Abstract:
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.
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