Proceedings of the American Mathematical Society

Abstract:

Of concern are some simple criteria about the convergence of sequences of positive linear operators and functionals in the framework of spaces of bounded functions which are continuous on a given subset of their domain.

Among other things some applications concerning the behaviour of the iterates of Bernstein operators defined both on $[0,1]$ and on the $d$-dimensional simplex and hypercube $(d\geq 1)$ are discussed.

A final section treats the behaviour of integrated arithmetic means with respect to a probability Borel measure on a convex compact subset $K$ of a locally convex space. As a consequence a general weak law of large numbers for sequences of $K-$valued random variables is derived.