LLuzin’s (N) and randomness reflection
Arno Pauly
Department of Computer ScienceSwansea UniversitySwansea, UK
Linda Westrick
Department of MathematicsPenn State UniversityUniversity Park, PA, US [email protected]
Liang Yu
Mathematical DepartmentNanjing UniversityNanjing City, China [email protected]
We show that a computable function f : R → R has Luzin’s property (N) if and only if itreflects ∆ ( O )-randomness, and if and only if it reflects O -Kurtz randomness, but reflectingMartin-L¨of randomness or weak-2-randomness does not suffice. Here a function f is said toreflect a randomness notion R if whenever f ( x ) is R -random, then x is R -random as well. Ifadditionally f is known to have bounded variation, then we show f has Luzin’s (N) if andonly if it reflects weak-2-randomness, and if and only if it reflects ∅ (cid:48) -Kurtz randomness. Thislinks classical real analysis with algorithmic randomness. We revisit a notion from classic real analysis, namely Luzin’s property (N), from the perspectiveof computability theory. A function f : R → R has Luzin’s (N), if the image of any (Lebesgue)null set under f has again measure 0. This concept was studied extensively by Luzin in histhesis [13]. For functions with bounded variation, this notion is just equivalent to absolutelycontinuous functions – but already for general continuous functions, Luzin’s (N) is a somewhatintricate property. A formal result amounting to this was obtained by Holick´y, Ponomarev,Zaj´jˇcek and Zelen´y, showing that the set of functions with Luzin’s (N) is Π -complete in thespace of continuous functions [9].From a computability-theoretic perspective, Luzin’s (N) is readily seen to be some kind ofrandomness reflection: By contraposition, it states that whenever f [ A ] has positive measure(i.e. contains a random point for a suitable notion of randomness), then A has positive measure,too (i.e. contains a random point). It thus seems plausible that for some suitable randomnessnotion, Luzin’s (N) for computable functions is equivalent to saying that whenever f ( x ) israndom, then so is x . Our main finding (Theorem 16) is that this is indeed the case, and that∆ ( O )-randomness is such a suitable randomness notion. An indication that this is a non-trivialresult is that our proof uses ingredients such as Friedman’s conjecture (turned into a theoremby Martin [8, 14, 25]).While the exploration of how randomness interacts with function application, and the generallinks to real analysis, has a long tradition (see e.g. the survey by Rute [21]), the concepts ofrandomness preservation (if x is random, so is f ( x )) and no-randomness-from-nothing (if y is random, then there is some random x ∈ f − ( y )) have received far more attention thanrandomness reflection. Our results not only fill this gap, but may shed a light on why randomnessreflection has been less popular: As the most natural notion of randomness reflection turns outto be ∆ ( O )-randomness reflection, we see that studying higher randomness is essential for thisendeavour. a r X i v : . [ m a t h . L O ] J un Randomness reflection
Our theorems and proofs generally refer to computability. However, we stress that since theresults relativize, one can obtain immediate consequences in classic real analysis. An exampleof this is Corollary 17, which recovers a theorem by Banach. More such examples can be foundin Section 8, where, by applying relativized computability method, we are able to prove someresults in classical analysis. While we are not aware of such consequences that would advance thestate of the art in real analysis, it is plausible that future use of our techniques could accomplishthis.
Overview of our paper
In Section 2 we do not discuss randomness reflection at all, butrather prove a result in higher randomness of independent interest. Theorem 1 is of the form “ifa somewhat random X is hyp-computed by a very random Y , then X is already very random”.It is the higher randomness analog of [15, Theorem 4.3] by Miller and the third author. Thisresult is a core ingredient of our main theorem.Section 3.1 contains the main theorem of our paper, the equivalence of Luzin’s (N) forcomputable functions with ∆ ( O )-randomness reflection. We consider higher Kurtz randomnessin Section 3.3, and show that for continuous functions f : R → R , Luzin’s (N) is equivalent tothe reflection of O -Kurtz randomness, and separate this from ∆ -Kurtz randomness reflection.In Subsection 3.4 we discuss the open questions raised by our main theorem: Just becauseLuzin’s (N) is equivalent to ∆ ( O )-randomness reflection does not mean that it cannot bealso equivalent to other notions of randomness reflection. For some notions, in particular forMartin-L¨of-randomness reflection and weak-2-randomness reflection, we provide a separationfrom Luzin’s (N) in Section 4.In Sections 5 and 6 we consider Luzin’s (N) for more restricted classes of functions, namelyfunctions with bounded variation and strictly increasing functions. Here Luzin’s (N) turns outto be equivalent to weak-2-randomness reflection, but we can still separate it from several otherrandomness-reflection-notion. These investigations tie in to a project by Bienvenu and Merkle[2] regarding how two computable measures being mutually absolutely continuous (i.e. havingthe same null sets) relates to randomness notions for these measures coinciding.In Section 7 we take a very generic look at the complexity of randomness reflection, and showthat the Π -hardness established for Luzin’s (N) in [9] applies to almost all other randomnessreflection notions, too.Section 8 contains a brief digression about functions where the image of null sets is smallin some other sense (countable or meagre). We prove these classical analysis results via variousclassical and higher computability methods.We then conclude in Section 9 with a discussion of how this line of investigation could becontinued in the future. Throughout, we assume familiarity with the theory of algorithmic randomness and higher ran-domness in particular. A standard references for the former are [6] and [17]. For the latter,readers may refer to [5]. We use standard computability-theoretic notation. The Lebesguemeasure is denoted by λ .Our goal in this section is to establish the following: . Pauly, L. Westrick & L. Yu Theorem 1.
Let Y be ∆ ( Z )-random and Π -random, let X be ∆ -random and let X ≤ h Y .Then X is ∆ ( Z )-random.This is a higher-randomness counterpart to [15, Theorem 4.3], and the proof proceeds byadapting both this and [15, Lemma 4.2]. We will use the theorem in the following form: Corollary 2.
