Vitali Lemma approach to differentiation on a time scale
Volume 162 / 2004
Studia Mathematica 162 (2004), 161-173
MSC: Primary 26A24, 28A15; Secondary 46G05, 39A05, 28A05.
DOI: 10.4064/sm162-2-4
Abstract
A new approach to differentiation on a time scale ${{\mathbb T}}$ is presented. We give a suitable generalization of the Vitali Lemma and apply it to prove that every increasing function $f:{{\mathbb T}}\rightarrow {\mathbb R}$ has a right derivative $f_{+}^{\prime } ( x) $ for $\mu _{\Delta } $-almost all $x\in {{\mathbb T}}$. Moreover, $\int _{[ a,b) }f_{+}^{\prime } ( x) \kern .16667em d\mu _{\Delta }\leq f ( b) -f ( a) .$
