# On a result of Maly, Preiss and Zajicek

### Anal. St. Univ. Ovidius Constanta 5 (1996), no. 1, 47-50.

#### Mathematical reviews subject classification: 26A24; 28A15.

## Abstract

Maly, Preiss and Zajicek proved the following result:

Let α ∈ (0,1) and let M ⊆ [a,b) such that a ∈ M; suppose that for any x ∈ M there is y ∈ (x,b] such that

μ*(M ∩ (x,y)) ≥ α (y-z);

Then there exists z ∈ (a,b] \ M such that

μ*(M ∩ (y,z)) ≥ α(z-y)

whenever y ∈ [a,z)

(see Corollary 2 of J. Maly, D. Preiss, and L.Zajicek, *An unusual monotonicity theorem with applications*, Proc. Amer. Math. Soc., **102** (1988), no. 4, 925-932).

The aim of this article is to find equivalent forms of this result.