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)


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].


Measurable Function Convex Function Bounded Variation Real Variable Radon Measure 
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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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