Page 43 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 4: Absolute Continuity

CONTENTS
Objectives

Introduction
4.1 Absolute Continuity
4.1.1 Absolute Continuous Function

4.1.2 Theorems and Solved Examples
4.2 Summary
4.3 Keywords
4.4 Review Questions

4.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define Absolute Continuous function.
 Solve problems on absolute continuity
 Understand the proofs of related theorems.

Introduction

It may happen that a continuous function f is differentiable almost everywhere on [0,1], its
derivative f’ is Lebesgue integrable, and nevertheless the integral of f’ differs from the increment
of f. For example, this happens for the Cantor function, which means that this function is not
absolutely continuous. Absolute continuity of functions is a smoothness property which is
stricter than continuity and uniform continuity.

4.1 Absolute Continuity

4.1.1 Absolute Continuous Function

A real-valued function f defined on [a,b] is said to be absolutely continuous on [a,b], if for an
arbitrary 0, however small, a, 0, such that

n n
f b r f a r whenever b r a r ,
r 1 r 1

where a b a b ... a b n i.e. a ’s and b ’s are forming finite collection
1 1 2 2 n i i
a ,b : i 1,2,...,n of pair-wise disjoint intervals.
i i
Obviously, every absolutely continuous function is continuous.

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