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Sergei Ovchinnikov - Measure, Integral, Derivative: A Course on Lebesgues Theory

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Measure, Integral, Derivative: A Course on Lebesgues Theory
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Sergei Ovchinnikov
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Springer
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Measure, Integral, Derivative: A Course on Lebesgues Theory: summary, description and annotation

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This classroom-tested text is intended for a one-semester course in Lebesgues theory. With over 180 exercises, the text takes an elementary approach, making it easily accessible to both upper-undergraduate- and lower-graduate-level students. The three main topics presented are measure, integration, and differentiation, and the only prerequisite is a course in elementary real analysis.

In order to keep the book self-contained, an introductory chapter is included with the intent to fill the gap between what the student may have learned before and what is required to fully understand the consequent text. Proofs of difficult results, such as the differentiability property of functions of bounded variations, are dissected into small steps in order to be accessible to students. With the exception of a few simple statements, all results are proven in the text. The presentation is elementary, where -algebras are not used in the text on measure theory and Dinis derivatives are not used in the chapter on differentiation. However, all the main results of Lebesgues theory are found in the book.

http://online.sfsu.edu/sergei/MID.htm

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Sergei Ovchinnikov Universitext Measure, Integral, Derivative 2013 A Course on Lebesgue's Theory 10.1007/978-1-4614-7196-7_1 Springer Science+Business Media New York 2013

1. Preliminaries

Sergei Ovchinnikov 1

(1)

Department of Mathematics, San Francisco State University, San Francisco, CA, USA

Abstract

Real analysis is a standard prerequisite for a course on Lebesgues theories of measure, integration, and derivative. The goal of this chapter is to bring readers with different backgrounds in real analysis to a common starting point. In no way the material here is a substitute for a systematic course in real analysis. Our intention is to fill the gaps between what some readers may have learned before and what is required to fully understand the material presented in the consequent chapters.

1.1 Sets and Functions

We write x A to denote the membership of an element x in a set A . If x does not belong to the set A , then we write x A . Two sets A and B are equal, A = B , if they contain the same elements, that is,

A set B is a subset of a set A , denoted by A B (equivalently, by B A ), if

Measure Integral Derivative A Course on Lebesgues Theory - image 2

Braces are frequently used to describe sets, so

Measure Integral Derivative A Course on Lebesgues Theory - image 3

denotes the set of all elements x for which the statement is true. For instance, the two element set {1,2} can be also described as

The operations of intersection union and relative complement are defined - photo 4

The operations of intersection , union , and (relative) complement are defined by

respectively, where A B is the difference between sets A and B .

There is a unique set , the empty set , such that x for any element x . The empty set is a subset of any set. A set consisting of a single element is called a singleton .

The Cartesian productA B of two sets A and B is the set of all ordered pairs ( a , b ) where a A and b B . Two ordered pairs ( a , b ) and ( a , b ) are equal if and only if a = a and b = b .

For two sets A and B , a subset f A B is said to be a function from A to B if for any element a A there is a unique element b B such that ( a , b ) f . We frequently write b = f ( a ) if ( a , b ) f and use the notation f : A B for the function f . The sets A and B are called the domain and codomain of the function f , respectively. For a subset A A the set

is the image of A under f The set f A is called the range of the function - photo 6

is the image of A under f . The set f ( A ) is called the range of the function f . The inverse imagef 1( B ) of a subset B B under f is defined by

If f A B the function f is said to be onto If for each b f A there - photo 7

If f ( A )= B , the function f is said to be onto . If for each b f ( A ) there is exactly one a A such that b = f ( a ), the function f is said to be one-to-one . A function f : A B is called a bijection if it is one-to-one and onto. In this case, we also say that f establishes a one-to-one correspondence between sets A and B . Given a bijection f : A B , for each element b B there is a unique element a A for which f ( a )= b . Thus the function

is well defined. We call this function, f 1 : B A , the inverse of f .

If A and B are sets of real numbers, then a function f : A B is called a real function . Real functions are the main object of study in real analysis.

For given sets A and J , a family { a i } i J of elements of A indexed by the set J (the index set ) is a function a : J A , that is, a i = a ( i ) for i J . The set { a i : i J } is the range of the function a . For Measure Integral Derivative A Course on Lebesgues Theory - image 9

Measure Integral Derivative A Course on Lebesgues Theory - image 9

and a A , we write Measure Integral Derivative A Course on Lebesgues Theory - image 10

if a = a i for some i J . If J is a subset of the index set J , then the family { a i } i J is called a subfamily of the family { a i } i J .

If the index set J is the set of natural numbers Measure Integral Derivative A Course on Lebesgues Theory - image 11

, a family Measure Integral Derivative A Course on Lebesgues Theory - image 12

is called a sequence of elements of the set A . It is customary to denote a sequence by ( a n ) or write it as

Measure Integral Derivative A Course on Lebesgues Theory - image 13

The element a n corresponding to the index n is called the n th term of the sequence. A sequence is an instance of a family. However, the former has a distinguished featureits index set Measure Integral Derivative A Course on Lebesgues Theory - image 14

Measure Integral Derivative A Course on Lebesgues Theory - image 14

is an ordered set. Thus the terms of a sequence ( a n ) are linearly ordered by their indices.

Let

Measure Integral Derivative A Course on Lebesgues Theory - image 15

be a family of sets, that is, each X i is a set. The intersection and union of are defined by and respectively Notations - photo 16

are defined by

and respectively Notations and - photo 17

and

respectively. Notations

and

are also common for these operations The following identities are known as - photo 21

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