PSEUDO-UNIFORM CONVERGENCE, A NONSTANDARD TREATMENT

We introduce and study the notion of pseudo-uniform convergence which is a weaker variant of quasi-uniform convergence. Applications include the following nonstandard characterization of weak convergence. Let X be an:infinite set, B(X) the Banach space of all bounded real-valued functions on X, {f(n) : n is an element of N} a bounded sequence in B( X), and f is an element of (X). Then the sequence converges weakly to f if and only if the convergence is pointwise on X-and, for each stric tly increasing function sigma : N --> N, each x is an element of X, a nd each n is an element of N-infinity, there is an unlimited m less t han or equal to n such that f*(sigma(m))(x) similar or equal to *f(x) .