西 南 交 通 大 学 学 报 第 55 卷 第 1 期 Vol. 55 No. 1 JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY 2020 年 2 月 Feb. 2020

ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.55.1.15

Research article

Mathematics

NEW DEFINITIONS OF SIGMA FIELD

适马场的新定义

Hind Fadhil Abbas Directorate of Education, Salah Eddin, Khaled Ibn Al Walid School Tikrit City, Iraq, [email protected]

Abstract The sigma field is a branch of mathematics that refers to the gathering of subsets includes sample space. It is useful in establishing a formal mathematical probability definition. The sets that occur in the sigma field are responsible for constituting various events in the presence of available sample space. In this paper, we focus on sigma fields. The main aim of this paper is to define the sigma field. Different types of sigma fields, along with the basic mathematics and examples, are also elaborated upon.

Keywords: Set Theory, Measure, Sigma Additivity, Sigma Ring, Borel σ Field

摘要 西格玛字段是数学的一个分支，它指的是包括样本空间在内的子集的聚集。在建立正式的数 学概率定义中很有用。出现在西格玛字段中的集合负责在存在可用样本空间的情况下构成各种事 件。在本文中，我们专注于西格玛字段。本文的主要目的是定义西格玛域。还阐述了不同类型的 西格玛字段，以及基本的数学和示例。

关键词: 集合论，度量，西格玛可加性，西格玛戒指，博雷尔 σ 场

I. INTRODUCTION Basically, there are three key points for Sigma fields is a branch of mathematics that is motivations for σ algebras, namely limits of set useful in many sectors, such as physics, biotech, manipulation, defining measures, and electronics, and communication. Basically, sigma management of fractional information that is fields are defined with the help of a sample space categorized by sets. There are various types of point. A Sigma field or Sigma is defined by the sigma algebras, such as sub σ algebras, separable Greek letter σ. σ algebra represents a set of types σ algebras, Borel and Lebesgue σ algebras, of algebras. combining σ algebras, and product σ algebras. In mathematics and the theory of probability, Algebra is generated from class, and then we σ-field or σ algebra on X set is a gathering Σ of look towards algebra that is generated from semi- all subsets of set X. It is included and closed algebra. Semi-algebra is a set of finitely under complement and countable unions. polynomial equalities and inequalities. The

2 Abbas / Journal of Southwest Jiaotong University / Vol. 55 No. 1 Feb. 2020 abstract measure theory, containing a brief B. Sample Space discussion of sigma algebra with the basics, is Sample space is nothing but the outcome of an well defined by various authors [1], [2]. Product experiment. Basically, there are three algebra is discussed with examples in reference components of the probability space. Probability numbers in [1]. The Borel and Lebesgue σ space is built with a specific situation or algebras are explained by Jagannathan [3]. In the experiments. Out of those three, sample space, present study, basic concepts of sigma, random variable values range, is one component. mathematical models, various types of sigma Remaining is a probability function (probability algebra, and examples are presented. spreading of a discrete random variable) and event space, includes all possible events. II. WRITING A PLAN Here we discuss the basic scenarios related to C. Motivation the sigma field. In short, we tried to focus on set Generally, there are three different key theory, sample space, motivation, measures, sub motivators available for sigma (σ) algebra, which σ-algebras, definitions and properties, types of are listed below: sigma algebra, sigma additivity, and sigma ring.  Measures definition;  Limits of sets manipulation; A. Set Theory  Managing fractional information Set theory is the foundation of mathematics. categorized by sets. Set theory generally refers to the collection of objects. We can just collect any type of object as D. Measurement part of a set that is relevant to mathematics. The function of the measure is to assign the Basically, set theory language is used to define non-negative value to a subset of x, which makes objects [4], [5]. Set theory is used for abstract precise sets. These sets are closed when algebra, discrete mathematics, and mathematical operations are carried out, except for the analysis. Simple set theory is shown in Figure 1. measurable sets. Here, we say that M is a measurable set. However, the complement of it is also a measurable set. These non-empty set collections with this property are termed sigma (σ) algebras [6]. A limit is an important aspect for set sequence. This limits closure for countable unions; intersections are most important. Here, set limits are defined followed by sigma (σ) algebras. 1. Consider a sequence s1, s2, s3 …which is a subset of X. (1) 2. This is the limit supremum. 3. The infimum limit for this sequence is (2)

