On a generalization of density topologies on the real line

Density topologies appear naturally in many considerations in real analysis. The most fundamental one is the classical density topology in the family of Lebesgue measurable sets. Namely, A⊂RA⊂R is open iff A⊂Φ(A)A⊂Φ(A), whereΦ(A)={x∈R:limh→0+⁡λ((A−x)∩[−h,h])2h=1} that is, when all points of A are Lebesgue density points of A.One of deeply studied problems in the last years (by, for example, M. Filipczak, J. Hejduk and R. Wiertelak) was the problem of weakening the definition of the Lebesgue density points (and, in turn, the definition of the operator Φ). Our paper deals with this problem – we try to define the most general (from a certain point of view) notion of density topology on the real line.