μ-statistical convergence and the space of functions μ-stat continuous on the segment

Abstract

In this work, the concept of a point μ-statistical density is defined. Basing on this notion, the concept of μ-statistical limit, generated by some Borel measure μ (·), is defined at a point. We also introduce the concept of μ-statistical fundamentality at a point, and prove its equivalence to the concept of μ-stat convergence. The classification of discontinuity points is transferred to this case. The appropriate space of μ-stat continuous functions on the segment with sup-norm is defined. It is proved that this space is a Banach space and the relationship between this space and the spaces of continuous and Lebesgue summable functions is considered.