Real Analysis Part 5 (Lebesgue Measure)

Measurable Sets and Lebesgue Measure

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Description

LEBESGUE MEASURE Part 1

'Measurable Sets and Lebesgue Measure'

In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.

And, A measurable set was defined to be a set in the system to which the extension can be realized; this extension is said to be the measure.

Contents of the Course:

In this Course you will learn about ' The Algebra of Sets' , 'Borel Sets' including ' 'Outer Measure'_ Outer measure of a set

                             _ Outer measure of an interval

A very important property ' Countable Subadditivity' is almost used in many theorems and Propositions. Then you will also learn about Measurable Sets, Family of Measurable sets and also the Existence of a Non Measurable set.


Next, the Definition of Lebesgue Measure and its importance in pairwise disjoint measurable sets, Infinite decreasing sequence of Measurable Sets. The Lemma that shows that Lebesgue measure is invariant under Translation modulo 1.


And, Finally at last ; you will learn about 'Measurable Functions' and its importance in extended real valued function having measurable Domain with Equivalent Properties.

                Also if two measurable real valued functions are having same domain with constant then you will learn the results that the sum, difference, product of these functions are also measurable, including a very Important Principles called

'LITTLEWOOD'S THREE PRINCIPLES'.

All the important theorems, Propositions, Lemma and Solved Examples are covered based on all above mentioned contents.


Thanks.


What You Will Learn!

  • Students will learn about Details of Outer Measure, Lebesgue Measure, Measurable Set and Measurable Functions including all expected Theorems and Prepositions.
  • The Lebesgue Measure is the standard method of allocating measure to subsets of n-dimensional Euclidean Space.
  • Algebra of sets, Borel Sets,Sigma Algebra , Smallest sigma algebra Family of Measurable Functions and Sets.
  • To learn that outer measure of an interval is its length and Countable Subadditivity and many more Results based on measurable sets and Lebesgue Measure.

Who Should Attend!

  • Graduate Bsc students, Msc. maths students, for UGC NET Entrance Exams