LEBESGUE THEORY OF INTEGRATION

The general theory of measure and integration was developed in 20^th century and is named after a French mathematician Henri Lebesgue. This theory is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. It is now used in different fields of mathematics such as probability theory, partial differential equations, functional analysis, harmonic analysis and dynamic systems.  Measures are the fundamental building blocks of probability theory and integration theory.

In chapter 1, an abstract concept of outer measure and Lebesgue measure is defined and their properties are derived in detail. It is observed that Outer measure is positive extended real number, Outer measure of empty set is zero and outer measure is monotonic function. Outer measure of singleton set is zero. Outer measure is countable sub-additive. Outer measure of countable set is zero. Outer measure of an interval is length of interval.

After defining outer measure, theory of Measurable sets and Boral sets is introduced and their properties are derived. It is observed that Finite and countable union and intersection of measurable sets is measurable, Difference of two measurable sets is measurable, Every Interval is measurable, Boral sets are measurable, Outer Measure is Translation Invariant.

Concept of algebras and σ- algebra of Lebesgue measurable sets are introduced in this chapter and it is observed that class of measurable sets is  –algebra. At the end of this chapter it is observed that for a set of positive measure, there exist a non-measurable subset in it. Cantor set is defined and it is found that Cantor Set is a closed, uncountable set of measure zero.

In chapter 2, Concept of measurable functions, Borel measurable function and Null Set is introduced and it is observed that a function is measurable inverse image of each open set under it is measurable set and A measurable real- valued function is approximately a simple function. Every countable set is null set. Subset of Null Set is Null Set. Countable union of Null Sets is Null Set. Linear combination of measurable functions is measurable. Square of measurable function is measurable. Product of measurable functions is measurable. Continuous real valued function defined on measurable domain is measurable. Constant function with measurable domain is measurable. Composition of measurable functions is measurable. Point wise limit of measurable functions is measurable. A function is simple if it measurable and takes only a finite numbers of values. At the end of this chapter very important Littlewood’s Three Principles deduced and it is concluded that a measurable real- valued function is approximately a simple function and real-valued measurable function is nearly continuous.

In chapter 3, Lebesgue Integration of some functions such as bounded function over a set of finite measure, non-negative measurable functions and general Lebesgue integral is defined and their properties such that Linearity and monotonicity of integration, Bounded convergence theorem, Fatou’s lemma, Monotone convergence theorem. The general Lebesgue integral and Lebesgue dominated convergence theorem, General Lebesgue dominated convergence theorem, countable additivity and continuity of integration. Step function takes finite number of values. Riemann integrable bounded function defined on a closed interval [a, b], Riemann and Lebesgue integrals are equal.  Every bounded measurable function on a set of finite measure is Lebesgue integrable.

In Chapter 4, Concepts of Vitali covering lemma, Lebesgue’s Theorem are discussed. Concept of Functions of Bounded variations is introduced and their properties are observed. . Every monotonic function defined on closed interval is function of bounded variation. If a function is not bounded, then it is not of bounded variation. Constant function defined on bounded and closed interval is function of bounded variation.

Concept of indefinite integral is introduced and it is observed that Indefinite integral of a function is continuous. Indefinite integral of a function is of bounded variation. Concept of Absolutely Continuous function is defined and it is observed that every absolutely continuous function is uniformly continuous. If a function is absolutely continuous, then it is of bounded variation. Every absolutely continues function is an Indefinite Integral.B

The material in this book has formed the basis of lecture and seminars for post graduate students of Mathematics. The presentation is easy and elaborated. The concepts and methods in this book are abstract. Some important examples where needed are also discussed.