  Sets of invariant measures are considered for continuous maps of a metric
compact set. We take Kantorovich metric to calculate distance between measures
and Hausdorff metrics to calculate distance between compact sets. Consider the
function that makes correspondence between a continuous map and the set of all
its Borel probability invariant measures. We demonstrate that a typical map is
a continuity point of that function. Using approaches of Takens' tolerance
stability theory we provide some corollaries that demonstrate that for a
typical map points are structurally stable in a statistical sense.
