论文标题
Lebesgue集成的分类推导
A categorical derivation of Lebesgue integration
论文作者
论文摘要
我们确定了简单的通用属性，这些属性唯一地描述了lebesgue $ l^p $空间。有两个主要定理。第一个指出，配备少量额外结构的Banach Space $ l^p [0，1] $是初始的。第二个指出，有限测量空间上的$ l^p $函数也是如此的额外结构。在这两种情况下，整合函数的普遍表征都会产生积分的独特表征。使用通用属性，我们开发了整合理论的一些基本要素。我们还陈述了表征序列空间$ \ ell^p $和$ c_0 $的通用属性，以及函数$ l^2 $在希尔伯特空间中取值。
We identify simple universal properties that uniquely characterize the Lebesgue $L^p$ spaces. There are two main theorems. The first states that the Banach space $L^p[0, 1]$, equipped with a small amount of extra structure, is initial as such. The second states that the $L^p$ functor on finite measure spaces, again with some extra structure, is also initial as such. In both cases, the universal characterization of the integrable functions produces a unique characterization of integration. Using the universal properties, we develop some of the basic elements of integration theory. We also state universal properties characterizing the sequence spaces $\ell^p$ and $c_0$, as well as the functor $L^2$ taking values in Hilbert spaces.
