一般连续图像的维数与测度
Dimension and measure for generic continuous images
摘要：
我们考虑由从任意不可数紧度量空间$X$到$\mathbb{R}^n$的连续函数组成的巴拿赫空间。关键的问题是“$f(X)$的一般层面是什么？我们考虑了两种不同的方法来回答它：Baire类别和患病率。在Baire类设置，我们证明，通常的包装和上盒尺寸尽可能大，$n$，但发现行为的豪斯多夫，较低的盒子和拓扑尺寸是相当微妙的。实际上，它们通常等于$n$的最小值和$X$的拓扑维数。我们还研究了$f(X)$的典型Hausdorff测度和填充测度，特别地，给出了它们为零、正且有限或无限的充要条件.将Baire分类结果与患病率设置中的结果进行比较是有趣的。因此，我们还讨论了Doughnard的结果的流行的拓扑维数的$f(X)$，并给出了一些简单的应用程序有关的流行维数的实值连续函数图的紧度量空间，使我们能够扩展最近的结果Bayart和Heurteaux。
英文摘要：
We consider the Banach space consisting of continuous functions from an arbitrary uncountable compact metric space, $X$, into $\mathbb{R}^n$. The key question is `what is the generic dimension of $f(X)$?' and we consider two different approaches to answering it: Baire category and prevalence. In the Baire category setting we prove that typically the packing and upper box dimensions are as large as possible, $n$, but find that the behaviour of the Hausdorff, lower box and topological dimensions is considerably more subtle. In fact, they are typically equal to the minimum of $n$ and the topological dimension of $X$. We also study the typical Hausdorff and packing measures of $f(X)$ and, in particular, give necessary and sufficient conditions for them to be zero, positive and finite, or infinite. It is interesting to compare the Baire category results with results in the prevalence setting. As such we also discuss a result of Dougherty on the prevalent topological dimension of $f(X)$ and give some simple applications concerning the prevalent dimensions of graphs of real-valued continuous functions on compact metric spaces, allowing us to extend a recent result of Bayart and Heurteaux.
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