论文标题
单变量均匀空间值的近似变化和点式选择原理
The approximate variation of univariate uniform space valued functions and pointwise selection principles
论文作者
论文摘要
令$ t \ subset \ mathbb {r} $和$（x，x，\ mathcal {u}）$是一个统一的空间，最多可计数的pseudometrics $ \ {d_p：p \ in \ mathcal {p} p} p \}给定$ f \ in x^t $（=所有功能的家族从$ t $到$ x $），$ f $的大约变化是两参数family $ \ { $ v _ {\ varepsilon，p}（f）$是约旦变体的最大下限$ v_p（g）$ t $ on $ t $相对于所有函数的$ d_p $ in x^t $ in x^t $ in x^t $ in x^t $中的$ d_p $，以便$ d_p（f（t），g leps，g lepsil $ t $ t。我们建立以下重点选择原则：如果函数的相对顺序紧凑序列$ \ {f_j \} _ {j = 1}^\ infty \ subset x^t $就是$ \ limsup_ { $ \ varepsilon> 0 $和$ p \ in \ mathcal {p} $，然后包含一个副序列，将$ t $以$ t $ cmanceed toce to cmanded tum to $ t $收敛到x^t $中的有限调节函数$ f \。我们通过适当的示例说明了这一结果，并在近似变化方面对受调节功能的表征$ f \表示。
Let $T\subset\mathbb{R}$ and $(X,\mathcal{U})$ be a uniform space with an at most countable gage of pseudometrics $\{d_p:p\in\mathcal{P}\}$ of the uniformity $\mathcal{U}$. Given $f\in X^T$ (=the family of all functions from $T$ into $X$), the approximate variation of $f$ is the two-parameter family $\{V_{\varepsilon,p}(f):\varepsilon>0,p\in\mathcal{P}\}$, where $V_{\varepsilon,p}(f)$ is the greatest lower bound of Jordan's variations $V_p(g)$ on $T$ with respect to $d_p$ of all functions $g\in X^T$ such that $d_p(f(t),g(t))\le\varepsilon$ for all $t\in T$. We establish the following pointwise selection principle: If a pointwise relatively sequentially compact sequence of functions $\{f_j\}_{j=1}^\infty\subset X^T$ is such that $\limsup_{j\to\infty}V_{\varepsilon,p}(f_j)<\infty$ for all $\varepsilon>0$ and $p\in\mathcal{P}$, then it contains a subsequence which converges pointwise on $T$ to a bounded regulated function $f\in X^T$. We illustrate this result by appropriate examples, and present a characterization of regulated functions $f\in X^T$ in terms of the approximate variation.
