...
  • 主题
高级搜索  >
历史搜索 清空历史
首页> 外文期刊>Topology and its applications >Borel measurability of separately continuous functions
【24h】

Borel measurability of separately continuous functions

机译：分别连续函数的Borel可测性

获取原文
获取原文并翻译 | 示例
           
页面导航
  • 摘要
  • 著录项
  • 相似文献
  • 相关主题

摘要

Lebesgue proved that every separately continuous function f : R x R→R is a pointwise limit of continuous functions. W. Rudin extended this by showing that if X is a metric space, then for any topological space Y, every separately continuous function f : X x Y→R is a pointwise limit of continuous functions. This statement can fail if we take for X an arbitrary linearly ordered space, even if X is separable. However, we show that if X is either a countable product of separable linearly ordered spaces, an arbitrary product of countably compact linearly ordered spaces, or the continuous image of an arbitrary product of compact linearly ordered spaces, and Y is any topological space, then every separately continuous function f: X x Y→ R is Borel measurable. In the case where X is a product of ordinals, we get stronger results. The results for countably compact linearly ordered spaces use some combinatorial properties of n -dimensional arrays of real numbers which are possibly of independent interest. We also give, under a cardinal arithmetic assumption, an example of a linearly ordered space X and a separately continuous function f: X x X→R which is not Borel measurable.
机译：Lebesgue证明，每个单独的连续函数f：R x R→R是连续函数的逐点极限。 W. Rudin通过证明如果X是一个度量空间，那么对于任何拓扑空间Y，每个单独的连续函数f：X x Y→R是连续函数的点极限。如果我们为X取任意线性排序的空间，即使X是可分离的，则该语句也会失败。但是，我们表明如果X是可分离线性有序空间的可数乘积，可数紧凑线性有序空间的任意乘积或紧凑线性有序空间的任意乘积的连续图像，而Y是任何拓扑空间，则每个单独的连续函数f：X x Y→R是Borel可测量的。在X是普通数的乘积的情况下，我们得到更强的结果。可数紧凑线性顺序空间的结果使用实数n维数组的某些组合属性，这些属性可能是独立关注的。在基数算术假设下，我们还给出了线性有序空间X和单独的连续函数f的示例：不能用Borel进行测量。

著录项

相似文献

  • 外文文献
  • 中文文献
  • 专利
获取原文

客服邮箱：kefu@zhangqiaokeyan.com

京公网安备：11010802029741号 ICP备案号：京ICP备15016152号-6 六维联合信息科技 (北京) 有限公司©版权所有

  • 客服微信

  • 服务号