Let Z ≥ O . If X is ∆ -random, Y is ∆ ( Z )-random and X ≤ h Y , then X is∆ ( Z )-random. Lemma 3.
Fix α ∈ O and e ∈ N . If X is ∆ -random, then: ∃ c ∀ n λ ( { Y | ϕ e ( Y ( α ) ) ∈ [ X (cid:22) n ] } ) < − n + c Proof.
Analogous to the proof of [15, Lemma 4.2]. Let H σ = { Y | ϕ e ( Y ( α ) ) ∈ [ σ ] } , and thenlet F i = { σ | λ ( H σ ) > −| σ | + i } . By construction, the H σ are uniformly ∆ α +1 (as subsets of { , } N ), and so the sets F i are uniformly ∆ α +2 (as subsets of 2 <ω ).A counting argument shows that λ ([ F i ]) < − i : Pick a prefix free set D ⊆ F i with [ D ] = [ F i ].Then: 1 ≥ λ ( (cid:91) σ ∈ D H σ ) = (cid:88) σ ∈ D λ ( H σ ) ≥ (cid:88) σ ∈ D −| σ | + i = 2 i λ ([ F i ])We see that ([ F i ]) i ∈ N is a Martin-L¨of test relative to ∅ α +2 . Since X is ∆ -random, there hasto be some c ∈ N with X / ∈ [ F c ]. This in turn means that ∀ n ∈ N X (cid:22) n / ∈ F c , which by definitionof F c is the desired claim. Fact 4 (Sacks [22]) . ∆ ( Z )-randomness (defined by being contained in no ∆ ( Z )-null sets) isequivalent to being ˆ Z -random for every ˆ Z ∈ ∆ ( Z ). Lemma 5.
Fix α ∈ O and e ∈ N . If X = ϕ e ( Y ( α ) ), X is ∆ -random and Y is ∆ ( Z )-random,then X is ∆ ( Z )-random. Proof.
We follow the proof of [15, Theorem 4.3]. Let c be the constant guaranteed for X byLemma 3. As in the proof of Lemma 3, let H σ = { W | ϕ e ( W ( α ) ) ∈ [ σ ] } . Let G σ = H σ if λ ( H σ ) < −| σ | + c and G σ = ∅ else. Note that G σ is still uniformly ∆ . The choice of c inparticular guarantees that Y ∈ G X (cid:22) n for each n ∈ N .Let ∩ n U n denote a Martin-L¨of test relative to ˆ Z for some ˆ Z ∈ ∆ ( Z ). By Fact 4, it sufficesto show that X (cid:54)∈ ∩ n U n . We set K i = (cid:83) σ ∈ U c + i G σ , and K = (cid:84) i ∈ N K i . We find that K is ∆ ( Z ).Moreover, we have that: λ ( K i ) ≤ (cid:88) σ ∈ U c + i λ ( G σ ) ≤ (cid:88) σ ∈ U c + i −| σ | + c ≤ − i Hence, it follows that λ ( K ) = 0, so for some i , Y (cid:54)∈ K i . Then X (cid:54)∈ U c + i , because Y ∈ G σ for all σ ≺ X . Fact 6 (Sacks [22]) . If Y is Π -random, then ω CK1 = ω CK ,Y . Proof of Theorem 1.
Since Y is Π -random, we know that X ≤ h Y implies the existence of some α ∈ O and e ∈ N such that X = ϕ e ( Y ( α ) ) (rather than merely α ∈ O Y ). We can thus invokeLemma 5 to conclude that X is ∆ ( Z )-random. Randomness reflection
As an aside, the requirement in Theorem 1 that Y be Π -random might be unexpected atfirst – it has no clear counterpart in [15, Theorem 4.3]. The following example, which is notneeded for anything else in the paper, shows that this assumption is necessary. Example 7.
There are ∆ -random X and ∆ ( Z )-random Y with X ≤ h Y but X is not ∆ ( Z )-random. In fact, we shall chose X = Z , and make X even Π -random. Proof.
Let Y be a ∆ -random satisfying Y ≥ h O . The existence of such a Y was shown in [4].Let X be Martin-L¨of random relative to Y ⊕ O while satisfying X ≤ h Y . This choice ensuresthat X is Π -random (so in particular ∆ -random).By van Lambalgen’s theorem relativized to ∅ ( α ) , if both X and Y are ∆ -random, then forany α < ω CK1 it holds that X is Y ⊕ ∅ ( α ) -random iff X ⊕ Y is ∅ ( α ) -random iff Y is X ⊕ ∅ ( α ) -random. Since by choice of X , we know that in particular X is Y ⊕ ∅ ( α ) -random, we concludethat Y is X ⊕ ∅ ( α ) -random.From ([3, Corollary 4.3]) it follows that for Π -random X and β < ω CK1 it holds that X ( β ) ≤ T X ⊕ ∅ ( β ) . (The conclusion above surely does not require full Π -randomness of X , but too muchprecision would take us afield.) Together with the above, this shows that Y is X ( α ) -random forevery α < ω CK1 . Since X is Π -random, by Fact 6 we have that ω CK1 = ω CK ,X , and thus that Y is Z -random for any Z ∈ ∆ ( X ). By Fact 4 this establishes Y to be ∆ ( X )-random. Buttrivially, X cannot be ∆ ( X )-random. Definition 8.
A function satisfies Luzin’s (N) iff the image of every null set is null.
Definition 9.
For any randomness notion R and a function f , we say that f reflects R -randomness if f ( x ) is R -random implies x is R -random for all x in the domain of f . By noting that the sets of points not Martin-L¨of random relative to some oracle are canonicalchoices of null sets, we obtain the following:
Proposition 10.
