The key theorem and the bounds on the rate of uniform convergence of learning theory on Sugeno measure space. (English) Zbl 1109.28010
Sugeno (1974) introduced \(\lambda\)-measures \(g_\lambda\) (for \(\lambda>-1\)) as a special kind of distorted probability measures, where the classical additivity rule was replaced by the \(\lambda\)-additivity, \(g_\lambda(A\cup B)= g_\lambda(A)+ g_\lambda(B)+ \lambda g_\lambda(A) g_\lambda(B)\) for any disjoint events \(A\), \(B\). The relation of \(\lambda\)-measures and probability measures allowed to introduce \(g_\lambda\) random variables and their distribution functions. The paper discusses several properties of \(\lambda\)-measure spaces, copying the similar properties of classical probability spaces, such as the Markov inequality, Khinchin law of large numbers, etc. Moreover, the authors investigate the bounds on the rate of uniform convergence of learning process on \(\lambda\)-measure space (in the paper called Sugeno measure space). Observe that several introduced results can be formulated for any distorted probability measure space.
Reviewer: Radko Mesiar (Bratislava)
MSC:
| 28A99 | Classical measure theory |
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