We report on an original formalization of measure and integration theory in the Coq proof assistant. We build the Lebesgue measure following a standard construction that had not yet been formalized in proof assistants based on dependent type theory: by extension of a measure over a semiring of sets. We achieve this formalization by leveraging on existing techniques from the Mathematical Components project. We explain how we extend Mathematical Components’ iterated operators and mathematical structures for analysis to provide support for infinite sums and extended real numbers. We introduce new mathematical structures for measure theory and incidentally provide an illustrative, concrete application of Hierarchy-Builder, a generic tool for the formalization of hierarchies of mathematical structures. This formalization of measure theory provides the basis for a new formalization of the Lebesgue integration compatible with the Mathematical Components project.

我们报告了 Coq 证明助手中测度和积分理论的原始形式化。我们按照尚未在基于依赖类型理论的证明助手中形式化的标准构造来构建勒贝格测度：通过在集合的半环上扩展测度。我们通过利用数学组件项目的现有技术来实现这种形式化。我们解释了如何扩展数学组件的迭代运算符和数学结构进行分析，以提供对无限和和扩展实数的支持。我们为测度论引入了新的数学结构，并顺便提供了 Hierarchy-Builder 的说明性具体应用，Hierarchy-Builder 是一种用于数学结构层次结构形式化的通用工具。