The following are equivalent for a computable function f : R → R :1. f satisfies Luzin’s (N)2. ∀ p ∈ { , } N ∃ q ∈ { , } N f ( x ) ∈ MLR( q ) ⇒ x ∈ MLR( p )3. ∀ p ∈ { , } N f ( x ) ∈ ∆ − random( p ) ⇒ x ∈ MLR( p )4. ∀ p ∈ { , } N f ( x ) ∈ ∆ − random( p ) ⇒ x ∈ ∆ − random( p ) Proof. . ⇔ . Each null set is contained in a set of the form MLR( q ) C for some oracle q ∈{ , } N . Luzin’s (N) is thus equivalent to saying that for any p there is a q with f [MLR( p ) C ] ⊆ MLR( q ) C . Taking the contrapositive yields (2).2 . ⇒ . Any Σ -null set is contained in a ∆ -null set ([22]). Thus it suffices to choose q assomething hyperarithmetical in p .3 . ⇒ . Trivial. . Pauly, L. Westrick & L. Yu . ⇒ . Assume that (4) fails, i.e. that there is some p ∈ { , } N and some x / ∈ ∆ − random( p )with f ( x ) ∈ ∆ − random( p ). But if x / ∈ ∆ − random( p ), then there is some q ≤ h p with x / ∈ MLR( q ), but f ( x ) ∈ ∆ − random( q ) = ∆ − random( p ), hence (3) is violated, too.4 . ⇒ . Trivial.
Corollary 11.
A computable function satisfying Luzin’s (N) reflects ∆ -randomness relativeto any oracle.We can now ask whether reflecting ∆ -randomness relative to some specific oracle alreadysuffices. Fact 12 ([14, 25]) . If A is an uncountable ∆ ( y )-class such that y ≤ h z for every z ∈ A , thenthere is some x ∈ A with O y ≤ h x . Fact 13 (Sacks [22]) . If O ≤ h x , then x is not ∆ ( O )-random. Corollary 14.
If computable f reflects ∆ ( O )-randomness, then for any ∆ ( O )-random y wefind that f − ( y ) is countable. Proof.
Assume that y is ∆ ( O )-random and f − ( y ) is uncountable. Then by Claim 12, there issome x ∈ f − ( y ) with O ≤ h x . By Fact 13, we find that x is not ∆ ( O )-random, contradictingthat f reflects ∆ ( O )-randomness. Observation 15.
The following are equivalent for computable f : R → R :1. For almost all y it holds that f − ( { y } ) is countable.2. For every ∆ -random y it holds that f − ( { y } ) is countable. Proof.
The implication 2 ⇒ { y | f − ( { y } ) is uncountable } is Σ . This holds because for a Σ -set A , being uncountable is equivalent to containing anelement which is not hyperarithmetic relative to A . Due to Kleene’s HYP-quantification the-orem, an existential quantifier over non-hyperarithmetic elements is equivalent an unrestrictedexistential quantifier. By assumption, it is a null set. Any Σ -null set is contained in a ∆ -nullset, so it is then contained in a ∆ -null set, and so cannot contain any ∆ -randoms. Theorem 16.
The following are equivalent for computable f : R → R :1. f satisfies Luzin’s (N)2. f reflects ∆ ( O )-randomness.3. f reflects ∆ -randomness and for almost all y , f − ( y ) is countable. Proof.
That (1) implies (2) follows from Proposition 10. To see that (2) implies (1), we showthat if f reflects ∆ ( O )-randomness, then it reflects ∆ ( r )-randomness for all r ≥ T O . This willbe enough because if f reflects ∆ ( r )-randomness for all r ≥ T O , then for any p ∈ { , } N , if f ( x ) ∈ MLR( O p ⊕O ), then f ( x ) is ∆ ( p ⊕ O )-random, so x is ∆ ( p ⊕ O )-random, so x ∈ MLR( p ).Thus f satisfies condition (2) of Proposition 10. Randomness reflection
So let y be ∆ ( r )-random for some r ≥ T O . By Corollary 14, f − ( y ) is countable. So if x ∈ f − ( y ), then x ≤ h y . Since f reflects ∆ ( O )-randomness, we know that x is ∆ -random.We can thus invoke Theorem 1 to conclude that x is ∆ ( r )-random, and have reached our goal.To see that (1) implies (3) we use Proposition 10 and Corollary 14. The proof that (3)implies (1) proceeds analogously to the proof that (2) implies (1), except that we conclude that f − ( { y } ) is countable from Observation 15 rather than Corollary 14. We obtain as a corollary a reproof of a theorem by Banach [1], cf. [23, Chapter IX, Theorem7.3]:
Corollary 17. If f is continuous and satisfies Luzin’s (N), then for almost all y we find that f − ( y ) is countable.The following generalization to measurable functions was also known, but we give a newproof. Corollary 18.
