Statistical convergence of Borel measurable functions at infinity via measure and modulus functions of order $\alpha$

Abstract

UDC 517.52; 517.5

We introduce and study a generalization of convergence for Borel-measurable functions at infinity by using a continuous version of density of order $\alpha$ based on the modulus functions (as defined in Section 2) with respect to a measure. Using this notion of density, we also define the corresponding statistical limit for Borel-measurable functions at infinity – independent of any measure – and show that it coincides with the classical limit of these functions at infinity. Furthermore, we identify and correct several inaccuracies in the results presented in [M. Altınok, M. Küçükaslan, A. K. Unay, J. Anal., 31, No. 2, 1487–1510 (2023)] and [B. Bilalov, S. Sadigova, Proc. Amer. Math. Soc., 143, No. 9, 3869–3878 (2015)] and propose a new condition for the measure that replaces the $\sigma$-finiteness assumption used in these works and ensures the validity of their results under our condition. In addition, we establish analogous results for the newly introduced concept.