It was generally believed that the study of probability theory was
started by Pascal and Fermat in 1654 when they succeeded in deriving the
exact probabilitiesforcertaingamblingproblem.
Greatprogresswasachievedwhen VonMisesinitializedtheconceptofsamplespace,
and?lledthegapebetween probability theory and measure theory in 1931. An
axiomatic foundation of
probabilitytheorywasgivenbyKolmogoro?inhisFoundationsofProbability
Theory of 1933. Since then, probability theory has been developed
steadily and has been widely applied in science and engineering.
Probability theory will be introduced in Chapter 2. Fuzzy set was
initiated by Zadeh via membership function in 1965, and was well
developed and applied in a wide variety of real problems. In order to
measure a fuzzy event, Zadeh proposed the concept of possibility measure
in 1978. Although possibility measure has been widely used, it has no
se- duality property. However, a self-dual measure is absolutely needed
in both theory and practice. In order to de?ne a self-dual measure, Liu
and Liu gave the concept of credibility measure in 2002. Credibility
theory is a branch of mathematics that studies the behavior of fuzzy
phenomena. An axiomatic foundation of credibility theory was given by
Liu in his Uncertainty Theory of 2004. Chapter 3 will provide the
credibility theory. Sometimes, fuzziness and randomness simultaneously
appear in a system.