1. If f satisfies Luzin’s (N) and there are a continuous function g and aBorel set A so that f agrees with g on A , then for almost every real y ∈ f ( A ) we find that f − ( y ) ∩ A is countable.2. If f satisfies Luzin’s (N) and is measurable, then for almost every real y we find that f − ( y ) is countable. Proof. (1). Fix a real x so that g is computable in x and A is ∆ ( x ). Assume that y is ∆ ( O x )-random and f − ( y ) ∩ A = g − ( y ) ∩ A is uncountable. Since A is ∆ ( x ), by Claim 12, then thereis some z ∈ g − ( y ) ∩ A = f − ( y ) ∩ A with O x ≤ h z ⊕ x . By Fact 13, we find that z is not∆ ( O x )-random, contradicting that g reflects ∆ ( O x )-randomness.(2). By Luzin’s theorem, there are a sequence Borel sets { A n } n ∈ ω and continuous functions { g n } n ∈ ω so that R \ (cid:83) n A n is null and f agrees with g n over A n for every n . As f has Luzin’s(N), also f [ R \ (cid:83) n A n ] is null, and thus can be ignored for our argument. For y / ∈ f [ R \ (cid:83) n A n ],we find that f − ( y ) ⊆ (cid:83) n ∈ N g − n ( y ). Since each set in the right-hand union is countable foralmost all y by Corollary 17, the union itself is countable for almost all y , proving the claim.Note that the Borelness of the set A in the corollary above cannot be replaced by “arbitraryset”, as it is consistent with ZFC that the corresponding statement is false. For example,assuming the continuums hypothesis (CH) or the even weaker Martin’s axiom (MA) suffices toconstruct a counterexample. We do not know whether the following proposition can be provedwithin ZFC. Proposition 19 (ZFC + MA) . There is a function f : [0 , → [0 ,
1] having Luzin’s (N) anda set A ⊆ [0 ,
1] such that f | A is a computable, f [ A ] is non-null, and for any y ∈ f [ A ] the set f − ( { y } ) ∩ A is uncountable. Proof.
We actually need only a weaker condition than MA for our construction, namely theequality cof( L ) = cov( L ) = non( L ) in Cicho´n’s diagram. Recall that cof( L ) is the least cardi-nality of a set R of null sets such that any null set is a subset of an element of R ; cov( L ) is theleast cardinal α such that [0 ,
1] is a union of α -many null sets, and non( L ) is the least cardinalof a non-null set. It is a consequence of MA that all these cardinals are 2 ℵ . As they all are . Pauly, L. Westrick & L. Yu κ denotethe value of these three invariants.First, we observe that κ = cof( L ) means that there exists a family ( z α ) α<κ such that a set A ⊆ { , } N is null iff A ⊆ MLR( z α ) C for some α < κ . Next, we point out that κ = cov( L )means that for any α < κ and family ( w β ) β<α there exists some u which is Martin-L¨of randomrelative to all w β .We start with a family ( z α ) α<κ as above, and then choose ( x α ) α<κ such that each x α isMartin-L¨of random relative to any z β for β ≤ α . We then choose another sequence ( y α ) α<κ suchthat y α is Martin-L¨of random relative to any x β ⊕ z γ for β, γ ≤ α . We identify { , } N with apositive measure subset of [0 ,
1] (a fat Cantor set), and then define A = { y β ⊕ x α | α ≤ β < κ } ,and f : [0 , → [0 ,
1] as f ( y β ⊕ x α ) = x α and f ( w ) = 0 for w / ∈ A .As f | A is just the projection, it is clear that it is computable. To see that f [ A ] is non-null,note that if it were null, it would need to be contained in MLR( z α ) C for some α < κ . But x α ∈ f [ A ] is explicitly chosen to prevent this. For any x α ∈ f [ A ], we find that f − ( { x α } ) ∩ A = { y β ⊕ x α | α ≤ β < κ } has cardinality κ , and κ is uncountable.It remains to argue that f has Luzin’s (N). As f is constant outside of A , we only need toconsider null sets B ⊆ A . Again invoking van Lambalgen’s theorem, we see that any y β ⊕ x α isMartin-L¨of random relative to z γ whenever γ ≤ α ≤ β . As such, each null B ⊆ A is containedin some { y β ⊕ x α | α ≤ β < γ } for γ < κ . It follows that f [ B ] ⊆ { x α | α < γ } has cardinalitystrictly below κ , and hence is null due to κ = non( L ).That all fibers are countable is just preservation of h -degrees: Observation 20.
For computable f : R → R the following are equivalent:1. f preserves h -degrees, i.e. ∀ x ∈ [0 , x ≡ h f ( x ).2. For all y ∈ R , f − ( y ) is countable. In his thesis [13], Luzin showed that if a continuous function f : R → R fails to have property( N ), then in fact there is a compact witness to this failure. For the reader’s convenience, wegive a proof of this fact below. Proposition 21.
Let f : R → R be continuous and map some null set to a non-null set. Thenthere is a compact subset A ⊆ R with λ ( A ) = 0 and λ ( f ( A )) > Proof.
Observe that a function f : R → R satisfies Luzin’s (N) if and only if its restriction f (cid:22) [ a, b ] satisfies Luzin’s (N) for every closed interval [ a, b ]. So without loss of generality, weassume that f fails Lusin’s (N) because µ ( A ) = 0 but µ ( f ( A )) = d > A ⊆ [ a, b ]. Asevery null set is contained in a Π -null set, without loss of generality we can assume A = ∩ n U n for some decreasing sequence of open sets U n . Each U n is itself equal to an increasing unionof closed sets. The idea is by picking big enough closed F n ⊆ U n , we can find a closed set (cid:84) n ∈ N F n ⊆ A whose image still has positive measure. How large to pick the F n ? Let F ⊆ U be large enough that µ ( f ( F ∩ A )) > d/
2. In general, if we have found ( F i ) i
2, then since A ⊆ U n , we can find closed F n ⊆ U n large enough that µ ( f ( ∩ i
