具有平移不变Borel测度的测度集
Measuring sets with translation invariant Borel measures
摘要：
继Davies，Elekes和Keleti之后，我们研究了测量集，即$\mathbb{R}$（或波兰群）中的Borel集$B$，其中有一个平移不变Borel测度赋予$B$正测度和σ-有限测度.我们调查哪些集可以被写为一个（不相交）工会的测量集。证明了第二类Borel空集$B\subset \mathbb{R}$大于任何空集$A\subset \mathbb{R}$，即存在$B=B_1\cup B_2$，$A=A_1\cup A_2$和规范函数$g_1, g_2$，使得Hausdorff测度满足$H^{g_i}(B_i)=1$和$H^{g_i}(A_i)=0$（$i=1,2$）.这意味着第二类的每一个波莱尔集都是两个可测集的并集。我们还提出了博雷尔和紧集$\mathbb{R}$这不是一个联盟的可数许多测量集。这分两步完成。首先，我们证明了非局部紧波兰群不是可数多个测量集的并集。然后，我们在一定的Banach空间中给出了$\mathbb{R}$的Borel和/或σ-紧可加子群，它不是可数多个可测集的并集.对于任意规范函数的每个Hausdorff测度，都存在零或非σ-有限的测度集.
英文摘要：
Following Davies, Elekes and Keleti, we study measured sets, i.e. Borel sets $B$ in $\mathbb{R}$ (or in a Polish group) for which there is a translation invariant Borel measure assigning positive and σ-finite measure to $B$. We investigate which sets can be written as a (disjoint) union of measured sets. We show that every Borel nullset $B\subset \mathbb{R}$ of the second category is larger than any nullset $A\subset \mathbb{R}$ in the sense that there are partitions $B=B_1\cup B_2$, $A=A_1\cup A_2$ and gauge functions $g_1, g_2$ such that the Hausdorff measures satisfy $H^{g_i}(B_i)=1$ and $H^{g_i}(A_i)=0$ ($i=1,2$). This implies that every Borel set of the second category is a union of two measured sets. We also present Borel and compact sets in $\mathbb{R}$ which are not a union of countably many measured sets. This is done in two steps. First we show that non-locally compact Polish groups are not a union of countably many measured sets. Then, to certain Banach spaces we associate a Borel and/or σ-compact additive subgroup of $\mathbb{R}$ which is not a union of countably many measured sets. It is also shown that there are measured sets which are null or non-σ-finite for every Hausdorff measure of arbitrary gauge function.
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