Uncertainty theory is a mathematical framework based on normality, monotonicity, self-duality, and countable subadditivity. It aims to provide a common foundation for probability theory, credibility theory, and chance theory. This chapter focuses on uncertain measure, uncertainty space, uncertain variable, uncertainty distribution, expected value, variance, moments, critical values, entropy, distance, characteristic function, and various types of convergence. Uncertain measure is defined as a set function M satisfying four axioms: normality, monotonicity, self-duality, and countable subadditivity. Normality requires that the measure of the entire set is 1. Monotonicity ensures that the measure of a subset is less than or equal to the measure of a superset. Self-duality implies that the measure of an event and its complement sum to 1. Countable subadditivity ensures that the measure of the union of countably many events is less than or equal to the sum of their measures. The chapter discusses the importance of self-duality and countable subadditivity over continuity and semicontinuity. It also highlights potential pathologies that can occur if these axioms are not satisfied. For example, without self-duality, a set function may assign the same value to all sets, which is not a valid measure. Similarly, without countable subadditivity, a set function may fail to satisfy the necessary properties for a valid measure. Definition 5.1 introduces the concept of uncertain measure, which includes probability measure, credibility measure, and chance measure as special cases. Example 5.2 illustrates an uncertain measure that does not satisfy the properties of probability or credibility measures but still satisfies the four axioms of uncertain measure.Uncertainty theory is a mathematical framework based on normality, monotonicity, self-duality, and countable subadditivity. It aims to provide a common foundation for probability theory, credibility theory, and chance theory. This chapter focuses on uncertain measure, uncertainty space, uncertain variable, uncertainty distribution, expected value, variance, moments, critical values, entropy, distance, characteristic function, and various types of convergence. Uncertain measure is defined as a set function M satisfying four axioms: normality, monotonicity, self-duality, and countable subadditivity. Normality requires that the measure of the entire set is 1. Monotonicity ensures that the measure of a subset is less than or equal to the measure of a superset. Self-duality implies that the measure of an event and its complement sum to 1. Countable subadditivity ensures that the measure of the union of countably many events is less than or equal to the sum of their measures. The chapter discusses the importance of self-duality and countable subadditivity over continuity and semicontinuity. It also highlights potential pathologies that can occur if these axioms are not satisfied. For example, without self-duality, a set function may assign the same value to all sets, which is not a valid measure. Similarly, without countable subadditivity, a set function may fail to satisfy the necessary properties for a valid measure. Definition 5.1 introduces the concept of uncertain measure, which includes probability measure, credibility measure, and chance measure as special cases. Example 5.2 illustrates an uncertain measure that does not satisfy the properties of probability or credibility measures but still satisfies the four axioms of uncertain measure.