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Motivated by risk measure, super-hedge pricing, and modeling uncertainty in finance, Shige Peng established the theory of sub-linear expectation. In this article, we derive two results of laws of large numbers in the framework of sub-linear expectations. One is the strong law of large numbers for the array of random variables, which satisfies non identical distributed and exponential negatively dependent under sub-linear expectation. The other is the weak law of large numbers for the array of random variables, which satisfies non identical distributed and Φ -negatively dependent under sub-linear expectation. These results include and extend some existing results. [ABSTRACT FROM AUTHOR]

In order to treat the degree of belief rationally, Baoding Liu created uncertainty theory. An uncertain variable, as a measurable function from an uncertainty space to the set of real numbers, is a basic concept in uncertainty theory. It is very meaningful to study its properties. Lusin's theorem is one of the most classical theorems in measure theory that reveals the close relationship between measurable and continuous functions, and has important significance. In this paper, we give three pairs of continuity conditions for uncertain measures, and present that every pair reveals duality, which is a kind of symmetry between objects. Furthermore, it is demonstrated that these continuity conditions are equivalent. And, we also prove that these three pairs of continuity conditions and the condition: if { Λ n } is a sequence of open sets and Λ n ↘ ∅ , then lim n → ∞ M { Λ n } = 0 are equivalent in compact metric spaces. It is shown that Lusin's theorem for uncertain variables holds if and only if the uncertain measure satisfies any of the above continuity conditions in a compact metric space. And, Lusin's theorem can be applied to uncertain variables with symmetric or asymmetric distributions. Finally, we provide several examples to illustrate applications of Lusin's theorem for uncertain variables. As far as we know, our results are new in uncertainty theory. [ABSTRACT FROM AUTHOR]