Geodesic
Singular Strictly Increasing Functions and a Problem on Partitions of Closed Intervals
Matematičeskie zametki, Tome 81 (2007) no. 3, pp. 341-347.

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We establish that the problem of constructing a strictly increasing singular function is equivalent to the problem of constructing subsets $\mathscr P$ and $\mathscr Q$ of a closed interval $[a;b]\subset\mathbb R$ such that (1) $\mathscr P\cap\mathscr Q=\varnothing$; (2) $\mathscr P\cup\mathscr Q=[a;b]$; (3) the Lebesgue measures of the intersections of $\mathscr P$ and $\mathscr Q$ with an arbitrary interval $J\subset[a;b]$ are positive.
Keywords: singular function, Cantor set, perfect set, heavily intermittent partition, Borel set, completely additive function.
Mots-clés : Lebesgue measurable set 
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I. S. Kats. Singular Strictly Increasing Functions and a Problem on Partitions of Closed Intervals. Matematičeskie zametki, Tome 81 (2007) no. 3, pp. 341-347. http://geodesic.mathdoc.fr/item/MZM_2007_81_3_a3/

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