$\mu$-statistical convergence and the space of functions $\mu$-stat continuous on the segment

Abstract

In this work, the concept of a point $\mu$-statistical density is defined. Basing on this notion, the concept of $\mu$-statistical limit, generated by some Borel measure $\mu\left(\cdot \right)$, is defined at a point. We also introduce the concept of $\mu$-statistical fundamentality at a point, and prove its equivalence to the concept of $\mu$-stat convergence. The classification of discontinuity points is transferred to this case. The appropriate space of $\mu$-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.