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Article 23

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.



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.