A BRIEF STUDY OF GENERAL MEASURESPACES AND INTEGRATIONA PROJECT REPORT SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCEINMATHEMATICAL SCIENCESByBishwa jit Gohain Roll No : MSM17036M.Sc. in MathematicsUNDER THE GUIDANCE OFDr. BIPUL KUMAR SARMAAssistant Professor, Department of Mathematical SciencesTezpur University, IndiaDEPARTMENT OF MATHEMATICAL SCIENCES, TEZPUR UNIVERSITY, TEZPUR.November, 2018.AcknowledgementsI am extremely grateful to my supervisor Dr. BIPUL KUMAR SARMASir for his constant guidance and expert comments during the entire course ofmy pro ject work. Without Sir’s help and valuable informations, I would nothave been able to complete this pro ject work successfully.
I also thankful to myfellow classmates and friends for their supports.Certi cateThis is to certify that the report entitled “A BRIEF STUDY OF GENERALMEASURE SPACES AND INTEGRATION ” submitted to the Departmentof Mathematical Sciences, Tezpur University is a record of pro ject work carriedout by BISHWAJIT GOHAIN (Enrollment no : 121 MSM17036) under mysupervision and guidance, in partial ful lment of the requirements for theaward of of the degree in Master of Science program in Mathematical Sciences.
(Dr. B. Kr. Sarma)Assistant ProfessorDepartment of Mathematical SciencesTezpur UniversityDate: Dec, 2018DeclarationI hereby declare that the report entitled ” A BRIEF STUDY OF GENERALMEASURE SPACES AND INTEGRATION” submitted by me is the resultof my own pro ject work carried out under the guidance of Dr. B. Kr. Sarma,Assistant Professor, Department of Mathematical Sciences, Tezpur University,India.(Bishwa jit Gohain)Department of Mathematical SciencesTezpur UniversityDate: Dec, 2018Contents1 AbstractIn this pro ject, we have to study general measure spaces , their propertiesand integration over it .In the rst section we introduce sigma algebras andtheir properties as basic concepts.The other sections include sign measure itsproperties,measure spaces and integrati0n. We also introduce here some mostimportant theorems such as hahn decomposition theorem, jordan decompositiontheorem,fatou lemma, monotone convergence theorem,radon nikodym theoremetc .12 IntroductionBasically our aim is to study about general measure space their propertiesand integration over it. In the section 3, we discuss sigma algebra and somebasic concepts of measure theory. In section 4 we study about the measure andmeasurable sets,and properties of measure space.In section 5,we discuss Signedmeasures , its theorem and the Hahn and Jordan decomposition theorem. In the last section we discuss integration over general measure spaces bythe above section to the consideration of measurable functions. Later westudy integration of non-negative measurable measurable functions,integrationof general measurable functions and some more important theorem RadonNikodym theorem etc.23 PRELIMINARIES3.1 algebras3.1.1 De nitionsDe nition 3.1.1. Let X be a non-empty set and a collection of subsetsof X. We call a algebra of subsets of X if it is non-empty, closed undercomplements and closed under countable unions. This means:(i) there exists at least one A X so that A 2 ,(ii) if A 2 , then A c2 , where A c= X A; and(iii) if A n2 for all n 2N, then 1Sn =1 An 184.108.40.206 ExamplesExample 3.1.2. The collection,X is a sigma algebra of subsets of X.Example 3.1.3. If E X is non-empty and di erent from X, then the collectionf , E, E c; X gis a sigma algebra of subsets of X.Example 3.1.4. P(X), the collection of all subsets of X, is a -algebra ofsubsets of X.3.1.3 PropositionsProposition 1. Every algebra of subsets of X contains at least the sets and X, it is closed under nite unions, under countable intersections, under nite intersections and under set-theoretic di erences.3Proposition 2.Letbe a -algebra of subsets of X and consider a nitesequence of fAng Nn =1 or an infnite sequencefAngin . Then there exists a nite sequence fBng Nn =1 or, respectively, an in nite sequencefBngin withthe properties:(i) B n An for all n = 1,…,N or, respectively,all n2N.(ii) NSn =1 Bn= NSn =1 Anor 1Sn =1 Bn= 1Sn =1 An respectively(iii) the B ns are pairwise disjoint.44 MEASURE SPACE4.1 De nitionsDe nition 4.1.1. The pair (X;) of a non-empty set X and a algebra ofsubsets of X is called a measurable space.De nition 4.1.2. Let (X,) be a measurable space. A mapping : ! [0, 1]is called a measure if(i) . ( )=0 and(ii) SnEn)=n 2N (En)for all pairwise disjoint fEngn 2N inDe nition 4.1.3. A triple(X; , ) consisting of a non-empty set, a -algebra on it and a measure on is called a measure space.4.2 RemarkRemark 4.2.1.(i) Sometimes a set function is called a mapping whose domain is some nonempty set A of subsets of some set X .(ii) If the condition 2. in the de nition of the measure is weakened so that it is only required that (E1[::: [En)= (E1)+ … + (En), for n 2N, and pairwisedisjoint E1; :::; En, we say that the mappingis a nitely-additive measure.If we want to stress that a mapping satis es the original requirement 2. forsequences of sets, we say that is countably additive or -additive .De nition 4.2.2. Let (X,) be a measurable space. A measure on the(X, ) is called :(i) a nite measure, if (X) kg is open and so measurable .6.1.3 TheoremsTheorem 6.1.4. Let (X,M) be a measurable space. Let c is any real numberand let f and g be real valued measurable functions on M. Then f + c , cf , f+ g , f – g and f.g are also measurable .Proof.(1) Since each k ,[x:f(x)+c >]=[x:f(x) >k-c] is a measurable set, so f + c ismeasurable.(2) If c=0, cf is measurable; otherwise if c 1 ]= [ x:f (x ) > c 1k]is measurable set and similarly for c k] only if f(x) >k-g(x) that is only if there exist a rational risuch that f(x)>ri> k-g(x), where fri,i=1,2,3,…gis an enumeration of Q . But then g(x) >k-riand so x 2[ x :f(x) >ri ] [ x :g(x) >k-ri] . Hence A B = S1i = 1 ([x:f(x) >ri] [x :g(x) >k-ri]) a measurable set. Since A clearly contains B we have A = Band so f + g is measurable .(4) f – g = f + (-g) is also measurable .(5) Since fg = 1/4((f + g) 2-(f – g) 2), so it is su cient to show that f 2ismeasurable whenever f is . If k k ] = R is measurable. If k 0 , [x :f 2(x) >k] = [x : f(x) >p k] [x :f(x)