The Key Theorem Of Statistical Learning Theory On Sugeno Measure Spaces

Statistical Learning Theory or SLT, which was based on probability space, is a small-sample statistics by Vapnik. This paper explores the SLT on a kind of nonadditive measure space-Sugeno measure space, which is more widely than probability space. Firstly, we discuss the properties of Sugeno measure and give the definitions of distribution function and expect value and variance under Sugeno measure. Based on above, we propose Chebyshev inequality and the law of large numbers on Sugeno measure space also. Secondly, we give the definitions of empirical risk functional, expected risk functional and the strict consistency of ERM principle on Sugeno measure space. Finally, we prove the key theorem of SLT on Sugeno measure space.