# 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.