Abstract: The key theorem plays an important role in the statistical learning theory. However, the researches about it at present focus on probability space and the samples are supposed to be noise-free. In consideration of these facts, the definitions of expected risk functional, empirical risk functional and empirical risk minimization principle with samples corrupted by noise on pan-space are proposed, and the key theorem of learning theory with samples corrupted by noise on pan-space is proposed and proved.

Key words: pan-space, pan-random variable, noise, empirical risk minimization principle, key theorem

摘要： 关键定理是统计学习理论的重要组成部分，但目前其研究主要集中在概率空间上且假设样本不受噪声的影响。鉴于此，提出了泛空间上样本受噪声影响的期望风险泛函、经验风险泛函以及经验风险最小化原则的定义，给出并证明了泛空间上样本受噪声影响的学习理论的关键定理。

关键词: 泛空间, 泛随机变量, 噪声, 经验风险最小化原则, 关键定理