Lebesgue density influences Hausdorff measure; large sets surface-like from many directions
Lebesgue density influences Hausdorff measure; large sets surface-like from many directions
复制标题
勒贝格密度影响豪斯多夫测度；
DOI：
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发表时间：
1978
期刊：
影响因子：
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通讯作者：
H. Fast
中科院分区：
文献类型：
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作者：
R. Davies;H. Fast
在几何测度论中典型的反例构造中，从某个初始集合出发，通过逐次约简得到一个递减的集合序列Fn，其交集具有一定的性质；要求∩Fn具有较大的Hausdorf F维。人们早就知道，这通常可以通过使每个Fn+1在Fn中足够“稠密”来实现。我们的第一个定理以一种精确的形式表达了这个直观的想法，我们认为这种形式既是新的，也是潜在的有用的，即使只是为了简化这种情况下的论述。我们的第二个定理正是使用这样的结构来解决最初激发这项工作的问题：ℝk中的Borel集是否具有Hausdorff维数k，但对于连续统-每个角度的许多方向在该方向上的每条直线上至多有一个点？这样的方向的集合必须有测量零，因为实际上在几乎所有的方向上，在1维的集合中都有满足Borel集的直线：这可以很容易地从Mattila[5]的定理6.6推导出来，该定理推广了Marstrand在k=2的情况下的结果[4]。
In a typical counter-example construction in geometric measure theory, starting from some initial set one obtains by successive reductions a decreasing sequence of sets F n , whose intersection has some required property; it is desired that ∩ F n shall have large Hausdorf F dimension. It has long been known that this can often be accomplished by making each F n+1 sufficiently “dense” in F n . Our first theorem expresses this intuitive idea in a precise form that we believe to be both new and potentially useful, if only for simplifying the exposition in such cases. Our second theorem uses just such a construction to solve the problem that originally stimulated this work: can a Borel set in ℝ k have Hausdorff dimension k and yet for continuum-many directions in every angle have at most one point on each line in that direction? The set of such directions must have measure zero, since in fact in almost all directions there are lines that meet the Borel set (of dimension k ) in a set of dimension 1: this can easily be deduced from Theorem 6.6 of Mattila [5], which generalized Marstrand's result [4] for the case k = 2.