论文标题
在波兰度量空间上的Borel功能几乎均匀的连续性上
On Almost Uniform Continuity of Borel Functions on Polish Metric Spaces
论文作者
论文摘要
我们表明，在任何给定的有限鲍尔测量空间上，环境空间是波兰度量空间，每个Borel真实价值的功能几乎都是一个有界有界的，均匀连续的功能，从某种意义上说，对于每一个$ \ varepsilon> 0 $，都有有界有界的，均匀的连续功能，以至于他们不同意他们的设置，他们都会同意$ <\ varepsilon $ <\ varepsilon $ <\ varepsilon $。特别是，该结果补充了在有限的ra尺测量空间上，其环境空间是局部紧凑的公制空间，该结果几乎均匀的连续性。 As direct applications in connection with some common modes of convergence, under our assumptions it holds that i) for every Borel real-valued function there is some sequence of bounded, uniformly continuous functions converging in measure to it, and ii) for every bounded, Borel real-valued function there is some sequence of bounded, uniformly continuous functions converging in $L^{p}$ to it.
We show that, on any given finite Borel measure space with the ambient space being a Polish metric space, every Borel real-valued function is almost a bounded, uniformly continuous function in the sense that for every $\varepsilon > 0$ there is some bounded, uniformly continuous function such that the set of points at which they would not agree has measure $< \varepsilon$. In particular, this result complements the known result of almost uniform continuity of Borel real-valued functions on a finite Radon measure space whose ambient space is a locally compact metric space. As direct applications in connection with some common modes of convergence, under our assumptions it holds that i) for every Borel real-valued function there is some sequence of bounded, uniformly continuous functions converging in measure to it, and ii) for every bounded, Borel real-valued function there is some sequence of bounded, uniformly continuous functions converging in $L^{p}$ to it.
