To overcome some problems with deep understanding of fuzzy values, certain learning finite automaton was put into a fuzzy environment. Previously, such a device has been studied in the probabilistic environment, where the classic technique of standard Markov chains was applicable. The new study became possible due to several previous results by the present author, namely the axiomatic of fuzzy evidence accumulation and the theory of generalized Markov chains. The mathematical results, obtained in the paper, prove that the learning automaton has the property of asymptotic optimality. We propose to use this property for measuring membership functions in case of values analogous to singletons or point functions. It is claimed that the obtained results might lead to a fuzzy value measurement procedure resembling statistics developed in probability area. © Springer International Publishing Switzerland 2016.