Advances in almost convergence

In this paper, we first give the concept of properly distributed sequence, and prove that it is almost convergent with F-limit expressed as a formal integral. Basing on these, we review the work of Feng and Li, which is shown to be a special case of our generalized theory. Then we generalize Banach limit to Banach limit functional, which is the minimum requirement to characterize strong almost convergence for bounded sequences in normed vector space. With this machinery, we show that Hajduković’s almost convergence and quasi-almost convergence are both equivalent to our strong almost convergence.