Topics on Measurable Functions of Real Variables

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].