A metric measure space is a complete, separable metric space equipped with a probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on the first space to the probability measure on the second. The resulting set of equivalence classes can be metrized with the Gromov-Prohorov metric of Greven, Pfaffelhuber and Winter. We consider the natural binary operation ⊞ on this space that takes two metric measure spaces and forms their Cartesian product equipped with the sum of the two metrics and the product of the two probability measures. We show that the metric measure spaces equipped with this operation form a cancellative, commutative, Polish semigroup with a translation invariant metric. There is an explicit family of continuous semicharacters that is extremely useful for, inter alia, establishing that there are no infinitely divisible elements and that each element has a unique factorization into prime elements. We investigate the interaction between the semigroup structure and the natural action of the positive real numbers on this space that arises from scaling the metric. For example, we show that for any given positive real numbers a, b, c the trivial space is the only space that satisfies a ⊞ b = c . We establish that there is no analogue of the law of large numbers: if X1, X2, … is an identically distributed independent sequence of random spaces, then no subsequence of [Formula: see text] converges in distribution unless each Xk is almost surely equal to the trivial space. We characterize the infinitely divisible probability measures and the Lévy processes on this semigroup, characterize the stable probability measures and establish a counterpart of the LePage representation for the latter class.

中文翻译：

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度量测度空间半群及其无限可分概率测度

度量空间是一个完备的、可分离的度量空间，它配备了一个完全支持的概率测度。如果两个这样的空间通过等距将第一个空间上的概率测度映射到第二个空间上的概率测度，它们与度量空间等距，则它们是等效的。等价类的结果集可以用 Greven、Pfaffelhuber 和 Winter 的 Gromov-Prohorov 度量来度量。我们考虑在这个空间上的自然二元运算⊞，它采用两个度量度量空间并形成它们的笛卡尔积，该乘积配备了两个度量的总和以及两个概率度量的乘积。我们表明，配备此操作的度量度量空间形成了具有平移不变度量的可取消、可交换、波兰语半群。有一个明确的连续半字符族，对于确定不存在无限可分的元素以及每个元素都具有唯一的质数元素分解等非常有用。我们研究了半群结构与正实数在这个空间上的自然作用之间的相互作用，这种作用是由缩放度量引起的。例如，我们证明对于任何给定的正实数 a, b, c 平凡空间是唯一满足 a ⊞ b = c 的空间。我们确定没有大数定律的类似物：如果 X1, X2, ... 是随机空间的同分布独立序列，那么 [公式：见正文] 的子序列在分布上不会收敛，除非每个 Xk 几乎肯定相等到琐碎的空间。