Uncertainty theory

Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.
Mathematical measures of the likelihood of an event being true include probability theory, capacity, fuzzy logic, possibility, and credibility, as well as uncertainty.
Four axioms.
Axiom 1. (Normality Axiom) formula_1.
Axiom 2. (Self-Duality Axiom) formula_2.
Axiom 3. (Countable Subadditivity Axiom) For every countable sequence of events Λ1, Λ2, ..., we have
Axiom 4. (Product Measure Axiom) Let formula_4 be uncertainty spaces for formula_5. Then the product uncertain measure formula_6 is an uncertain measure on the product σ-algebra satisfying
Principle. (Maximum Uncertainty Principle) For any event, if there are multiple reasonable values that an uncertain measure may take, then the value as close to 0.5 as possible is assigned to the event.
Uncertain variables.
An uncertain variable is a measurable function ξ from an uncertainty space formula_8 to the set of real numbers, i.e., for any Borel set B of real numbers, the set 
formula_9 is an event.
Uncertainty distribution.
Uncertainty distribution is inducted to describe uncertain variables.
Definition:The uncertainty distribution formula_10 of an uncertain variable ξ is defined by formula_11.
Theorem(Peng and Iwamura, "Sufficient and Necessary Condition for Uncertainty Distribution") A function formula_10 is an uncertain distribution if and only if it is an increasing function except formula_13 and formula_14.
Independence.
Definition: The uncertain variables formula_15 are said to be independent if 
for any Borel sets formula_17 of real numbers.
Theorem 1: The uncertain variables formula_15 are independent if 
for any Borel sets formula_17 of real numbers.
Theorem 2: Let formula_15 be independent uncertain variables, and formula_22 measurable functions. Then formula_23 are independent uncertain variables.
Theorem 3: Let formula_24 be uncertainty distributions of independent uncertain variables formula_25 respectively, and formula_26 the joint uncertainty distribution of uncertain vector formula_27. If formula_15 are independent, then we have 
for any real numbers formula_30.
Operational law.
Theorem: Let formula_15 be independent uncertain variables, and formula_32 a measurable function. Then formula_33 is an uncertain variable such that
where formula_35 are Borel sets, and formula_36 means formula_37 for anyformula_38.
Expected Value.
Definition: Let formula_39 be an uncertain variable. Then the expected value of formula_39 is defined by 
provided that at least one of the two integrals is finite.
Theorem 1: Let formula_39 be an uncertain variable with uncertainty distribution formula_26. If the expected value exists, then 
Theorem 2: Let formula_39 be an uncertain variable with regular uncertainty distribution formula_26. If the expected value exists, then 
Theorem 3: Let formula_39 and formula_49 be independent uncertain variables with finite expected values. Then for any real numbers formula_50 and formula_51, we have
Variance.
Definition: Let formula_39 be an uncertain variable with finite expected value formula_54. Then the variance of formula_39 is defined by 
Theorem: If formula_39 be an uncertain variable with finite expected value, formula_50 and formula_51 are real numbers, then
Critical value.
Definition: Let formula_39 be an uncertain variable, and formula_62. Then
is called the α-optimistic value to formula_39, and 
is called the α-pessimistic value to formula_39.
Theorem 1: Let formula_39 be an uncertain variable with regular uncertainty distribution formula_26. Then its α-optimistic value and α-pessimistic value are 
Theorem 2: Let formula_39 be an uncertain variable, and formula_62. Then we have
Theorem 3: Suppose that formula_39 and formula_49 are independent uncertain variables, and formula_62. Then we have
formula_80,
formula_81,
formula_82,
formula_83,
formula_84,
formula_85.
Entropy.
Definition: Let formula_39 be an uncertain variable with uncertainty distribution formula_26. Then its entropy is defined by
where formula_89.
Theorem 1("Dai and Chen"): Let formula_39 be an uncertain variable with regular uncertainty distribution formula_26. Then
Theorem 2: Let formula_39 and formula_49 be independent uncertain variables. Then for any real numbers formula_50 and formula_51, we have
Theorem 3: Let formula_39 be an uncertain variable whose uncertainty distribution is arbitrary but the expected value formula_54 and variance formula_100. Then
Inequalities.
Theorem 1("Liu", Markov Inequality): Let formula_39 be an uncertain variable. Then for any given numbers formula_103 and formula_104, we have
Theorem 2 ("Liu", Chebyshev Inequality) Let formula_39 be an uncertain variable whose variance formula_107 exists. Then for any given number formula_103, we have
Theorem 3 ("Liu", Holder's Inequality) Let formula_110 and formula_111 be positive numbers with formula_112, and let formula_39 and formula_49 be independent uncertain variables with formula_115 and formula_116. Then we have
Theorem 4:(Liu [127], Minkowski Inequality) Let formula_110 be a real number with formula_119, and let formula_39 and formula_49 be independent uncertain variables with formula_122 and formula_116. Then we have
Convergence concept.
Definition 1: Suppose that formula_125 are uncertain variables defined on the uncertainty space formula_8. The sequence formula_127 is said to be convergent a.s. to formula_39 if there exists an event formula_129 with formula_130 such that
for every formula_132. In that case we write formula_133,a.s.
Definition 2: Suppose that formula_125 are uncertain variables. We say that the sequence formula_127 converges in measure to formula_39 if
for every formula_138.
Definition 3: Suppose that formula_125 are uncertain variables with finite expected values. We say that the sequence formula_127 converges in mean to formula_39 if
Definition 4: Suppose that formula_143 are uncertainty distributions of uncertain variables formula_125, respectively. We say that the sequence formula_127 converges in distribution to formula_39 if formula_147 at any continuity point of formula_26.
Theorem 1: Convergence in Mean formula_149 Convergence in Measure formula_149 Convergence in Distribution. 
However, Convergence in Mean formula_151 Convergence Almost Surely formula_151 Convergence in Distribution.
Conditional uncertainty.
Definition 1: Let formula_8 be an uncertainty space, and formula_154. Then the conditional uncertain measure of A given B is defined by
Theorem 1: Let formula_8 be an uncertainty space, and B an event with formula_158. Then M{·|B} defined by Definition 1 is an uncertain measure, and formula_159is an uncertainty space.
Definition 2: Let formula_39 be an uncertain variable on formula_8. A conditional uncertain variable of formula_39 given B is a measurable function formula_163 from the conditional uncertainty space formula_164 to the set of real numbers such that
Definition 3: The conditional uncertainty distribution formula_166 of an uncertain variable formula_39 given B is defined by
provided that formula_169.
Theorem 2: Let formula_39 be an uncertain variable with regular uncertainty distribution formula_171, and formula_172 a real number with formula_173. Then the conditional uncertainty distribution of formula_39 given formula_175 is
Theorem 3: Let formula_39 be an uncertain variable with regular uncertainty distribution formula_171, and formula_172 a real number with formula_180. Then the conditional uncertainty distribution of formula_39 given formula_182 is
Definition 4: Let formula_39 be an uncertain variable. Then the conditional expected value of formula_39 given B is defined by
provided that at least one of the two integrals is finite.