论文标题
在$ \ mathbb上
On $\mathbb{F}_2^ω$-affine-exchangeable probability measures
论文作者
论文摘要
对于任何标准的Borel空间$ b $，令$ \ Mathcal {p}（b）$表示$ b $上的Borel概率度量的空间。与交换性理论中的一个困难问题有关，以及与算术组合主义者有关，奥斯汀提出了描述载体 - 交换可能性衡量标准的结构，该问题是由矢量空间索引$ \ Mathbb $ \ Mathbb {f} f} _2^ω$，即衡量的。 $ \ MATHCAL {p}（b^{\ Mathbb {f} _2^ω}）$在$ b^{\ Mathbb {f} _2^ω}上的坐标排列下是不变的，这是所有由$ \ mathbbbb {f} $ f} $ f} $ f} $ _2^^=^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^$ _2^^=我们通过描述此类偏好交换措施的空间的极端要点来回答这个问题。我们证明，在每种措施中都有一个单一的结构，即是一个随机的无限多维多维数据集（使用适合特定过滤的HAAR测量进行采样），这是一个是2次整数的可数功率的组。实际上，每一个极端的仿射 - 分离度量$ \ mathcal {p}（b^{\ mathbb {f} _2^ω}）$都是从$ \ mathcal {p}（b）$ yathcal {p}（b）$ - 由此随机构图的该组中的值的功能。此结果的后果包括描述$ \ Mathcal {p}（b^{\ Mathbb {f} _2^ω}）$ offine-extchangable措施的描述（当$ b $是一个紧凑型公位时），显示了这个convex集合是一个bauerers bauerers。我们还获得了对向量空间上（紧凑型 - 金属空间值）函数（$ \ mathbb {f} _2^n $ as $ n \ to \ infty $的（compact-metric空间值）函数（compact-metric空间值）函数的限制（紧凑型 - 金属空间值）的对应关系。通过这种对应关系，我们将上述组建立为对任何此类序列有效的一般极限域。
For any standard Borel space $B$, let $\mathcal{P}(B)$ denote the space of Borel probability measures on $B$. In relation to a difficult problem of Aldous in exchangeability theory, and in connection with arithmetic combinatorics, Austin raised the question of describing the structure of affine-exchangeable probability measures on product spaces indexed by the vector space $\mathbb{F}_2^ω$, i.e., the measures in $\mathcal{P}(B^{\mathbb{F}_2^ω})$ that are invariant under the coordinate permutations on $B^{\mathbb{F}_2^ω}$ induced by all affine automorphisms of $\mathbb{F}_2^ω$. We answer this question by describing the extreme points of the space of such affine-exchangeable measures. We prove that there is a single structure underlying every such measure, namely, a random infinite-dimensional cube (sampled using Haar measure adapted to a specific filtration) on a group that is a countable power of the 2-adic integers. Indeed, every extreme affine-exchangeable measure in $\mathcal{P}(B^{\mathbb{F}_2^ω})$ is obtained from a $\mathcal{P}(B)$-valued function on this group, by a vertex-wise composition with this random cube. The consequences of this result include a description of the convex set of affine-exchangeable measures in $\mathcal{P}(B^{\mathbb{F}_2^ω})$ equipped with the vague topology (when $B$ is a compact metric space), showing that this convex set is a Bauer simplex. We also obtain a correspondence between affine-exchangeability and limits of convergent sequences of (compact-metric-space valued) functions on vector spaces $\mathbb{F}_2^n$ as $n\to\infty$. Via this correspondence, we establish the above-mentioned group as a general limit domain valid for any such sequence.
