  We adapt the concept of $\mathcal{K}-$convergence of Young measures to the
sequences of approximate solutions resulting from numerical schemes. We obtain
new results on pointwise convergence of numerical solutions in the case when
solutions of the limit continuous problem possess minimal regularity. We apply
the abstract theory to a finite volume method for the isentropic Euler system
describing the motion of a compressible inviscid fluid. The result can be seen
as a nonlinear version of the fundamental Lax equivalence theorem.
