A Besicovitch-Morse function preserving Lebesgue measure is constructed and shown to form a meager set among continuous measure-preserving maps.
Saks, On the functions of Besicovitch in the space of continuous fu nctions, Funda- menta Mathematicae 19 (1932), 211–219
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A Besicovitch-Morse function preserving the Lebesgue measure
A Besicovitch-Morse function preserving Lebesgue measure is constructed and shown to form a meager set among continuous measure-preserving maps.