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

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 \ud\uds(1,1)(x), s(1,2)(x) s(1,3)(x) ... \ud\uds(2,1)(x), s(2,2)(x) s(2,3)(x) ... \ud\uds(3,1)(x), s(3,2)(x) s(3,3)(x) ... \ud\udhave the following properties: \ud\ud1. for each (m, n) s(m,n)(x) is continuous in A; \ud\ud2. for each x in A we have P - lim(m,n) s(m,n)(x) = s(x); \ud\ud3. s(x) is continuous in A; \ud\ud4. there exists M such that for all (m, n) and all x in A vertical bar s(m,n)(x)vertical bar <= M. \ud\udThen there exists a T - transformation such that \ud\udP - lim(m,n) sigma(m,n)(x) = s(x) uniformly in A \ud\udand to that end we obtain the following. In order that the transformation be such that \ud\udP - lim(s -> s0(S);t -> t0(T)) sigma(s;t;x) = 0 \ud\uduniformly with respect x for every double sequence of continuous functions (s(m,n)(x)) define over A such that s(m,n)(x) is bounded over A and for all (m, n) and P - lim(m,n) s(m,n)(x) = 0 over A it is necessary and sufficient that \ud\udP- lim(s -> s0(S);t -> t0(T)) Sigma(infinity,infinity)(k,l=1,1) vertical bar a(k,l)(s, t)vertical bar =