Abstract: It is proved that the UCEM property of a family of measurable functions F implies that F is totally bounded; the UCEMUP property and PAC learnability still preserve when the family of probabilities is replaced by its closure. And a concept class C is constructed to show that every PAC algorithm of C would require a super-polynomial number of samples. Finally, the learnability of a concept class C with respect to the probability measures P and its convex hull C(P) is discussed and a mistake of [1] is corrected.

Key words: UCEM property, PAC learnability, PUAC learnability, totally bounded

摘要： 证明了如果函数族F具有UCEM性质，那么F是完全有界的。此外如果F关于概率族P是PAC可学习的或具有UCEM性质，则F关于P的闭包也具有同样的性质。构造了一个非多项式可学习的例子，说明了PAC可学习的概念族可以有任意的复杂性。最后讨论了概念族C关于概率族P及其凸包C(P)的可学习性，并纠正了文[1]的一点错误。

关键词: UCEM性质, PAC可学习, PUAC可学习, 完全有界