ABSTRACT. The paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator Φ J generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273-286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.

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Abstract. The density topologies with respect to measure and category are motivation to consider the density topologies with respect to invariant σ-ideals on R. The properties of such topologies, including the separation axioms, are studied. NotationBy R we shall denote the set of all reals numbers and by N the set of positive integers. Let l stand for Lebesgue measure. The capitals L and L denote the σ-algebra of all Lebesgue measurable sets in R and the σ-ideal of all Lebesgue null sets. The natural topology on R is denoted by T 0 . If T is a topology on R, then we fix the notation: B(T ) -the σ-algebra of all Borel sets with respect to T , Ba(T ) -the σ-algebra of all sets having the Baire property with respect to T , K(T ) -the σ-ideal of all meager sets with respect to T .2000 Mathematics Subject Classification. 28A05, 54A10. Key words and phrases. Density point, density topology, the separation axioms, invariant ideals and algebras.

The paper concerns the topologies introduced in the family of sets having the Baire property in a topological space (X, τ) and in the family generated by the sets having the Baire property and given a proper σ-ideal containing τ-meager sets. The regularity property of such topologies is investigated.

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