  In this paper, we give some $W^{1,p}$ limit theorems for the total scalar
curvature. More precisely, we show that the lower bound of the total scalar
curvatures on a closed manifold is preserved under the $W^{1, p}$ convergence
of the Riemannian metrics for sufficiently large $p$ provided that each scalar
curvature is nonnegative. We also give a similar type of theorem for a sequence
of metrics that converges to a metric only in the $C^{0}$ sense under some
additional assumptions. Moreover, we give some counterexamples to this theorem
on the Euclidean space.
