On Kantorovich-Stieltjes operators

Let ν be a finite Borel measure on [0,1] The Kantorovich-Stieltjes polynomials are defined by $$K_n v = (n + 1)sumnolimits_{k = 0}^n {(int_{l_{k,n} } {dv)N_{k,n} } } {text{ (}}n in {text{N),}}$$ where (N_{k,n} (x) = (_k^n )x^k (1 - x)^{n - k} (x in [0,1],k = 1,2,...,n)) are the basic Bernstein polynomials and (I_{k,n} : = [frac{k}{{n + 1}},frac{{k + 1}}{{n + 1}}](k = 0,1,...,n;n in {text{N)}}) . We prove that the maximal operator of the sequence (K _n) is of weak type and the sequence of polynomials (K ν) converges a.e. on [0,1] to the Radon-Nikodym derivative of the absolutely continuous part of ν.