Luzin's (N) and randomness reflection

Abstract: We show that a computable real-valued function f has Luzin's property (N) if and only if it reflects Pi^1_1-randomness, if and only if it reflects Delta^1_1-randomness relative to Kleene's O, and if and only if it reflects Kurtz randomness relative to Kleene's O. 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 Kurtz randomness relative to 0'. This links classical real analysis with algorithmic randomness. Joint with Arno Pauly and Liang Yu.

logic

Audience: researchers in the topic

Computability theory and applications

Series comments: Description: Computability theory, logic

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