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We prove that arbitrary superharmonic functions and superparabolic functionsrelated to the $p$-Laplace and the $p$-parabolic equations are locally obtainedas limits of supersolutions with desired convergence properties of thecorresponding Riesz measures. As an application we show that a family ofuniformly bounded supersolutions to the $p$-parabolic equation contains asubsequence that converges to a supersolution.

Local approximation of superharmonic and superparabolic functions in nonlinear potential theory