PSYCHOLOGY AND EDUCATION (2021) 58(3): 2242-2247 ISSN: 00333077

Measure Theory of Premeasures and Measures with Extension

Hassan Hussien Ebrahim1, Hind Fadhil Abbas2, Slah Al Deen/SAMMARA3 1 College of computer and /Tikrit university 3 Msc mathematics science , Teacher, Directorate of Education Salah Eddin/ Khaled Ibn Al Walid School/ Tikrit City

ABSTRACT theory is one of the important scenarios in mathematics as it has better properties as compared to theory. We aimed to focus on the premeasures and measures construction along with their extension. Initially the was used at undergraduate mathematical study but recently there are many measures are available such as Borel, Gauss Measure, Euler Measure, Hausdorff Measure, Haar Measure and many more.In construction of few measures the Carath´eodory extension theorem as well as Carath´eodory-Hahn theorem was solicited subsequently. Similarly the Dynkin’s π − λ theorem was also used in construction purpose. This paper illustrated the mathematical analysis for construction of measures and premeasures. Also extensions of measures and premeasures were studied in this paper. Keywords σ-algebra, Premeasure, Measure, Extensions of measure and premeasure, Carath´eodory theorem.

Introduction measure. Here travail examples follows this like measure counting (It defines measure of set S Measure theory deals with the branch of along with present no of elements of set S. Due to mathematics. It plays an important role in of this scenario measurable subset are comes in mathematical analysis where it is familiar with the picture which defines the measure on subset basic notion and various statements. It does not collection for all present subsets. Measurable much important to have detailed analysis proof if subsets require forming the σ algebra. It means used. Consider the example, anyone should has a that finite unions, complements for measurable particular place where he or she could visited so at subsets and finite intersection are measurable [3]. time it is not necessary to have any detailed proof. Measure theory is permissible or can be applying 1.1 Measures: to the limiting processes but is not applicable in In field of mathematics, Lots of systematic ways case of weaker setups such as Riemann integration are available for assigning the number for suitable theory. Basically measure is the way to assign subset which are made from that set, instinctively to the subsets of R. The all set collection elucidated its size. In this sensation, measures are which allowed to measure must 𝜎 algebra. Very the generalization of hypothesis of area, length important thing is , the measure is used it for and volume [4].Fig 1 shows the measure i.e. integrating the [1]. formation of set by considering various parameter. Recent measure theories are remarkably very powerful as compared to previous theory as extra ordinary types of sets enough regular for measured, but still it has some limitations. Everybody is aware of length, area and volume concepts which are applicable specifically to reasonable regular set. Proper realization class of the sets allows its measure indispensable which further leads to existing non measurable sets. [2]. Measures Assigns the zero to empty set and then it be the additive. Measure of the large subset is breakdown or divided into finite number of small disjoint subsets .Additive aspect describes the sum Fig 1: Measure of measures of small subset. Generally anyone Term measures have specific technical definition wants to assign a predictable size to every subset which usually involves sigma algebra. It is for given set with satisfaction of the axioms of

2242 www.psychologyandeducation.net PSYCHOLOGY AND EDUCATION (2021) 58(3): 2242-2247 ISSN: 00333077 strenuous to comprehend. However technical simplicity and no confusion, 휇 is measure on P or statement is very important as it is base for any measure on Q [7] and [8]. mathematical analysis which includes some 1.2 Premeasures: slippery sets of objectives of calculus. Let’s consider example. Every definition or statements It is a function, in some part of sensation, of integrals are based on particular measure. precursor used for open, fair, regardless outcome is depends on Jordan measure result and honest measure for given space. The whereas Lebesgue integral is depends on the fundamental theorem is present in measure theory Lebesgue measure. Measure theory is nothing but which states that premeasures can be extends to the study of the measures with their application to measure. the integration [5]. Let Z set and 퐴 ⊂ 푃(푍) is algebra. Premeasure on The non-negative real function from the delta ring algebra 퐴 is mapped 𝜌: 퐴 → [0, ∞] which follows D such that satisfies 푠(∅) = 0 푤ℎ푒푟푒 ∅ 푟푒푝푟푒푠푒푛푡푠 푒푚푝푡푦 푠푒푡. 1. 𝜌(∅) = 0

