  We prove that arbitrary superharmonic functions and superparabolic functions
related to the $p$-Laplace and the $p$-parabolic equations are locally obtained
as limits of supersolutions with desired convergence properties of the
corresponding Riesz measures. As an application we show that a family of
uniformly bounded supersolutions to the $p$-parabolic equation contains a
subsequence that converges to a supersolution.
