LUZIN'S (N) AND RANDOMNESS REFLECTION

We show that a computable function has Luzin's property (N) if and only if it reflects -randomness, if and only if it reflects -randomness, and if and only if it reflects -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 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 -Kurtz randomness. This links classical real analysis with algorithmic randomness.