And 2. 퐼푓 (푃푖)푖∈퐼푁퐴 and 푃 = ⋃푖∈퐼푁푃푖 ∈ 퐴 then

푚(퐵) = ∑ 푚(퐵푛) For Countable or finite collection of pairwise disjoint set 퐵푛 in D such 𝜌(푃) = ∑ 𝜌(푃푖) that 퐵 =∪ 퐵푛 is also present in D. If D is 𝜎 finite 푖∈퐼푁 and m become bounded, then m is extending Let’s assume that R is the ring of subset (which is uniquely to measure defined on 𝜎 Algebra closed under relative complement and union) of generated by D. The m is known as probability fixes set say X. Here 휇0 : 푅 and R varies from 0 to measure on set Z if m (Z) =1 and D is 𝜎 Algebra. ∞ is set function. 휇0 termed as premeasure if it Basically measure is real value function on power satisfies bellows conditions: set P(S) of infinite set X. Probability measure satisfies the bellows property. 휇0(∅) = 0 … … … … … … … … … … (2) 1. 휇(∅) = 0 푎푛푑 휇(푆) = 1 It represents the empty sets, 2. 퐼푓 푍 ⊆ 푌 푡ℎ푒푛 휇(푧) ≤ 휇(푦), 𝜎 And for every finite or countable sequence {푃푛}푛∈푁 ⊆ 푅 of the pairwise disjoint set 3. 휇({푝} = 0 푓표푟 푎푙푙 푝 휖 푋 and its union lies in set R, 4. 퐼푓 푆 , 푛 = 0,1,2,3 … … … … … … Are pairwise ∞ 푛 휇0(⋃푛=1푃푛) disjoint set, then ∞

∞ = ∑ 휇0(푃푛) … … … … … … … … … … (3) ∞ 휇(⋃푛=0푍푛) = ∑ 휇푍푛 … … … … … … … … (1) 푛=1 푛=0 This property is known as 𝜎 additivity [7] and [8]. The measure m can extend by completion. A The addition or composition of two measurable Subset of set with measure zero forms functions with the continuous function must be 훿 푟𝑖푛푔 퐽.From set J, the set D changes so again measurable. All the sensible settlements are 훿 푟𝑖푛푔 퐷푐 which make set D will completed with correct but proven details or proof of that reference to obtained measure m. statement occupies large space. The measure (m) is a completed if 퐷 = 퐷푐.If I.3 Prerequisites mathematical Details measure is not completed then is extending to 퐷 푐 Definitions1.3.1: Let assume P is the set. A by satisfying the condition 푚((푃 ∖ 푄)⋃(푄 ∖ collection of m of subset of P is known as 𝜎 푃)) = 푚(푃) 푤ℎ푒푟푒 푃휖퐷 푎푛푑 푄휖 퐺 [6]. algebra in P, if it satisfies the following condition The triple (P, Q, 휇 ) of the non-empty set P. The 𝜎 1. 푃 ∈ 푚 algebra of subset of P and a measure 휇 on Q is termed measure space. Two conditions such as 2. 푃 − 퐴 ∈ 푚 푓표푟 퐴 ∈ 푚