Figure 1. Set theory III. SIGMA (Σ OR Σ) FIELD The term sample space is uncountable. Due to A subset indicates all the possible this uncountability, it is difficult to find a observations. This partial information is problem in case of assignment of probabilities to characterized by smaller σ-algebra, and this is a all possible subsets. Hence, it is necessary to subset of principal sigma (σ) algebra. implement with small algebra. Some examples Consider that X is any set, and P(X) is a are listed below. The sigma (σ) algebra estimated power set. The subset is known by collecting subsets of given sample space, as σ algebra. This satisfies the following which is termed the smallest sigma (σ) algebra, properties: contains the collection [3]. ● X is a part of sigma and it is considered a universal set.

3

● Sigma (σ) is closed for complementation, s ∈ S meaning S is in the Sigma field, and it is also a If P ∈ S then Pc ∈ S complement. If P ∈ S and Q ∈ S then P ∪ Q ∈ S ● Sigma (σ) is closed for unions, meaning Now here we discuss the natural sigma field s1,s2 ,s3,… is in the Sigma field, and S = s1 U s2 or algebra. This is useful in many special types of U s3….. cases of sets. Every sigma (σ) algebra is semi-ring, a set two binary operators together. Hence, if a set of E. Dynkin’s π-λ Theorem semi-ring is needed, then it is adequate to show The sigma field covers the Dynkin system as that it is sigma (σ) algebra. well as the π system. The Dynkin system is also Sigma algebras could also be made by known as the λ system. Here we focus on the available arbitrary sets. It helps to develop Dynkin system. probability space [7]. Theorem: Dynkin’s π-λ theorem is very useful in supporting the property of specific sigma (σ) A. Theorem algebras. In π system, it is the gathering of Consider set X as an intersection of all the subsets of set x having many close and finite sigma (σ) algebras, Si, containing X means, x ∈ X intersections. The λ-system set is the gathering of = > x ∈ Si, for all i is itself sigma (σ) algebra. It is subsets that contain x and it is locked underneath denoted by σ (X). This is known as the generation countable unions of all disjoint subsets and of sigma (σ) algebra by set X. complements, whereas the Dynkin system is straightforward in nature. Dynkin's π-λ Theorem B. Sigma Additivity is applicable for all σ (P) sets where it reduces Additivity or specifically finite additivity and the checking of σ (P) arbitrary sets, which countable additivity or sigma additivity are going shrinks the computational time. to be explained with a subset of available sets, Definition 1: Sigma (σ) field or generation of which are abstracted by properties like volume, algebra by sets family: Consider F to be a family length, and area when many objects are available. of sets, then the sigma field generated by F is Simple additivity is weak compared to sigma denoted as σ (F). It is the intersection of complete additivity because in sigma additivity minor sigma algebras of F. This is the smallest sigma specification, decimals are also considered. field or algebra having all sets in Set F. Example: Consider A = [0, 1] and F = {[0, .3], C. Sigma Ring [.5, 1]} = {b1, b2}. When a set is made by a nonempty collection Then of sets, if this set is closed for a countable union σ (F) = {ϕ, b1, b2, b3, b1 ∪ b2, b1 ∪ b3, b2 ∪ and its relative complementation, then it is b3, A} where b3 = (.3, .5). known as a sigma (σ) ring. It is also known as a Here 8 sets are present in σ (F). pronounced sigma (σ) ring. F. Borel and Lebesgue Sigma Field D. Sigma Field Definitions Consider S to be countable. Σ (σ) algebra or σ Sigma fields are generated by set families. field is used as the power set P(s). Hence, all sets Sigma fields are divided into many types, such as become defensible meaning allowable. ℝ is the generation of sigma field based on arbitrary set of real numbers. Here we generate the sigma family, cylinder set, random variable, stochastic field by all interval collection. Sometimes it is process and function, Borel and Lebesgue sigma known as Borel σ algebra or Borel σ field field, and product sigma field. Generally, σ field because it is named after Emil Borel [3]. or σ-algebras are important concepts and play Definition 2: Borel σ-algebra: The Borel fundamental scenarios for given probability in sigma field or Borel sigma (σ) algebra is case of random experiments. Consider S is the indicated by B. A topological (X; τ) space is the universal set. Many times, it is difficult to sigma (σ) algebra created by τ which belongs to summarize all the subsets of the set S if set S is the family of open sets. Elements of this set are uncountable. Hence, it is difficult to maintain known as Borel sets. consistency and we need to collect all permissible Lemma: Let B = {(p; q): p < q}. Then σ (B) = subsets closed under the set operation. This BR Borel field which is produced because of the results in some essential definitions. open intervals B family. Here, BR may be all Let’s consider s is the collection of subsets for probable open intervals. It takes their tributes. It the given set S. Then s is called a field or algebra also takes arbitrary unions. BR is continuing with if it satisfies these conditions: a large range of intervals which includes open,

