Ileana Bucur
The Technical University of Civil Engineering, Bucharest, Romania

The famous Korovkin Theorem asserts that if a sequence $(T_n)_n$ of positive linear operators on the Banach space $C$ of all continuous real functions defined on the interval $[0,1]$ are such that the sequences $(T_n(1))_n, (T_n(x))_n, (T_n(x^2))_n$ are uniformly convergent to the functions $1, x$, respectively $x^2$, then the sequence $(T_n(f))_n$ converges uniformly to $f$ for any $f\in C$.

We give a large generalization of this result for a set $A$ of real, bounded, measurable functions on $X$ and a sequence $(T_n)_n$ of kernels on $X$ such that the sequence $(T_n(a))_n$ converges simply or uniformly to a function $a$ for all $a\in A$. We show that the same approximation holds for all measurable functions controlled by $A$. An application to excessive and fine topology is given.