The theory of integration over infinite-dimensional spaces is known to encounter serious difficulties. Categorical ideas seem to arise naturally on the path to a remedy. Such an approach was suggested and initiated by Segal in his pioneering article (Segal, Bull Am Math Soc 71:419-489, 1965). In our paper we follow his ideas from a different perspective, slightly more categorical, and strongly inspired by the point-free topology. First, we develop a general (point-free) concept ofmeasurability (extending the standard Lebesgue integration when applying to the classical σ- algebra). Second (and here we have a major difference from the classical theory), we prove that every finite-additive function μ with values in [0, 1] can be extended to a measure on an abstract σ-algebra; this correspondence is functorial and yields uniqueness. As an example we show that the Segal space can be characterized by completely canonical data. Furthermore, from our results it follows that a satisfactory point-free integration arises everywhere where we have a finite-additive probability function on a Boolean algebra.
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机译：已知在无穷维空间上进行积分的理论会遇到严重的困难。在寻求补救的道路上，分类观念似乎自然而然地出现了。 Segal在其开创性文章（Segal，Bull Am Math Soc 71：419-489，1965）中提出并提出了这种方法。在我们的论文中，我们从不同的角度关注他的观点，他的观点更为明确，并受到无点拓扑的强烈启发。首先，我们开发了一个通用的（无点）可度量性概念（将适用于经典σ代数的标准Lebesgue积分进行了扩展）。其次（这与经典理论有很大的不同），我们证明了[0，1]中每个值的有限可加函数μ都可以扩展为抽象σ代数上的一个测度;这种对应是功能性的，并产生唯一性。作为示例，我们表明Segal空间可以用完全规范的数据来表征。此外，从我们的结果可以得出，在布尔代数上具有有限可加概率函数的任何地方都会出现令人满意的无点积分。
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