The fundamental result of Li, Long, and Srinivasan on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, combinatorics and data analysis (described as `the pinnacle of a long sequence of papers'). The goal of this paper is to give a simpler, self-contained, modular proof of this result for finite set systems. The only ingredient we assume is the standard Chernoff's concentration bound. This makes the proof accessible to a wider audience, readers not familiar with techniques from statistical learning theory, and makes it possible to be covered in a single self-contained lecture in an algorithms course.

中文翻译：

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最优近似的简单证明

Li、Long 和 Srinivasan 在集合系统近似方面的基本成果已成为跨多个社区的关键工具，例如学习理论、算法、组合学和数据分析（被描述为“一长串论文的顶峰”）。本文的目标是为有限集系统的这个结果提供一个更简单的、独立的、模块化的证明。我们假设的唯一成分是标准的切尔诺夫浓度界限。这使得更广泛的受众、不熟悉统计学习理论技术的读者可以访问该证明，并且可以在算法课程的单个独立讲座中进行介绍。