Integral inequalities for integrals of differences of subharmonic functions against Borel measures on balls in multidimensional Euclidean spaces are obtained. These integrals are estimated from above in terms of the product of the Nevanlinna characteristic of the function and various characteristics of the Borel measure and its support. The main theorem, which is a criterion concerning such estimates, presents several equivalent statements of different character. All results are new for the logarithms of the moduli of meromorphic functions on discs in the complex plane. They cover all preceding results, which go back to the classical small arcs lemma of Edrei and Fuchs, as special cases. Integrals against Borel measures with support on fractal sets are also allowed; in this case estimates are in terms of the Hausdorff measure and Hausdorff content of the support of the measure. Special cases of functions on the whole complex plane or space, or in the unit disc or ball, which are important for applications are distinguished, as well as cases involving integration against arc length over subsets of Lipschitz curves and against surface area over subsets of Lipschitz hypersurfaces. Bibliography: 42 titles.