  In this work we construct subdivision schemes refining general subsets of R^n
and study their applications to the approximation of set-valued functions.
Differently from previous works on set-valued approximation, our methods are
developed and analyzed in the metric space of Lebesgue measurable sets endowed
with the symmetric difference metric. The construction of the set-valued
subdivision schemes is based on a new weighted average of two sets, which is
defined for positive weights (corresponding to interpolation) and also when one
weight is negative (corresponding to extrapolation).
  Using the new average with positive weights, we adapt to sets spline
subdivision schemes computed by the Lane-Riesenfeld algorithm, which requires
only averages of pairs of numbers. The averages of numbers are then replaced by
the new averages of pairs of sets. Among other features of the resulting
set-valued subdivision schemes, we prove their monotonicity preservation
property. Using the new weighted average of sets with both positive and
negative weights, we adapt to sets the 4-point interpolatory subdivision
scheme. Finally we discuss the extension of the results obtained in the metric
spaces of sets, to general metric spaces endowed with an averaging operation
satisfying certain properties.
