On Almost Convergent and Statistically Convergent Subsequences

There are two well-known non-matrix summability methods which we will consider, namely “almost convergence” and “statistical convergence”. The results presented in this paper will be of two types, dealing with Lebesgue measure and Baire category. Establishing a one-to-one correspondence between the interval (0; 1] and the collection of all subsequences of a given sequence s = (s _n), we will examine the measure and category of the set of all almost convergent subsequences of (s _n). Similar questions for statistical and lacunary statistical convergence are considered. Results on rearrangements of sequences are also presented. This revised version was published online in June 2006 with corrections to the Cover Date.