ABSTRACT

In the [3] is proven that almost every, in terms of measure , A P subsequence ( ) x S of double sequence S converges to L in the Pringsheim’s sense, if and only if sequence S uniformly statistically converges to L. In this paper, it is proven that analogue is valid and for lacunary uniformly statistical convergence. Almost every, in terms of measure , A P subsequence ( ) x S of double sequence S converges to L in the Pringsheim’s sense, if and only if sequence S lacunary uniformly statistically converges to L. This is not true for measure P. Almost every, in terms of measure P, subsequence ( ) x S of double sequence S of 0’s and 1’s is not almost uniformly statistically convergent, if is sequence S lacunary uniformly statistically convergent and divergent in the Pringsheim’s sense. FIKRET ČUNJALO 26

In the [3] is proven that sequence Sij uniformly statistically converges to L if and only if it there is a subset A of the set N × N uniform density zero and subsequence S (x) defined by, Sij (x) = Sij for (i, j) ∈ Ac, converges to L, in the Pringsheim’s sense. In this paper it is proven that analog is valid for subsequence S (x) provided that for each N and i 6 N ∨ j 6 N is a set of all Sij (x) finite set. Is generally valid: If the subsequence S (x) uniformly statistically converges to L, then, there is a subset A of the set N× N uniform density zero and subsequence S (y) defined by, Sij (y) = Sij (x) for (i, j) ∈ Ac, converges to L, in the Pringsheim’s sense. If there is a subset A of the set N × N uniform density zero and subsequence S (y) defined by, Sij (y) = Sij (x) for (i, j) ∈ Ac, such that lim i→∞ ( lim j→∞ Sij (y)) = L, then, the subsequence S (x) uniformly statistically converges to L.

Almost-convergence of double sequences (subsequences) is equivalent to almost Cauchy condition. If the set of all almost convergent subsequences of a sequence S = Snm is of the second category, then S is convergent in the simple sense. For the sequence S = Snm which almost converges to L, Lebesgue measure of the set of all its subsequences which almost converge to L is either 1 or 0. .