2, andtherefore µ ( ∩ n f ( ∩ i 2. Claim: ∩ n f ( ∩ i Given oracle Z , suppose that x is not Z -Kurtz random because x ∈ F , where F is a Z -computable compact set of measure 0. Then f ( F ) is also a Z -computable compact set, whichhas measure 0 because f has Luzin’s (N). Therefore, f ( x ) is also not Z -Kurtz random.An immediate consequence is that any f with Luzin’s (N) also reflects ∆ -Kurtz randomnessrelative to every oracle. In general, Kurtz randomness reflection for stronger oracles implies itfor weaker ones. Proposition 23. If a continuous function f : R → R reflects Z -Kurtz randomness, then f reflects X -Kurtz randomness for every X ≤ T Z . Proof. Assume that f reflects Z -Kurtz randomness. Suppose x is not X -Kurtz random. Let P be an X -computable compact null set with x ∈ P . Then f ( P ) is an X -computable compact set,which is null because P is also Z -computable. Therefore, f ( x ) is not X -Kurtz random.Additionally, any witness to the failure of Luzin’s (N) also provides an oracle relative towhich Kurtz randomness reflection fails. Proposition 24. Suppose that f : R → R and A ⊆ R is a Z -computable compact null set with λ ( f ( A )) > 0. Then f does not reflect Z -Kurtz randomness. Proof. Since f ( A ) is also Z -computable and has positive measure, it must contain some Z -Kurtzrandom y . There is some x ∈ A with f ( x ) = y , but since A is a Z -computable compact null set,it cannot contain any Z -Kurtz randoms. Hence, f does not reflect Z -Kurtz randomness.We can thus characterize Luzin’s (N) in terms of Kurtz randomness reflection. Theorem 25. The following are equivalent for computable f : R → R :1. f has Luzin’s (N).2. For every O -computable compact set A with λ ( A ) = 0 also λ ( f ( A )) = 0.3. f reflects O -Kurtz randomness. Proof. . ⇒ . Trivial.2 . ⇒ . We observe that given computable f : R → R and number n , the set { A ⊆ [ − n, n ] compact | λ ( A ) = 0 ∧ λ ( f ( A )) ≥ − n } is a Π -subset of the Polish space of compact subsets of [ − n, n ]. By Proposition 21, if f fails Luzin’s (N), this set is non-empty for some n . If it is non-empty, it must have an O -computable element by Kleene’s basis theorem. . Pauly, L. Westrick & L. Yu . ⇒ . By Proposition 22.3 . ⇒ . By Proposition 24.We also see that ∆ -Kurtz randomness reflection does not suffice for a characterization. Lemma 26. Reflecting ∆ -Kurtz randomness is a Σ -property of continuous f : R → R . Proof. By Proposition 24, reflecting ∆ -Kurtz randomness is equivalent to the statement thatfor any ∆ compact set A with λ ( A ) = 0 we have λ ( f ( A )) = 0. By Kleene’s HYP quantificationtheorem [11, 12], a universal quantification over ∆ can be replaced by an existential quantifi-cation over Baire space. That λ ( A ) = 0 implies λ ( f ( A )) = 0 is a ∆ -statement for given f and A . Corollary 27. Reflecting ∆ -Kurtz randomness is strictly weaker than Luzin’s (N). Proof. By Proposition 22, Luzin’s (N) implies ∆ -Kurtz randomness reflection. By Lemma 26reflecting ∆ -Kurtz randomness is a Σ -property. But it was shown in [9] that Luzin’s (N) isΠ -complete for continuous functions. Thus the two notions cannot coincide. Theorems 16 and 25 tell us that ∆ ( O )-randomness reflection and O -Kurtz randomness reflectioneach characterize Luzin’s (N) for computable functions. This does not rule out that other kindsof randomness reflection could also characterize Luzin’s (N). In the next section we shall see thatnone of MLR-reflection, W2R-reflection, or MLR( ∅ (cid:48) )-reflection imply Luzin’s (N) for arbitrarycomputable functions (Corollary 31). Because reflection asks for the same level of randomnesson both sides, there are no completely trivial implications between the Π -type randomnessreflection notions. Indeed, results in [2] suggest that the implication structure between Π -type randomness reflection notions may have little relation to the implication structure betweennotions of randomness. However, the most interesting open question seems to be: Open Question 28. Can a computable function reflect ∆ -randomness but fail Luzin’s (N)?By Theorem 16 any such example would need to have a positive measure of fibers beinguncountable, which is incompatible with most niceness conditions. We also do not know theanswer to the above question if ∆ -randomness is replaced with Martin-L¨of randomness relativeto ∅ ( α ) for any α ≥ f which fails Luzin’s (N) must see that failure witnessedby a O -computable compact set. The proof shows that such a set can also be chosen hyper-arithemetically low, by applying Gandy basis theorem in place of the Kleene. On the otherhand, Corollary 27 shows that a function which fails Luzin’s (N) need not have a hyperarith-metic compact witness. Indeed, one can obtain specific examples of this separation by feedingpseudo-well-orders into the Π -completeness construction of [9]. Thus the results for compactwitnesses are rather tight overall.The situation for the minimum complexity of Π witnesses is less well understood. Theproof of Corollary 31 shows that a computable function may fail Luzin’s (N) while still mappingall rapidly null Π ( ∅ (cid:48) ) sets to null sets. That is, the set MLR( ∅ (cid:48) ) C is mapped to a null set.0 Randomness reflection Open Question 29. Can a computable function map W R C to a null set but fail Luzin’s (N)?Equivalently, can a computable function