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∞ 3.⋃푛=1푋푛 ∈ Remark 1.5.2: Property of 휇 used in proposition 푚 푓표푟 푓𝑖푛𝑖푡푒 푐표푙푙푒푐푡𝑖표푛 푤ℎ푒푟푒 푛 푣푎푟푦𝑖푛푔 푓푟표푚 1 푡표 ∞1.5.1 is known as monotonicity of µ. [9] and [10]. Remark: 1.5.3 휇(푃) < ∞ ,this condition is extra Definition1.3.2: s is the nonempty subset of set P which is given in proposition 1.5.1 because is termed as semi ring where X and Y belongs to s sometimes it is happens that 휇(푄 − 푃) < ∞ but then (푋 ∩ 푌) ∈ 푠 . There are also presence of both 휇(푃)푎푛푑 휇(푄) are equal to ∞.Let’s see the 푛 finite disjoint collection {퐾푙}푙=1 of set in s having example, 푃 = 푅 , 퐴 = 퐵(푅) and 휇 is the length of 푛 condition 푋 − 푌 = ⋃푙=1퐾푙 measure on P. Consider Q= 푅,P=Set of irrational Here collection of s of subsets of P is known as element in 푅.Then 푄 − 푃 = 푄 is the finite subset ring of set. It is closed with reference to evolution of 푅,hence 휇(푄 − 푃) = 0.But 휇(푄) = 휇(푃) = ∞ of finite unions and corresponding complements [ 15]. and hence with reference to evolution of finite or I.6 Construction of Premeasure: countable intersection [2] and [11]. Proposition1.6.1: Consider 푠 ⊂ 2푧 is the semi Definition 1.3.3: From the measurable space, algebra,퐴 = 퐴(푠)푎푛푑 휇 ∶ 퐴 → [0, ∞] is the finite mean a pair (P, m) consist of set P and 𝜎 algebra additive measure. Hence µ is the premeasure on 퐴 m of subsets of P.A subset Z of P is termed as if µ is the sub additive on s. measurable with reference to m supplied Z ∈ m Proof: If µ is the premeasure on set 퐴 then µ [12] and [13]. becomes sigma (σ) additive. It becomes sub Definition1.3.4: From the measure space(푃, 푄, 휇), additive on set s. means a measurable space (P, Q) together defines Suppose 푃 = ∑∞ 푃 with 푃 ∈ 퐴 and each the measure 휇 on Q [13] and [14]. 푛=1 푛 element 푃푛 ∈ 퐴 which can be written as Definition 1.3.5: A countable additive unique se 푘 function s which is defined on delta ring or 푃 = ∑ 퐸 푤𝑖푡ℎ 퐸 ∈ 푠 … … … … … (4) algebra 퐴 is known as premeasure on 퐴 and 푗 푗 푠(∅) = 0 [12] and [13]. 푗=1 And Definition 1.3.6: A set function 휇∗ = 2푧 → [0, ∞] is known as supplied 푁푛 ∗ 휇 (∅) = 0 and it is finitely single or unique [13]. 푃푛 = ∑ 퐸푛,푖 푤𝑖푡ℎ 퐸,푖 ∈ 푠 … … … … … … (5) I.4 Types of measure: 푖=1 Then get the equation There are number of measure are present such as ∞ Radon Measure, Borel Measure, Gauss Measure, 퐸푗 = 푃 ∩ 퐸푗 = ∑푛=1 푃푛 ∩ 퐸푗 = Euler Measure, Hausdorff Measure, Haar ∞ 푁푛 ∑푛=1 ∑ 퐸푛,푖 ∩ 퐸푗 … … … … … . (6) Measure, Integral, Helson-Szegö Measure, 푖=1 Lebesgue Measure, Jordan Measure, Mahler This equation is finite union and using assumption Measure, Liouville Measure, Minkowski ∞ 푁푛 Measure, Probability Measure, Wiener Measure 휇(퐸 ) ≤ ∑ ∑ 휇(퐸 and Ergodic Measure [7] and [8]. 푗 푛,푖 푛=1 푖=1 I.5 Properties of measure: ∩ 퐸푗) … … … … … … . . (7) Proposition 1.5.1: Consider (푃, 푄, 휇) is a Adding this equation on the j with using countable measure space. P and Q belongs to 퐴 in such way additivity property of µ shows that 푃 ⊆ 푄 and hence 휇(푃) ⊆ 휇(푄).In case of addition, The subset Q satisfies 휇(푃) < ∞ then 휇(푄 − 푃) ⊆ 휇(푄) − 휇(푃). Proof: The set P and 푄 − 푃(= 푄 ∩ 푃푐) are belongs to 퐴 and disjoints. Now then use the aspect countable or finite additivity of the 휇.

