Abstract
As mentioned in the previous chapter, the problem of reconstructing a function from its derivative leads to the concept of absolute continuity. We begin this chapter with the standard results from introductory texts. Lebesgue points are then introduced. Next we give an extensive collection of sufficient conditions on the derivative to ensure absolute continuity of a function; the hypotheses of these theorems are considerably weaker than those found in introductory texts. Of major importance in establishing absolute continuity is to find conditions on a function or its derivative to ensure that the function possesses the property of mapping null sets into null sets. A counterexample of this property is given in Chapter 8. Emphasis is placed on the relationship between bounded variation and absolute continuity, first by giving conditions to ensure that a function of bounded variation is absolutely continuous; again, the property of mapping null sets into null sets plays an important role. We then introduce singular functions, leading to a decomposition of a function of bounded variation into the sum of three functions, one of which is absolutely continuous. Singular functions will be discussed extensively in Chapter 8. Integration by parts and change of variable results are presented. The chapter concludes with a section on the rectifiability of curves; in this section, we consider vector-valued functions. Rectifiability will be revisited in Chapter 9.
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© 1996 Springer-Verlag New York, Inc.
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Kannan, R., Krueger, C.K. (1996). Absolute Continuity. In: Advanced Analysis. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8474-8_8
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DOI: https://doi.org/10.1007/978-1-4613-8474-8_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-94642-9
Online ISBN: 978-1-4613-8474-8
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