Borel structures for function spaces
Borel structures for function spaces
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功能空间的 Borel 结构
DOI：
10.1215/ijm/1255631584
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发表时间：
1961
影响因子：
0.6
通讯作者：
R. Aumann
中科院分区：
文献类型：
--
作者：
R. Aumann
如果x和y是拓扑空间，则yx表示从x到y的所有连续映射的集合。对于yx上的给定拓扑，我们可能会询问yx是否自然映射> <x-+ y yx（f，f，x） ）f（x）是连续的；如果是这样，则据说YX上的拓扑是可以接受的[1]。总是有可能找到可接受的拓扑。例如，YX上的离散拓扑总是可以接受的。此外，当X在局部紧凑时，YX具有独特的最小可接受拓扑。这是熟悉的“紧凑型”拓扑。几位作者已经对这些功能空间拓扑的相关问题进行了大量研究[1，4]。当X和Y是Borel空间而不是拓扑空间时，我们对类似情况感兴趣。在这种情况下，我们将YX定义为从X到Y的所有Borel映射的集合。不幸的是，事实证明，即使对于某些最简单的Borel空间，也无法在YX上定义Borel结构，因此这是Borel映射。即使我们将离散结构强加于YX，通常也不是Borel。作为替代品，我们可能会问自己以下问题：“对于哪个子集F y的f，可以将borel结构强加在f上，以便如果> <x将是borel？说出适当的结构？ “ Borel结构”，而F而不是F> <X。f的结构r将被称为borel for f bore
If X and Y are topological spaces, then yX denotes the set of all continuous mappings from X into Y. For a given topology on yX, we may ask whether yX the natural mapping >< X --+ Y defined by (f, x) f(x) is continuous; if it is, then the topology on yX is said to be admissible [1]. It is always possible to find an admissible topology; for instance, the discrete topology on YX is always admissible. Moreover, when X is locally compact, YX has a unique smallest admissible topology; this is the familiar "compactopen" topology. These and related questions concerning topologies for function spaces have been investigated in considerable detail by several authors [1, 4]. We are interested in the analogous situation when X and Y are Borel spaces rather than topological spaces; in this case we define YX as the set of all Borel mappings from X into Y. Unfortunately, it turns out that even for some of the simplest Borel spaces, it is impossible to define a Borel structure on YX so that is a Borel mapping; even if we impose the discrete structure on YX, will in general not be Borel. As a substitute, we may ask ourselves the following questions" For which subsets F of Y is it possible to impose a Borel structure on F so that IF >< X will be Borel? If is is possible for a given F, what can we say about the appropriate structures? In particular, is there always a smallest such structure (corresponding to the compact-open topology) ? Let us introduce some terminology. We will write "space" instead of "Borel space", "structure" instead of "Borel structure", and F instead of F >< X. A structure R on F for which F is Borel will be called admissible; a subset F of yX on which it is possible to impose an admissible structure is