Abstract In this paper, we propose three types of absolute continuity for monotone measures and present some of their basic properties. By means of these three types of absolute continuity, we establish generalized Egoroff's theorem, generalized Riesz's theorem and generalized Lebesgue's theorem in the framework involving the ordered pair of monotone measures. The Egoroff theorem, the Riesz theorem and the Lebesgue theorem in the traditional sense concerning a unique monotone measure are extended to the general case. These three generalized convergence theorems include as special cases several previous versions of Egoroff-like theorem, Riesz-like theorem and Lebesgue-like theorem for monotone measures, respectively.

中文翻译：

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单调测度的广义收敛定理

摘要 在本文中，我们提出了单调测度的三种绝对连续性，并介绍了它们的一些基本性质。借助这三种绝对连续性，我们在涉及有序单调测度对的框架内建立了广义Egoroff定理、广义Riesz定理和广义Lebesgue定理。将传统意义上的关于唯一单调测度的 Egoroff 定理、Riesz 定理和 Lebesgue 定理推广到一般情况。这三个广义收敛定理分别包括几个以前版本的类 Egoroff 定理、类 Riesz 定理和类 Lebesgue 定理的特例，用于单调测度。