Some Properties of Continuous Uncertain Measure

There are various types of uncertainty in the real world. This is a motivation to investigate the behavior of uncertain phenomena. Random phenomena is one class of objective uncertain phenomena which has been well studied. Probability theory is an efficient tool to study the behavior of random phenomena. Besides randomness, fuzziness is a basic type of subjective uncertainty initiated by Zadeh. In order to measure a fuzzy event, Liu and Liu introduced credibility measure. Credibility theory founded by Liu and refined by Liu is a branch of mathematics for studying the behavior of fuzzy phenomena. A survey of credibility theory was given by Liu. In many cases, fuzziness and randomness simultaneously appear in a system. In order to describe this phenomena, a fuzzy random variable was introduced by Kwakernaak as a random element taking “fuzzy variable” values. This concept was then developed by several researchers such as Puri and Ralescu, Kruse and Meyer, and Liu and Liu according to different requirements of measurability. The concept of chance measure of fuzzy random event was first given by Liu, Gao and Liu. In addition, a random fuzzy variable was proposed by Liu as a fuzzy element taking “random variable” values. The chance measure for random fuzzy event and expected value for random fuzzy variables were defined by Liu, Liu and Liu, respectively. More generally, a hybrid variable was introduced by Liu as a measurable function from a chance space to the set of real numbers,