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푘 Hence it becomes unambiguous for defining 휇(푃) 휇(푃) = ∑ 휇(퐸푗) by 푗=1 휇(푃) = 휇(푃 ) … … … … … … … (13) 푘 ∞ 푁푛 푛 For some sequence {푃 }∞ ⊂ 퐴 such way 푃 ↑ 푃 ≤ ∑ ∑ ∑ 휇(퐸푛,푖 푛 푛=1 푛 with the help of definition, continuity of measure 푗=1 푛=1 푖=1 휇 is understandable and monotonic of 휇 following ∩ 퐸푗) … … … … … … … . . (8) the equation 11. ∞ ∞ 푘 ∞ 푁푛 Consider 푃, 푄휖퐴휎 and {푃푛}푛=1 and {푄푛}푛=1 are the sequence in 퐴 such way 푃 ↑ 푃 and 푄 ↑ 푄 for = ∑ ∑ ∑ 휇(퐸 ∩ 퐸 ) 푛 푛 푛,푖 푗 푛 → ∞ .It passes through identity 푗=1 푛=1 푖=1 휇(푃 ∪ 푄 ) + 휇(푃 ∩ 푄 ) 푛 푛 푛 푛 = 휇(푃 ) ∞ 푁푛 푛 + 휇(푄푛) … … … … … … . . (14) = ∑ ∑ 휇(퐸푛,푖) 푛=1 푖=1 Here 휇 is finitely additive on the 퐴휎. ∞ Let consider {푃푛}푛=1 is any sequence in the ∞ ∞ 퐴 .select the {푃 } ⊂ 퐴 which follows the 휎 푛,푖 푖=1 = ∑ 휇푃푛 … … … … … … … … … . . (9) 푃푛,푖 ↑ 푃푛 since 𝑖 → ∞. 푛=1 Then we get , Hence sub additivity of µ on the 퐴 is proved. 푁 푁 I.7 Construction of measure. 푁 휇(⋃푛 푃푛,푁) ≤ ∑ 휇(푃푛,푁) ≤ ∑ 휇(푃푛) As µ is the premeasure on the 퐴 where 퐴 → [0, ∞] 푛=1 푛=1 ∞ ∞ if 휇(퐴푛) ↑ 휇(푃) for {푃푛}푛=1 ⊂ 퐴 like 푃푛 ↑ 푃 ∈ 퐴 If the 휇(푃) < ∞ then 휇 is the premeasure on ≤ ∑ 휇(푃푛) … … … … … … (15) ∞ algebra 퐴 if 휇(퐴푛) ↓ 0 for {푃푛}푛=1 ⊂ 퐴 푛=1 since 푃 < ∅. 푁 푁 푛 Here 퐴 ∋ ⋃푛=1푃푛,푁 ↑ ⋃푛=1푃푛휖퐴휎 .If 푁 → ∞, the Proposition1.7.1: Consider 휇 a premeasure on previous equation holds the bellows equation of algebra 퐴, the n the 휇 has unique extension to sub-additivity function 퐴 , It satisfies the continuity, ∞ 휎 휇(⋃푛=1푃푛) monotonicity, strong additivity, sub additivity and ∞ sigma additivity. ≤ ∑ 휇(푃푛) … … … … … … … … (16) 푛=1 Proof: Assume that P, Q are the sets in the 퐴휎 such way 푃 ⊂ 푄 .Consider {푃 }∞ 푎푛푑 {푃 }∞ ∞ 푛 푛=1 푛 푛=1 Further assumption {푃푛,푖} ⊂ 퐴휎 is disjoints are queue in 퐴 as 푃 ↑ 푃 and 푄 ↑ 푄 where 푛 → 푖=1 푛 푛 sequence with countable additivity along with ∞.Since 푄 ∩ 푃 ↑ 푃 where 푚 → ∞.The 푚 푛 푛 monotonic property of µ on algebra퐴 . So finally continuity for measure 휇 on 퐴. 휎 get the equation 휇((푃푛) = 휇(푄푚 ∩ 푃푛) ≤ 휇 (푄푚) … … … … … (10) ∞ 푁 푁 Consider 푛 → ∞ , ∑ 휇(푃푛) = ∑ 휇(푃푛) = 휇(⋃푛=1푃푛) 푛=1 푛=1 휇( 푃푛) ≤ 휇 (푄푚) … … … … … … . (11) ∞ ≤ 휇(⋃푛=1푃푛) … … … … … … . (17) With the help of above equation Q=P, suggested, Consider µ is the finite premeasure on 푍 휇( 푃푛) = 휇 (푄푚) … … … … … … . (12) algebra, 퐴 ⊂ 2 , and 푃 ∈ 퐴훿 ∩ 퐴휎.So we have two conditions 푃, 푃푐 ∈ 퐴 and 푍 = 푃 ∪ 푃푐 which Whenever 휎 follows the 휇(푍) = 휇(푃) + 휇(푃푐). 푃 ↑ 푃 And 푄 ↑ 푄. 푛 푛

