An approach to construct the singular monotone functions by using Markov chains
A nondecreasing function \(f\) is singular if it is continuous and \(f'(x) = 0\) almost everywhere with respect to the Lebesgue measure. The Cantor function is the best known example of nondecreasing singular functions. Example of strictly increasing singular functions have been constructed by many authors. See, for example, \textit{G. Freilich} [Am. Math. Mon. 80, 918-919 (1973; Zbl 0278.26010)], \textit{L. Takács} [Am. Math. Mon. 85, 35-37 (1978; Zbl 0394.26005)], \textit{F. Riesz} and \textit{B. Sz.-Nagy} [``Functional analysis'' (1956; Zbl 0070.10902)]. In this paper, the author presents a method for constructing singular monotone functions by using Markov chains. His approach is based on the strong laws of Markov chains.
scientific article; zbMATH DE number 1234924