4 Abbas / Journal of Southwest Jiaotong University / Vol. 55 No. 1 Feb. 2020 half-open and closed intervals. It also includes Definition 4: Separable sigma (σ) field or disjoint intervals. It is simply the collection of all algebras: The Separable sigma (σ) field possible intervals. separable is a space which is nothing but the Assume a sequence s1, s2, s3, …. sn and Si to metric space having metric be the sigma field or σ algebra of subset of si for P, Q belongs to . Any σ field generated via where i ∈ {1, 2, ..., n}. gathered sets is separable. If the Lebesgue σ field So product set is is separable, it is measurable similar to a Borel. If the family consists of an empty set, then this set s = s × s × s × …. × s (3) 1 2 3 n is known as trivial σ field over set X. It is also Now we use the sigma field or sigma algebra known as a minimal sigma field. The power set which is generated by gathering all product of set x is known as discrete sigma field or subsets algebra. Definition 5: Product sigma (σ) field or S = σ [(g1 × g2 × ··· × gn: gi ∈ Si for i ∈ {1, 2, ..., n})] (4) algebra: Consider if (A1, 1 ) and (A2, 2 ) are the two measurable spaces. The sigma field for We can write this for infinite products. the respective product space A1 × A2 is known as If [s1, s2, s3,….] an infinite sequence of sets is product sigma (σ) field or algebra. It is defined in obtained and set Si becomes sigma (σ) algebra of mathematical form as follows: given subset si for i ∈ ℕ+. 1× 2 = σ ({p1 × p2: p1 ∈ 1, p2 ∈ 2}) (7) For s = s1 × s2 × s3 ×… the product set now uses sigma (σ) field or sigma (σ) algebra which is Here it is found that the terms in brackets {p1 generated by the grouping of cylinder sets. × p2: p1 ∈ 1 , p2 ∈ 2} represents the π system. S = σ ({g1 × g2 × ··· × gn × Sn + 1 × sn + 2 × Definition 6: Sigma (σ) field or algebra ···: (n ∈ ℕ+) (5) generated by cylinder sets: Consider if set X is formed by real number. where gi ∈ Si for i ∈ {1, 2, ..., n}}) The combination of product construction from (8) the earlier discussion about ℝ as ℝn and the This set consists of real value functions. B ( ) generation of sigma (σ) field is done with the represents a Borel subset of set R. The cylinder help of all possible product intervals. This is subset of set X is finite in nature and defined as: known as Borel σ-algebra for ℝn [7], [8]. Example: The Borel Sigma Algebra is defined (9) on a topological space (A, O) and is C = σ(O). Then, with each term Theorem: The Borel sigma (σ) algebra on R is σ(L) and then generates sigma algebra. Those classes of sets L are as below (10) i. L1 = {(p, q); p ≤ q} representing the π system. This system generates ii. L2 = {(p, q]; p ≤ q} the sigma field . iii. L3 = {[p, q); p ≤ q} iv. L4 = {[p, q]; p ≤ q} Then the family of subsets is v. L5 = set [all open subsets of R] vi. L6 = set [all closed subsets of R] (11) Definition 3: Lebesgue sigma field or algebra: This is a field that generates the cylinder As it is unique measure λ on (A, BR) which sigma field or algebra for the set X. satisfies With the help of product topology of , Λ ([p, q]) = q – p (6) which is restricted to set X, this sigma field is responsible to get sub algebra for the Borel For each finite interval [p, q], −∞ < p ≤ q < ∞. Sigma Field. Here is the set of natural numbers, This is known as the Lebesgue measure. while X is the set of real numbers. Here it is If we put the restriction to λ with ([0, 1], adequate to assume a cylinder set. B[0,1]) measurable space, then λ becomes probability measure. Examples: For Cartesian product having intervals [p, q] x [r, s] is Lebesgue Measurable. λ = (q − p) (s − r) its Lebesgue measure. Then λ is (12) the set that contains rational numbers having an For this, the non-decreasing sequence of the interval R. Hence λ = 0 [9]. sigma field is

