\textit{I. N. Pesin} [Dokl. Akad. Nauk SSSR 187, 740-742 (1969; Zbl 0215.128)] formulated theorems on existence, equicontinuity and completeness for classes with integral restrictions of a certain form. \textit{G. David} [Ann. Acad. Sci. Fenn., Ser. AI 13, No. 1, 25-70 (1988; Zbl 0651.30015)] proved the theorem on existence and uniqueness for the normalized homeomorphic solutions of the Beltrami equation with certain restrictions on their complex characteristics. Considering an equivalent form of these restrictions it is observed that the foregoing results can be extended to mappings that are quasiconformal in the mean. The semicontinuity of a deformation of the homeomorphisms from the Sobolev classes is established here. This in particular generalizes the Strebel inequality [see, \textit{K. Strebel}, Comment. Math. Helv. 44, 469-475 (1969; Zbl 0184.310)] to the deformations with locally summable majorants. As the main result of the paper the author gives a criterion of compactness for the mappings quasiconformal in the mean.