Takahashi and Murakami introduced a theoretical framework to define and evaluate a measure of information gained through a biometric matching system, called the Biometric System Entropy or BSE. The BSE enables us to understand and evaluate the personal identification capability of biometric information from an information theoretical point of view. However, there are limitations when evaluating the BSE for actual systems and biometric information; (1) the BSE cannot be applied to evaluation of biometric information entropy of individuals, (2) it requires a strong and unrealistic assumption regarding statistical distributions of biometric information. In this paper, we generalize the theory of the BSE and give a new measure of biometric information so that we can evaluate both individual and average entropy of any kind of biometric information and verification system without unrealistic assumptions.