The key theorem and the bounds on the rate of uniform convergence of learning theory on Sugeno measure space

摘要

Some properties of Sugeno measure are further discussed, which is a kind of typical nonadditive measure. The definitions and properties of g_(lambda) random variable and its distribution function, expected value, and variance are then presented. Markov inequality, Chebyshev's inequality and the Khinchine's Law of Large Numbers on Sugeno measure space are also proven. Furthermore, the concepts of empirical risk functional, expected risk functional and the strict consistency of ERM principle on Sugenomeasure space are proposed. According to these properties and concepts, the key theorem of learning theory, the bounds on the rate of convergence of learning process and the relations between these bounds and capacity of the set of functions on Sugeno measure space are given.