To obtain the highest confidence on the correction of numerical simulation programs implementing the finite element method, one has to formalize the mathematical notions and results that allow to establish the soundness of the method. Sobolev spaces are the correct framework in which most partial derivative equations may be stated and solved. These functional spaces are built on integration and measure theory. Hence, this chapter in functional analysis is a mandatory theoretical cornerstone for the definition of the finite element method. The purpose of this document is to provide the formal proof community with very detailed pen-and-paper proofs of the main results from integration and measure theory.

中文翻译：

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Lebesgue整合。详细证明将在Coq中正式化

为了在实施有限元方法的数值模拟程序的校正中获得最高的置信度，必须形式化数学概念和结果以建立方法的可靠性。Sobolev空间是正确的框架，在其中可以陈述和求解大多数偏导数方程。这些功能空间基于集成和度量理论。因此，功能分析的这一章是定义有限元方法的必不可少的理论基石。本文档的目的是为形式证明社区提供非常详细的纸笔证明，以证明集成和度量理论的主要结果。