Let (X, , P) be a probability space and n, n >= 1, a sequence of classes of measurable complex-valued functions on (X, , P). Under a weak metric entropy condition on n and sup {||g||[infinity]: g [set membership, variant] n}, Glivenko-Cantelli theorems are established for the classes n with respect to the probability measure P; i.e., limn --> [infinity] supg [set membership, variant] n ||[integral operator] g(dPn - dP)|| = 0 a.s. The result is applied to kernel density estimation and a law of the logarithm is derived for the maximal deviation between a kernel density estimator and its expected value, improving upon and generalizing the recent results of W. Stute (Ann. Probab. 10 (1982), 414-422). This result is also used to derive improved rates of uniform convergence for the empirical characteristic function.