5

(13) of Science Frontier Research, 11 (2). Available from Definition 7: Generation of σ-algebra by a random variable: Consider the probability space https://studyres.com/doc/1873138/on- (Ω, σ, P). If S: Ω is a Borel sigma field or topological-sets-and-spaces---global-journal- algebra on real number Rm, which is measurable, of-science. then S is called a random variable. [5] SAYED, A.F. (2017) On The Sigma field or algebra generated by S is characterizations of some types of fuzzy soft sets fuzzy soft bitopological spaces. Journal )} (14) of Advances in Mathematics and Computer Definition 8: Generation of σ-algebra by Science, 24 (3), pp. 1-12. random process: Consider the probability space [6] DˇZAMONJA, M. and KUNEN, K. (Ω, σ, P). is a real value function set on T. If (1995) Properties of the class of measure G: Ω is cylinder sigma algebra σ(Fx), and is separable compact spaces. Fundamenta measurable, then the term G is known as a Mathematicae, 147, pp. 261-276. random process. It is also called a stochastic process. The [7] RANA, I.K. (2017) Algebra and sigma generated by the G is Sigma Algebra of Subsets of a Set. Bombay: Department of Mathematics, Indian Institute (15) of Technology. Here, the inverse of the cylinder set is used to [8] KHACHATRYAN, Y. (2012) Borel generate the sigma field or algebra. Sets are Ramsey. [9] MEISTERS, G.H. (1997) Lebesgue IV. CONCLUSION Measure on the Real Line. Sigma fields or algebras are subset of fields. This means all sigma (σ) algebras are algebras, but vice versa is not true. Sigma fields must be : closed and may be union of sets or may be a 参考文 finite. Here, in this paper, we studied the various [1] PAPADIMITRAKIS，M。（2004）关 concepts related to sigma fields, such as set 于测度理论的注释。雷斯蒙，伊拉克利 theory, sample space, motivation, measure, sub 翁：克里特大学数学系。 σ-algebras, sigma additivity, and sigma rings. Also, we mathematically discuss the definition [2] CHRISTOPHER，H.（2011）抽象测度 and properties of various types of sigma algebra. 理论讲义。美国佐治亚州亚特兰大。 [3] JAGANNATHAN，K.（2015）博雷尔 ACKNOWLEDGMENTS 集和勒贝格测度，电气工程师的概率基础。 The Author is thankful to Directorate of ， ， ， Education, Salah Eddin, Khaled Ibn Al Walid [4] CHAUDHARY M.P. KUMAR V. School for their kind support. 和 CHOWDHARY，S.（2015）关于拓扑 集和空间。全球科学前沿研究杂志，11 REFERENCES （ 2 ） 。 可 从 https://studyres.com/doc/1873138/on- [1] PAPADIMITRAKIS, M. (2004) topological-sets-and-spaces---global-journal- Notes on Measure Theory. Rethymnon, of-science 获取。 Heraklion: Department of Mathematics, [5] SAYED，A.F.（2017）关于某些类型 University of Crete. 的模糊软集的表征模糊软位空间。数学与 [2] CHRISTOPHER, H. (2011) Lecture 算机科学 展， （ ），第 。 Notes on Abstract Measure Theory. Atlanta, 计 进 24 3 1-12 页 Georgia. [6] DˇZAMONJA ， M. 和 KUNEN ， K. [3] JAGANNATHAN, K. (2015) Borel （1995）度量可分离紧凑型空间的性质。 Sets and Lebesgue Measure, Probability 数学基础，147，第 261-276 页。 Foundations for Electrical Engineers. [7] RANA，I.K。（2017）集合子集的代 [4] CHAUDHARY, M.P., KUMAR, V., 数和西格玛代数。孟买：印度理工学院数 and CHOWDHARY, S. (2015) On 学系。 topological sets and spaces. Global Journal

6 Abbas / Journal of Southwest Jiaotong University / Vol. 55 No. 1 Feb. 2020

[8] KHACHATRYAN，Y.（2012）博雷尔 套装是拉姆西。 [9] MEISTERS，G.H.（1997）实线上的勒 贝格测度。

1