可数标号图空间上的积分与测度
Integration and measures on the space of countable labelled graphs
摘要：
在本文中，我们开发了一个严格的基础上的整合和措施的研究空间$\mathscr{G}(V)$的所有图定义在一个可数的标记顶点集$V$。本文首先研究了图空间上几个相互关联的$σ$ -代数和一个大的概率测度族。然后，我们专注于一个“并矢”汉明距离函数$\left\| \cdot \right\|_{ψ,2}$，这是非常有用的分化$\mathscr{G}(V)$的研究。函数$\left\| \cdot \right\|_{ψ,2}$被证明是从无限图的子集到圆的Haar测度保持双射（具有Haar/勒贝格测度），从而自然地识别了这两个空间.因此，我们建立了一个“变量变换“的公式，使转移的Riemann-Lebesgue理论$\mathbb{R}$到图空间$\mathscr{G}(V)$。这也补充了以前的工作，其中牛顿-莱布尼茨微分理论从真正的线转移到$\mathscr{G}(V)$可数$V$。最后，我们证明了$\mathscr{G}(V)$的Pontryagin对偶，并刻画了$\mathscr{G}(V)$上的正定函数.
英文摘要：
In this paper we develop a rigorous foundation for the study of integration and measures on the space $\mathscr{G}(V)$ of all graphs defined on a countable labelled vertex set $V$. We first study several interrelated $σ$-algebras and a large family of probability measures on graph space. We then focus on a "dyadic" Hamming distance function $\left\| \cdot \right\|_{ψ,2}$, which was very useful in the study of differentiation on $\mathscr{G}(V)$. The function $\left\| \cdot \right\|_{ψ,2}$ is shown to be a Haar measure-preserving bijection from the subset of infinite graphs to the circle (with the Haar/Lebesgue measure), thereby naturally identifying the two spaces. As a consequence, we establish a "change of variables" formula that enables the transfer of the Riemann-Lebesgue theory on $\mathbb{R}$ to graph space $\mathscr{G}(V)$. This also complements previous work in which a theory of Newton-Leibnitz differentiation was transferred from the real line to $\mathscr{G}(V)$ for countable $V$. Finally, we identify the Pontryagin dual of $\mathscr{G}(V)$, and characterize the positive definite functions on $\mathscr{G}(V)$.
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