On Foran's property (M) and its relations to Lusin's Property (N)
Real Analysis Exchange 9 (1984), 558-563.
Mathematical reviews subject classification: 26A30
Abstract
In An extension of the Denjoy integral (Proc. Amer. Math. Soc., 49 (1975), 359-365), J. Foran has constructed a continuous function F on [0,1] satisfying the Foran property (M), but which does not satisfy Lusin's property (N).
In Theorem 3 we show that such a function may be obtained by using a result due to S. Mazurkiewicz more than fifty years ago (see Sur les fonctions qui satisfont a la condition (N). Fund. Math., 16 (1930), 348-352).
Our strategy will be the following: we shall construct a continuous function F having on [0,1] Lusin's property (N) such that for any linear nonconstant function g, the function F+g has the Foran property (M), but does not have Lusin's property (N).
Our idea is inspired by Mazurkiewicz' result, that asserts the existence of a continuous function F having on [0,1] Lusin's property (N) such that F+g has the property (N) for no linear nonconstant function g.
As it will become apparent, Mazurkiewicz' paper anticipates implicitly the Foran property (M).
In Theorem 2 of our paper we indicate a new, shorter way to obtain the above mentioned Mazurkiewicz Theorem.
