctor of Philosophy in Mathematics
UNDER THE SUPERVISION OF
DR. H.S. NAYAL
Associate Professor
Department of mathematics
Govt. Postgraduate College, Ranikhet (Almora)
UTTARAKHAND.
2015
Summary
The thesis entitled,
“STUDY OF FUZZY MEASURE AND SOME
PROPERTIES OF NULL-ADDITIVE FUZZY
MEASURE”
is divided into the following five chapters:
CHAPTER-1
Introduction
CHAPTER-2
Possibility Theory versus Probability Theory in Fuzzy
Measure Theory
CHAPTER-3
Properties of Null-Additive and Absolute Continuity of a
Fuzzy Measure
CHAPTER-4
Properties of Strong Regularity of Fuzzy Measure on
Metric Space
CHAPTER-5
Continuous, auto-continuous and completeness of fuzzy
measure
Fuzzy set theory, describing fuzziness mathematically for the
first time introduced by L. A. Zadeh in 1965, [60]. In 19 th
century, mathematician defined the concepts of sets and functions
to represent a problem. Measure theory was developed in
successive stage during the late 19 th and early 20th centuries by
Emile Borel, Henri Lebesgue, Johann Radon and Maurice
Frechet, among others by [100], [18], [31].
The main applications of measures are in the foundations of the
Lebesgue integral, [64] in Andrey Kolmogorov’s axiomatisation
of probability theory and in ergodic theory [42]. The central
concept of fuzzy measure theory is the fuzzy measure which was
introduced by Choquet in 1953 by [44] and independently
defined by Sugeno in 1974 by [72] in the context of fuzzy
integrals. The name “Theory of possibility was coined by Zadeh
[59], who was inspired by a paper by Gaines and Kohout [8].
Recently, many authors have investigated different type of nonadditive set functions, as sub-measures, k-triangular set
functions, ⊥−¿ decomposable measures [96], null-additive set
functions and others. The range of the null-additive set functions
introduced by Wang [109]. Suzuki [45] introduced and
investigated atoms of fuzzy measures and Pap [26] introduced
and discussed atoms of null-additive set functions.
In the present thesis we deal with the study of fuzzy measure and
null-additive fuzzy measure which have wide applications in
computer, information technology, manufacturing systems,
production, communication, transportation systems etc.
The details of chapter-wise organization of the whole thesis are
as follows:
Chapter-1: Introduction:
In this chapter we discuss some important fundamental concepts
and definitions related to our research work. The concept of
fuzzy measure provide us with a broad framework within which
it is convenient to introduce and examine possibility theory [44],
a theory that is closely connected with fuzzy set theory and pays
an important role in some of its applications [37]. Thus with the
advancement of the research areas the range of mathematical
tools has expanded considerably with the use of fuzzy measure
[26].
CHAPTER-2: Possibility Theory versus Probability Theory
in Fuzzy Measure Theory:
The aim of this chapter is to discuss the differences and
similarities between probability measures and possibilities
measures [21]. This chapter thus establishes a new link between
probability and possibility theories. We discuss some
mathematical properties of probability theory and possibility
theory. These basic properties help us to compare probability
theory and possibility theory [30]. Some of properties define the
similarity between probability and possibility [17].
Secondly we discuss some variants of fuzzy subsets entropies
and the linking relations of these quantities with indicate a
similarity of many important characteristics of fuzzy subsets
entropies and Shannon’s entropy [10]. We analyze some basic
properties of entropy of fuzzy measure [61].
CHAPTER-3: Properties of Null-Additive and Absolute
Continuity of a Fuzzy Measure:
In this chapter we discussed the connection between two types of
absolute continuity of a fuzzy measure with respect to a given
null additive fuzzy measure [107]. The notion of absolute
continuity allows one to obtain generalizations of the relationship
between the two central operations of calculus, differentiation
and integration, expressed by the fundamental theorem of
calculus in the framework of Riemann integration [78]. Such
generalizations are often formulated in terms of Lebesgue
decomposition type for null additive fuzzy measure [65]. We
proposed generalization of the universal set. At last we discussed
some properties of fuzzy measure [96].
CHAPTER- 4: Properties of Strong Regularity of Fuzzy
Measure on Metric Space:
We discussed the regularity of a null- additive fuzzy measure and
proved Egoroff’s theorem and Lusin’s theorem for fuzzy
measures on a metric space [110]. The Egoroff’s theorem and
Lusin’s theorem in the classical measure theory are important and
useful for discussion of convergence and continuity of
measurable functions [106]. We discussed further regularity
properties of null-additive fuzzy measure on metric spaces. We
studied some properties of inner/ outer regularity and the
regularity of fuzzy measure under null-additive condition [56].
Also the strong regularity of fuzzy measure discussed on
complete separable metric spaces.
In this chapter, we shall investigate strong regularity of fuzzy
measure on metric spaces. Under the null-additivity, weekly nulladditivity and converse null-additivity condition, we shall discuss
the relation among the inner regularity, the outer regularity and
the strong regularity of fuzzy measure [82].
CHAPTER-5: Continuous, Auto-Continuous and Completeness
of Fuzzy Measure:
We show that any uniformly auto-continuous finite fuzzy
measure is equivalent to a sub-additive finite fuzzy measure in
the sense of absolute continuity to each other [109]. We discuss
σ −¿
finite and purely atomic fuzzy measures [83]. In this case
the uniform auto-continuity can be replaced with the nulladditive [106]. The concepts of two structured characteristics that
are the pseudo-metric generating property and the auto-continuity
play important roles in the fuzzy measure theory [81].
By mean of the asymptotic structural characteristics of fuzzy
measure we discuss the convergence in fuzzy measure space and
some results [105], such as Lebesgue’s theorem, Riesz’s theorem,
Egoroff’s theorem and their generalizations, of fuzzy measure
theory [106]. In this chapter we studied some theorems on the
completeness of fuzzy measure space and some properties of
fuzzy measure and convergence in measure [108].
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