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From all above observations µ is extended to the Consider 𝜌: 퐴 → [0, ∞] is the premeasure on function on 퐴훿 ∪ 퐴휎 with definition of algebra 퐴 of Z. 푀 = 〈퐴〉 is the algebra generating 푐 퐴. 휇(푃) ≔ 휇(푍) − 휇(푃 ) for all values of 푃 ∈ 퐴훿 [16]. 1. It is special measure 휇: 푀 → [0, ∞] such I.8 Outer measures from premeasure: way that휇|푀 = 𝜌. Theorem 1.8.1: Consider 휇∗ is the outer measure 2. If 푣: 푀 → [0, ∞] such way 푣|푀 = 𝜌 then on 2푧.Then collection m of sets are measurable 푣(푃) < 휇(푃) For set 푃 ∈ 푀 and 푣(푃) < with reference to 휇∗ is the sigma (𝜎) algebra. If 휇 휇(푃) 𝑖푓 휇(푃) < ∞ . is the limitations of 휇∗ to m, then (P,Q, 휇) is entire Proof: From previous discussion it is clear that 휇 measure space [7] and [13]. can be obtained by restriction of 푀.Now consider I.9 Extension of measure: 푣 is one more measure on 푀, such way 푣|퐴 = It is the easiest way to define measure on the basis 휇.There are few things about u and v. of small collection of and stating the 푃푖 ∈ 퐴 and 퐸 = ⋃푖∈퐼푁푃푖 Then with consistency condition which allows extending approximation of 휇(퐸) in the form of sum measure to power set. Power set is nothing but the ∑ 휇(푃 ) ,this is get for all elements of probability measures on possible uncountable and 푖∈퐼푁 푖 퐸 ∈ 푀 infinite sets such as power set of naturals. 푣(퐸) = 휇(퐸)……………. (18) Theorem1.9.1: Carath´eodory’s definition Let 휇(퐸) < ∞ and 퐸 ∈ 푀 ,with approximation For the outer measure휇∗: 2푍 → [0, ∞], it is subset 푃 ∈ 퐴 on the basis of definition of 𝜌0,will implies E of Z measurable supplied for each subset P of Z, 푖 휇(푃) = 푣(퐸)1 ≤ 휇∗(푃) = 휇∗(푃 ∩ 퐸) + 휇∗(푃 ∩ 퐸푐) [2],[14] and[15]. So finally get 푣(퐸) = 휇(푃) [19]. The outer measure 푚∗ on Z is known as metric There are some other extension theorems also outer measure with respect to Carath´eodory if it used such as Dynkin’s π − λ theorem. In this there satisfies, are two systems comes in picture i.e. π system and λ system. Collection of the subsets is closed in 푚∗(푃 ∪ 푃 ) = 푚∗(푃 ) + 푚∗(푃 ) 1 2 1 2 nature for the finite or countable intersections in Whenever 푑(푃 , 푃 ) [17]. 1 2 case of π system while collection of subset is Theorem1.9.2: (Carath´eodory’s Extension closed having disjoints and complementation Theorem) finite unions in case of λ system. So the class Consider µ is measure on algebra. µ∗ is outer having both π system and λ system becomes measure introduced by µ.Then limitation 휇 of µ∗ 𝜎 algebra [20].Carath´eodory-Hahn Theorem is important in case of product measure. to µ∗ is measurable sets the extension of µ to 𝜎 algebra contains with 퐴.It the µ is 𝜎 finite or I.10 Conclusion: countable, then 휇 is only measure on small 𝜎 From the above discussion, it is concluded that algebra contains with 퐴 which is the extension of Measures have most formal properties than the µ [13] and [18]. sets. In this paper, we have studied the basics of measure and the premeasures along with their 1.9.3 Extension of premeasure: The construction having mathematical analysis. Also Carath´eodory-Hahn Theorem focused the extensions for measures and Let consider µ: s→ [0, ∞] is premeasure on premeasures using Carath´eodory theorem. Little semiring s of subset of Z. Then Carath´eodory bit basic part of Dynkin’s π − λ theorem and measure 휇 introduces by µ is the extension of µ. Carath´eodory-Hahn Theorem is highlighted. This Moreover, if µ is the 𝜎 finite =휇.The 휇 is unique complete review paper introduces the measure theory. measure on 𝜎 algebra of µ∗ is measurable sets which extends µ [13].

