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Real Analysis pp 171-219 | Cite as

Topics on Measurable Functions of Real Variables

  • Emmanuele DiBenedetto
Part of the Birkhäuser Advanced Texts book series (BAT)

Abstract

Letfbe a real-valued function defined and bounded in some interval \(\left[ {a,b} \right] \subset \mathbb{R}\). Denote by
$$\mathcal{P} \equiv \left\{ {a = {{x}_{0}} < {{x}_{1}} < \cdots < {{x}_{n}} = b} \right\}$$
a partition of[a,b]and set
$${{v}_{f}}\left[ {a,b} \right] = \begin{array}{*{20}{c}} {\sup } \\ {\text{P}} \\ \end{array} \sum\limits_{{i = 1}}^{n} {\left| {f\left( {{{x}_{i}} - f\left( {{{x}_{{i - 1}}}} \right)} \right)} \right|}$$
This number, finite or infinite, is called thetotal variationoffin[a,b].IfVf [a,b]is finite, the functionfis said to be ofbounded variationin[a,b].

Keywords

Measurable Function Convex Function Bounded Variation Real Variable Radon Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Emmanuele DiBenedetto
    • 1
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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