Luzin（N）与随机性反射
Luzin's (N) and randomness reflection
摘要：
证明了可计算函数$f:\mathbb R\rightarrow\mathbb R$具有Luzin性质（N）当且仅当它反映$Π^1_1$ -随机数，当且仅当它反映$Δ^1_1(\mathcal O)$ -随机数，当且仅当它反映$\mathcal O$ -Kurtz随机数，但反映Martin-Löf随机数或弱2-随机数是不够的.这里，如果每当$f(x)$是$R$ -随机的，则$x$也是$R$ -随机的，则函数$f$被称为反映随机性概念$R$。如果另外已知$f$具有有界变差，则我们证明$f$具有Luzin（N）当且仅当它反映弱2-随机性，并且当且仅当它反映$\emptyset'$ -Kurtz随机性。这将经典的实分析与算法随机性联系起来。
英文摘要：
We show that a computable function $f:\mathbb R\rightarrow\mathbb R$ has Luzin's property (N) if and only if it reflects $Π^1_1$-randomnes, if and only if it reflects $Δ^1_1(\mathcal O)$-randomness, and if and only if it reflects $\mathcal O$-Kurtz randomness, but reflecting Martin-Löf randomness or weak-2-randomness does not suffice. Here a function $f$ is said to reflect a randomness notion $R$ if whenever $f(x)$ is $R$-random, then $x$ is $R$-random as well. If additionally $f$ is known to have bounded variation, then we show $f$ has Luzin's (N) if and only if it reflects weak-2-randomness, and if and only if it reflects $\emptyset'$-Kurtz randomness. This links classical real analysis with algorithmic randomness.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
