Abstract: Some definitions and properties of pan-random variable and its distribution，pan-expectation and pan-variance on pan-space are introduced，and Chebyshev’inequality and the Khinchine’s strong law of large numbers on pan-space are proved.The definitions of expected risk functional，empirical risk functional and the empirical risk minimization inductive principle on pan-space are proposed，and then the key theorem of learning theory on pan-space is proved.As a result，the key theorems of learning theory on probability measure space and possibility space are unified and extended to pan-space.

Key words: pan-space, pan additive measure, the empirical risk minimization inductive principle, the key theorems

摘要： 给出泛空间上泛随机变量及其分布函数、泛期望和泛方差的定义和性质，证明泛空间上的Chebyshev不等式和Khinchine大数定律；给出泛空间上期望风险泛函、经验风险泛函以及经验风险最小化原则严格一致收敛的定义，证明了泛空间上学习理论的关键定理，把概率空间和可能性测度空间上的学习理论的关键定理统一推广到了泛空间上。

关键词: 泛空间, 泛可加测度, 经验风险最小化原则, 关键定理