map all null Π ( ∅ (cid:48) ) sets to null sets while failing Luzin’s(N)?We note that the functions produced by the Π -completeness construction of [9] are of nohelp because the failure of Lusin’s (N), when it occurs, is witnessed by an effectively null Π set. We present a construction of a computable function that violates Luzin’s (N), and yet ispiecewise-linear in a neighborhood of every point that is not MLR( ∅ (cid:48) ). Here, we say that f is piecewise-linear in a neighborhood of x , if there are rationals a < x < b such that f | [ a,x ] and f | [ x,b ] are linear functions. Computable piecewise-linear functions reflect essentially all kinds ofrandomness. Theorem 30. For each Π ( ∅ (cid:48) )-set A ⊆ [0 , 1] there is a computable function f : [0 , → [0 , x ∈ [0 , \ A , f is piecewise-linear on a neighbourhood of x .2. For every ε > 0, there is a null Π ( ∅ (cid:48)(cid:48) ) set B ⊆ A such that λ ( f [ B ]) ≥ λ ( A ) − ε . Corollary 31. There is a computable function f : [0 , → [0 , 1] that reflects ML-randomness,weak-2-randomness and ML( ∅ (cid:48) )-randomness, yet does not have Luzin’s (N), nor reflects weak-3-randomness. Proof. Let A be the complement of the first component of a universal ML( ∅ (cid:48) )-test. Then λ ( A ) > . We invoke Theorem 30 on A and (cid:15) = . The resulting function is the desired one: If x ∈ [0 , ∅ (cid:48) )-random, then x / ∈ A , f is piecewise-linear on a neighborhood of x , and thus f ( x )is not ML( ∅ (cid:48) )-random. As such, we conclude that whenever f ( x ) is ML( ∅ (cid:48) )-random, then so is x (same for the other notions).Since we can choose the witness B as being Π ( ∅ (cid:48)(cid:48) ), it is also Π ( ∅ (cid:48) ), and thus containsonly elements which are not weak-3-random. Since f [ B ] has positive measure, it contains aweak-3-random – hence f does not reflect weak-3-randomness.We remark that this is the strongest result possible for the strategy we are using. We aremaking sure that f reflects MLR( ∅ (cid:48) )-randomness by making f piecewise-linear in a neighborhoodof every non-MLR( ∅ (cid:48) ) point. However, the following proposition shows that the set of pointswhere f can be this simple has a descriptive complexity of Σ ( ∅ (cid:48) ). But the weak-3-non-randomsare not contained in any Σ ( ∅ (cid:48) ) set except [0 , Proposition 32. Let f : R → R be computable. The set of points where f is locally piecewise-linear is Σ ( ∅ (cid:48) ). Proof. We consider the property 2L of a function f and an interval [ a, b ] that there is some x ∈ [ a, b ] such that both f | [ a,x ] and f | [ x,b ] are linear. We first claim that this is a Π -property.To this, we observe that 2L is equivalent to: ∀ n ∈ N ∃ i ≤ n (cid:18) ∀ j ∈ { , . . . , n − } \ { i − , i, i + 1 } f ( a + jn ) − f ( a + j + 1 n ) = f ( a + j + 1 n ) − f ( a + j + 2 n ) (cid:19) Next, we observe that f is locally piecewise-linear in x iff there is some rational interval( a, b ) (cid:51) x such that f has property 2L on [ a, b ]. Using ∅ (cid:48) , we can enumerate all these intervals. . Pauly, L. Westrick & L. Yu L such that d ( f ( x ) , f ( y )) ≤ Ld ( x, y ) ≤ L d ( f ( x ) , f ( y )) for all x, y in the domain. Since computable locally bi-Lipschitzfunctions preserve and reflect all kinds of randomness, Another way for f to ensure a givennotion of randomness reflection is by being locally bi-Lipschitz on the non-random points forthat notion. However, we still get a Σ ( ∅ (cid:48) )-set of suitable points. Proposition 33. Let f : R → R be computable. The set of points where f is locally bi-Lipschitzis Σ ( ∅ (cid:48) ). Proof. The following is a co-c.e. property in a, b ∈ Q and L ∈ N and f ∈ C ( R , R ): ∀ x, y ∈ [ a, b ] d ( x, y ) ≤ Ld ( f ( x ) , f ( y )) ≤ L d ( x, y )We obtain the set of points where f is locally bi-Lipschitz by taking the union of all ( a, b ) havingthe property above for some L ∈ N – access to ∅ (cid:48) suffices to get such an enumeration. Before diving into the details of the proof of Theorem 30 we give a high-level sketch of what isgoing on. Consider first the case where A is Π . Then A = ∩ n A n , where each A n is a finiteunion of closed intervals and A n +1 ⊆ A n . We iteratively define a sequence of piecewise linearfunctions f , f , . . . , where f : [0 , → [0 , 1] is the identity, and f n is obtained from f n − byperforming a “tripling” operation on those line segments of f n − which are contained in A n . Inorder to “triple” a line segment, we replace it by a zig-zag of three line segments each of whichhas triple the slope of the original. (See Figure 1 ) We want to make the sequence ( f n ) convergein the supremum norm, so before tripling we add invisible break points to f n − so that none ofits linear pieces are more than 2 − n tall. Letting f be the limit function, observe that if x (cid:54)∈ A n ,then f coincides with f n on a neighborhood of x , and thus f is linear on a neighborhood of x .On the other hand, A is then exactly the set of points where we tripled infinitely often.Next we describe how to find a closed set B ⊆ A such that µ ( f ( B )) ≥ µ ( A ). (The ε in thestatement of the theorem exists in order to bring down the descriptive complexity of B , but wecan ignore it for now.) We want B ⊆ A , so of course we throw out of B any interval that leaves A . Also, every time we perform a tripling, we choose two-thirds of the tripled