Summary: Let \(\nu\) be a finite Borel measure on \([0,1]\). The Kantorovich-Stieltjes polynomials are defined by \[ K_ n\nu= (n+1) \sum_{k=0}^ n \biggl( \int_{I_{k,n}}d\nu \biggr) N_{k,n} \qquad (n\in\mathbb{N}), \] where \(N_{k,n}(x)= {n \choose k} x^ k(1-x)^{n-k}\) (\(x\in[0,1]\), \(k=1,2,\dots,n\)) are the basic Bernstein polynomials and \(I_{k,n}:= \bigl[{1\over {n+1}}, {{k+1} \over {n+1}}\bigr]\) (\(k=0,1,\dots,n\); \(n\in\mathbb{N}\)). We prove that the maximal operator of the sequence \((K_ n)\) is of weak type and the sequence of polynomial \((K_ n\nu)\) converges a.e. on \([0,1]\) to the Radon-Nikodym derivative of the absolute continuous part of \(\nu\).