  We study the approximation of univariate and multivariate set-valued
functions (SVFs) by the adaptation to SVFs of positive samples-based
approximation operators for real-valued functions. To this end, we introduce a
new weighted average of several sets and study its properties. The
approximation results are obtained in the space of Lebesgue measurable sets
with the symmetric difference metric.
  In particular, we apply the new average of sets to adapt to SVFs the
classical Bernstein approximation operators, and obtain a set-valued analog of
the Weierstrass approximation theorem. The rate of approximation of H\"older
continuous SVFs by Bernstein operators is studied and shown to be
asymptotically equal to that for real-valued functions. Finally, the results
obtained in the metric space of sets are generalized to metric spaces endowed
with an average satisfying certain properties.