interval to throwout of B . We do this so that the one-third which we keep has maximal measure of intersectionwith A . Observe that µ ( B ) = 0.Here is why µ ( f ( B )) ≥ µ ( A ). Let B = [0 , 1] and let B n denote the set of points that remainin B at the end of stage n . By induction, f n (cid:22) B n is injective (except possibly at break points)and µ ( f n ( B n ∩ A )) ≥ µ ( A ). The key to the induction is that by the choice of thirds, we alwayshave 3 n µ ( B n ∩ A ) ≥ µ ( A ), and since f n has slope ± n on all of B n and is essentially injective, µ ( f n ( B n ∩ A )) = 3 n µ ( B n ∩ A ). It now follows that µ ( f n ( B n )) ≥ µ ( A ) for all n . Furthermore, sincethe continuous image of a compact set is uniformly compact, we cannot have µ ( f ( B )) < µ ( A ),for this would have been witnessed already for some µ ( f n ( B n )). This completes the sketch forthe case where A is Π .If A is Π ( ∅ (cid:48) ), we can do essentially the same construction, tripling on the stage- n approxi-mation to A n instead of A n itself. Any interval which is going to leave A eventually leaves theapproximations for good, so the key features of the above argument are maintained even as thestructure of the triplings gets more complicated.2 Randomness reflection The remainder of this section is devoted to the preparation for the proof of Theorem 30, andthe proof itself. Lemma 34. Given a Π ( ∅ (cid:48) )-set A ⊆ [0 , 1] and some open U ⊇ A we can compute some open V with A ⊆ V ⊆ U such that d ( V, U C ) > Proof. Since U C and A are disjoint closed sets, there is some N ∈ N such that d ( U C , A ) > − N .If we actually had access to A , we could compute a suitable N . Since A is computable from ∅ (cid:48) , we can compute N with finitely many mindchanges. The monotonicity of correctness heremeans we can actually obtain suitable N ∈ N < . We now obtain V by enumerating an interval( a, b ) into V once we have learned that U covers [ a − − N , b + 2 − N ] (which is semidecidable in U ∈ O ( R ) and N ∈ N < ).For an interval [ a, b ], let T ([ a, b ]) = [ a, a + b − a ], T ([ a, b ]) = [ a + b − a , a +2 b − a ] and T ([ a, b ]) =[ a + 2 b − a , b ]. Lemma 35. Let A be a Π ( ∅ (cid:48) ) set. Then there is a computable double-sequence ( I kn ) k,n ∈ N ofclosed intervals with the following properties:1. A = (cid:84) n ∈ N (cid:83) k ∈ N I kn .2. I kn and I (cid:96)n intersect in at most one point.3. For m < n , we find that (cid:83) k ∈ N I kn has positive distance to the complement of (cid:83) k ∈ N I km .4. ∀ k, n ∈ N | I kn | ≥ | I k +1 n | 5. Fix n > 0. For each k there are (cid:96) , i such that | I kn | < − | I (cid:96)n − | and I kn ⊆ T i ( I (cid:96)n − ). Proof. Any Π ( ∅ (cid:48) ) is in particular Π , and thus has Π -approximation A = (cid:84) n ∈ N U n . We invokeLemma 34 inductively first on A and U to obtain V , then on A and U ∩ V to obtain V ,and so on. This will ensure Condition (3). We can effectively write any open set V n ⊆ [0 , 1] asa union of closed intervals such that the pairwise intersections contain at most one point. Tomake Conditions (4,5) work it suffices to subdivide intervals sufficiently much. Definition 36. An interval J is well-located relative to ( I kn ) k,n ∈ N , if for all k, n one of thefollowing hold:1. | J ∩ I kn | ≤ J ⊇ I kn J ⊆ T i ( I kn ) for some i ∈ { , , } For well-located J , let its depth be the greatest n such that J ⊆ I kn for some k . We call twowell-located intervals J , J peers, if whenever J b (cid:40) I kn for both b ∈ { , } and some k, n , thenthere is one i ∈ { , , } such that J b ⊆ T i ( I kn ) for both b ∈ { , } .Note that our requirements for the ( I kn ) k,n ∈ N in Lemma 35 in particular ensure that each I k n is well-located relative to ( I kn ) n,k ∈ N . . Pauly, L. Westrick & L. Yu Definition 37. We are given a double-sequence ( I kn ) k,n ∈ N for a set A as in Lemma 35 and ε > 0. Let N n ∈ N be chosen sufficiently large such that λ ( (cid:83) k>N n I kn ) < − n − n − ε . Let b k,n ∈ { , , } be chosen such that λ ( T b k,n ( I kn ) ∩ A ) + 3 − n − n − − k ε ≥ λ ( T c ( I kn ) ∩ A ) for all k, n ∈ N and c ∈ { , , } . Let (cid:96) n,k and i n,k be the witnesses for Condition 5 in Lemma 35. Wethen inductively define I ε = { I k | k ≤ N } and: I εn = { I kn | k ≤ N n ∧ I (cid:96) n,k n − ∈ I n − ∧ b (cid:96) n,k , ( n − = i n,k } In words, the intervals in I εn are those on the n -th level which occur inside a particular thirdof their parent intervals on the n − A . By construction, the intervals in I n are pairwise peers. Lemma 38. Starting with a Π ( ∅ (cid:48) )-set A , we can compute the sets I εn relative to ∅ (cid:48)(cid:48) . Proof. As the double-sequence ( I kn ) k,n ∈ N is computable, we can obtain the sufficiently large N n by using ∅ (cid:48) . We have T c ( I kn ) ∩ A available to us as Π ( ∅ (cid:48) )-sets, so ∅ (cid:48)(cid:48) lets us compute λ ( T c ( I kn ) ∩ A ) ∈ R . Then getting the choices for the b k,n right can be done computably. Thewitnesses (cid:96) n,k , i n,k can also be found computably. Lemma 39. − n ≥ λ ( (cid:83) I εn ) ≥ − n ( λ ( A ) − ε ) Proof. We prove both inequalities by induction. For the first, the base case is trivial. Forthe induction step, we note that ( (cid:83) I εn +1 ) ⊆ (cid:83) I kn ∈ I εn T b k,n ( I kn ), and that λ (cid:16)(cid:83) I kn ∈ I εn T b k,n ( I kn ) (cid:17) = λ ( (cid:83) I εn ).For the second