Abstract: The key theorem plays an important role in the statistical learning theory.However，the researches about it at present mainly focus on real random variable and the samples which are supposed to be noise-free.In this paper，the definitions of complex g_λ variable and primary norm are introduced.Then，the definitions of the empirical risk functional，the expected risk functional and empirical risk minimization principle about g_λ samples corrupted by noise are proposed.Finally，the key theorem of learning theory about complex g_λ samples corrupted by noise is proposed and proved.The investigations help lay essential theoretical foundations for the systematic and comprehensive development of the statistical learning theory of complex g_λ samples.

Key words: complex g_λ variable, primary norm, noise, empirical risk minimization principle, the key theorem

摘要： 关键定理是统计学习理论的重要组成部分。但是，目前的研究主要集中在实随机变量且样本不受噪声影响。引入了复g_λ随机变量、准范数的定义，提出了受噪声影响的复g_λ样本的经验风险泛函、期望风险泛函以及经验风险最小化原则严格一致性的定义；给出并证明了受噪声影响的复g_λ样本的学习理论的关键定理，为系统建立基于复g_λ样本的统计学习理论奠定了理论基础。

关键词: 复g_λ随机变量, 准范数, 噪声, 经验风险最小化原则, 关键定理