The sub-key theorem on credibility measure space

In 1970s, Vladimir N. Vapnik proposed statistical learning theory. The theory is considered as optimum theory on small samples statistical estimation and prediction learning. It has more systematically investigated the rational conditions of the empirical risk minimization discipline and the relations between the empirical risk and the expected risk on finite samples. In fact, the key theorem of learning theory plays an important role in statistical learning theory. Its importance results in paving the way for the subsequent theories and applications. However, some theories and definitions only suit to fixed probability measure. These restricted conditions reduce the applied range of theorem. In this paper, we will generalize the applied range by means of changing the probability measure space into credibility measure space. In new measure space, we give new concepts and new theorem on classical theoretical foundation.