On double sequences of continuous functions having continuous P-limits II

Abstract

The goal of this paper is to relax the conditions of the following theorem: Let A be a compact closed set; let the double sequence of function s1,1 (x), s1,2 (x) s1,3 (x) . . . s2,1 (x), s2,2 (x) s2,3 (x) . . . s3,1 (x), s3,2 (x) s3,3 (x) . . . have the following properties: 1. for each (m, n) sm,n (x) is continuous in A; 2. for each x in A we have P - limm,n sm,n (x) = s(x); 3. s(x) is continuous in A; 4. there exists M such that for all (m, n) and all x in A |sm,n (x)| ? M. Then there exists a T -transformation such that P lim ?m,n (x) = s(x) uniformly in A m,n and to that end we obtain the following. In order that the transformation be such that P - lim s?s0 (S);t?t0 (T) ?(s; t; x) = 0 uniformly with respect x for every double sequence of continuous functions (sm,n (x)) define over A such that sm,n (x) is bounded over A and for all (m, n) and P - limm,n sm,n (x) = 0 over A it is necessary and sufficient that P - lim ?,? ? |ak,l (s, t)| = 0. s?s0 (S);t?t0 (T) k,l=1,1 .