  For the qualitative results of pointwise and uniform approximation obtained
in \cite{Gal-Opris}, we present general quantitative estimates in terms of the
modulus of continuity and in terms of a $K$-functional, for the generalized
multivariate Bernstein-Durrmeyer operator $M_{n, \Gamma_{n, x}}$, written in
terms of the Choquet integral with respect to a family of monotone and
submodular set functions, $\Gamma_{n, x}$, on the standard $d$-dimensional
simplex. When $\Gamma_{n, x}$ reduces to two elements, one a Choquet submodular
set function and the other one a Borel measure, for suitable modified
Bernstein-Durrmeyer operators, univariate $L^{p}$-approximations, $p\ge 1$,
with estimates in terms of a $K$-functional are proved. In the particular cases
when $d=1$ and the Choquet integral is taken with respect to some concrete
possibility measures, the pointwise estimate in terms of the modulus of
continuity is detailed. Some simple concrete examples of operators improving
the classical error estimates are presented. Potential applications to
practical methods dealing with data, like learning theory and regression
models, also are mentioned.
