论文标题
通过非平凡的简单函数统一连续函数的近似
Uniform Approximation of Continuous Functions by Nontrivial Simple Functions
论文作者
论文摘要
我们证明，在给定的紧凑型公制空间上的每个非负连续实现函数都是某些非负简单函数序列的统一限制是开放集的指标的线性组合。在这里，非平凡性是相对于可测量函数的近似简单函数的标准选择，其中一个人失去了对所指定的可测量集的控制。因此，通过增加非负简单函数，可以改善紧凑型度量空间的非负连续实现功能，可以改善有限的非负可测量实现函数的标准均匀近似值。半连续函数和平滑函数也有一些有趣的后果。
We prove that every nonnegative continuous real-valued function on a given compact metric space is the uniform limit of some increasing sequence of nonnegative simple functions being linear combinations of indicators of open sets; here the nontriviality is relative to the standard choice(s) of approximating simple functions for measurable functions, where one loses control over the indicated measurable sets. Thus the standard uniform approximation of bounded nonnegative measurable real-valued functions by increasing nonnegative simple functions may be improved for nonnegative continuous real-valued functions on compact metric spaces. There are also some interesting consequences regarding semi-continuous functions and smooth functions.
