On Kantorovich-Stieltjes operators

Let ν be a finite Borel measure on 0,1] The Kantorovich-Stieltjes polynomials are defined by $$K_n v = (n + 1)\sum\nolimits_{k = 0}^n {(\int_{l_{k,n} } {dv)N_{k,n} } } {\text{ (}}n \in {\text{N),}}$$ 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.