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References [20] Sayan Mukherjee, Extension of measure, [1] Book by Stein and Shakarchi ,Notes on “A pp.1-7. Very Brief Review Of Measure Theory”. [2] Fremlin D. H. Measure Theory. Volume 1. Torres Fremlin; 2000. [3] Ayekple Y.E , William O.D. and and Amevialor J.(2014). A Review of the Construction of Particular Measures. Physical Science International Journal,4(8),pp1211-1217. [4] Measure, Encyclopedia of Mathematics. [5] Czyz, J. Paradoxes of Measures and Dimensions Originating in Felix Hausdorff's Ideas. Singapore: World Scientific, 1994. [6] Jech, T. J. Set Theory, 2nd ed. Berlin: Springer-Verlag, p. 295, 1997. [7] Stanisław Łowjasiewicz, translated by Lawden G. H. An Introduction to the Theory of Real Functions. Wiley and Sons Ltd; 1988. [8] M. Papadimitrakis (2004). Notes on Measure Theory. Department of Mathematics,University of Crete ,pp.1-261. [9] Cohn D. L. Measure Theory. Birkh¨auser; 1980. [10] Weiz¨acker H. V. Basic Measure Theory. Revised translation. [11] Rudin W. Principles of Mathematical Analysis. Third edition. McGraw Hill; 1976. [12] Dshalalow Jewgeni H. Real Analysis, An Introduction to the Theory of Real Functions and Integration. Chapman & Hall/CRC; 2001. [13] Royden H. L, Fitzpatrick P. M. Real Analysis. Fourth edition, Prentice Hall; 2010. [14] Bartle R. G. The Elements of Integration and Lebesgue Measure. Wiley and Sons Inc; 1995. [15] Shyamashree Upadhyay (2012). Measure theory (MA 550) lecture notes. IIT Guwahati. [16] Bruce K. Driver(2006). Math 280 (Probability Theory) Lecture Notes.pp.1-59. [17] Wheeden R. L., Zygmund A. Measure and Integral, An Introduction to Real Analysis. MarcelDekker Inc; 1977. [18] Halmos P. R. Measure Theory. Springer- Verlag New York Inc; 1974. [19] Notes on Measure Theory and Complex Analysis, Math 103,Fall 2018.

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