完全决定的Borel集与可测性
Completely determined Borel sets and measurability
摘要：
我们考虑陈述$\mathsf{C\text-DM}$的逆数学强度：“每一个完全确定的Borel集都是可测的。“在$\mathsf{WWKL}_0$上，我们得到了类似于先前研究的范畴情况的以下结果。$\mathsf{C\text-DM}$严格介于$\mathsf{ATR}_0$和$\mathsf{L}_{ω_1,ω}\text-\mathsf{CA}$之间。只要$M\subseteq 2^ω$是$\mathsf{C\text-DM}$的$ω$模型的二阶部分，则对于每个$Z \in M$，存在一个$R \in M$，使得$R$相对于$Z$是$Δ^1_1$ -随机的。另一方面，如果没有$\mathsf{WWKL}_0$，所有集合的测度都为零（根据$\mathsf{C\text-DM}$进行测量），因此毫无意义地得出$\neg \mathsf{WWKL}_0$意味着$\mathsf{C\text-DM}$优于$\mathsf{RCA}_0$。
英文摘要：
We consider the reverse math strength of the statement $\mathsf{C\text-DM}$:"Every completely determined Borel set is measurable." Over $\mathsf{WWKL}_0$, we obtain the following results analogous to the previously studied category case. $\mathsf{C\text-DM}$ lies strictly between $\mathsf{ATR}_0$ and $\mathsf{L}_{ω_1,ω}\text-\mathsf{CA}$. Whenever $M\subseteq 2^ω$ is the second-order part of an $ω$-model of $\mathsf{C\text-DM}$, then for every $Z \in M$, there is a $R \in M$ such that $R$ is $Δ^1_1$-random relative to $Z$. On the other hand, without $\mathsf{WWKL}_0$, all sets have measure zero (as measured according to $\mathsf{C\text-DM}$), and it follows vacuously that $\neg \mathsf{WWKL}_0$ implies $\mathsf{C\text-DM}$ over $\mathsf{RCA}_0$.
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