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2004-2005 On a space of Besicovitch functions
Jozef Bobok
Real Anal. Exchange 30(1): 173-182 (2004-2005).


Let $C([0,1])$ be the set of all continuous functions mapping the unit interval $[0,1]$ into itself, equipped with the metric $\rho$ of uniform convergence (and the induced topology $\tau$). A function $f\in C([0,1])$ is called Besicovitch if it is nowhere one-sided differentiable (finite or infinite). For the Lebesgue measure $\lambda$ we define the set $B(\lambda)\subset C([0,1])$ by $$B(\lambda)=\{f\vert~\forall~{\rm Borel}~A\subset [0,1]\colon~\lambda(A)=\lambda(f^{-1}(A))\text{ and }f \text{ is Besicovitch}\}.$$ We construct a set $X\subset B(\lambda)$ such that the space $(X,\tau\vert X)$ is homeomorphic to the product topological space $(\prod_{i=0}^{\infty}[0,1),\mu)$.


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Jozef Bobok. "On a space of Besicovitch functions." Real Anal. Exchange 30 (1) 173 - 182, 2004-2005.


Published: 2004-2005
First available in Project Euclid: 27 July 2005

zbMATH: 1060.26009
MathSciNet: MR2127523

Primary: 26A18 , 37E05

Keywords: Besicovitch function , complete metric space , continuum

Rights: Copyright © 2004 Michigan State University Press

Vol.30 • No. 1 • 2004-2005
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