Estimates of approximation by Kantorovich type operators in terms of the second modulus of continuity

Approximation of bounded measurable functions on the segment $[0, 1]$ by Kantorovich type operators
$$ B_n=\sum_{j=0}^nC_n^jx^j(1-x)^{n-j}F_{n, j}, $$
where the $F_{n, j}$ are functionals produced by probability measures with sufficiently small supports is considered. The error of approximation is estimated in terms of the second modulus of continuity. The result is sharp.