inequality, we prove a stronger claim, namely that λ ( A ∩ (cid:83) I εn ) ≥ − n ( λ ( A ) − (1 − − n − ) ε ). The base case follows from (cid:83) k ∈ N I k ⊇ A together with the choice of N . We thenobserve that A ∩ ( (cid:83) I εn +1 ) = A ∩ (cid:83) { I kn +1 | ∃ I (cid:96)n ∈ I εn ∃ i ∈ { , , } I kn +1 ⊆ T i ( I (cid:96)n ) } . By definitionof b (cid:96),n in Definition 37, this also means that λ ( A ∩ ( (cid:83) I εn +1 )) + 3 − n − n − ε ≥ − λ ( A ∩ (cid:83) { I kn +1 |∃ I (cid:96)n ∈ I εn I kn +1 ⊆ T b (cid:96),n ( I (cid:96)n ) } ). The set on the right hand side differs from (cid:83) I εn only by thefact that in the latter, we restrict to (cid:96) ≤ N n . By the induction hypothesis together with theguarantee that λ ( (cid:83) k>N n I kn ) < − n − n − ε we get the desired claim. Lemma 40. For a sequence ( I kn ) k,n ∈ N as in Lemma 35 and x / ∈ A , it holds that there existssome a < x < b and some N ∈ N such that I kn ∩ ( a, b ) = ∅ for every n > N . Proof. If x / ∈ A , then there is some N with x / ∈ (cid:83) k ∈ N I kN . By Condition (3) of Lemma 35, wehave that x has positive distance to (cid:83) k ∈ N I kN +1 , hence there exists an interval ( a, b ) around x disjoint from (cid:83) k ∈ N I kN +1 , and by monotonicity, also from (cid:83) k ∈ N I kn for every n > N . Proof of Theorem 30. Preparation: We note that for each Π -set A and each n ∈ N , there isa Π ( ∅ (cid:48) )-set C with C ⊆ A and λ ( C ) ≥ λ ( A ) − − n . We can assume w.l.o.g. that A is alreadyΠ ( ∅ (cid:48) ). We then obtain a computable double-sequence ( I kn ) k,n ∈ N as in Lemma 35. Construction: We obtain our function f as the limit of functions f N,K for N, K ∈ N . f , isthe identity on [0 , f N,K takes into account only intervals I kn with n ≤ N and | I kn | > − K , of which there are only finitely many (and by monotonicity of the enumerations,we can be sure that we have found them all). We process intervals with smaller n first, andreplace the linear growth f currently has on I kn by a triple as shown in Figure 1. Property 54 Randomness reflection Figure 1: Demonstrating the interative construction of the function f in the proof of Theorem30from Lemma 35 ensures that through the process, the function is linear on each interval I kn yetto be processed. We then define f N := lim K →∞ f N,K and f := lim N →∞ f N .That the first limit has a computable rate of convergence follows from the monotonicity of | I kn | in k . Since the size of the intervals shrinks sufficiently fast compared to the potential growthrates of f N , we see that we also do have a computable rate of convergence of ( f N ) N ∈ N . Property 1: If x / ∈ A , we can invoke Lemma 40 to obtain a neighbourhood U of x thatis disjoint from any I kn for n > N . But that ensures that f | U is 3 N -Lipschitz, and and bypotentially restricting the interval further we can make f locally bi-Lipschitz. Lemma 41. 1. Let J be well-located at depth N . Then f [ J ] = f N [ J ].2. Let J be well-located at depth n . Then λ ( f [ J ]) = 3 n λ ( J ).3. Let J , J be peer well-located intervals. Then | f [ J ] ∩ f [ J ] | ≤ Proof. 1. First, we observe that for any M > N it holds that f M [ J ] = f N [ J ], since allmodifications based on intervals I kn with n > N will affect f | J at most by locally replacingthe shape of the graph with a different shape having the same range. It remains to argue . Pauly, L. Westrick & L. Yu f M [ J ] is preserved by limits. Since [0 , 1] is compact andHausdorff, we find that A ([0 , ∼ = K ([0 , g [ J ] ∈ A ([0 , g and A ∈ A ([0 , g [ A ] ∈ V ([0 , A ∈ V ([0 , A ([0 , ∧ V ([0 , λ ( f n [ J ]) = 3 n λ ( J ) instead. Now ( f n ) | J is just a linear functionwith slope 3 n , which yields the claim.3. If J , J are peers and well-located, and J ⊆ I kn but J (cid:42) I kn , then I kn and J are alsopeers. It thus suffices to prove the claim for the case where J = I k n +1 and J = I k n +1 .These are both contained in the same T i ( I (cid:96)n ), and ( f n ) | T i ( I (cid:96)n ) is a linear function. Since | J ∩ J | ≤ | f n [ J ] ∩ f n [ J ] | ≤ 1. By (1 . ), this already yields the claim. Property 2: We obtain the desired set B as B = (cid:84) n ∈ N ( (cid:83) I εn ). Since each I εn is a finitecollection of closed intervals, B is indeed closed. Since the intersection is nested and λ ( (cid:83) I εn ) ≤ − n by the first part of Lemma 39, we conclude that λ ( B ) = 0. Since the intervals in I εn arewell-located and pairwise peers, we know that λ ( f ([ (cid:83) I εn ])) = 3 n λ ( (cid:83) I εn ) by Lemma 41 2&3.Invoking the second inequality from Lemma 39 then lets us conclude λ ( f ([ (cid:83) I εn ])) ≥ ( λ ( A ) − ε ).Since this estimate holds for every stage of a nested intersection of compact sets, it follows that λ ( f [ B ]) ≥ λ ( A ) − ε as desired. That B is obtainable by an oracle of the claimed strength followsfrom Lemma 38. We recall the definitions of absolute continuity and bounded variation: Definition 42. A function f : [0 , → R is absolutely continuous , if for every ε > x < y < x < y . . . < x k < y k there is a δ > i ≤ k | y i − x i | < δ ⇒ Σ i ≤ k | f ( y i ) − f ( x i ) | < ε. Definition 43. A function f : [0 , → R has bounded variation, if there is some bound M ∈ N such that for any k ∈ N and any x < x < . . . < x k it holds thatΣ i Fact 44 (see [23], Theorem VII.6.7) . A continuous function is absolutely continuous iff it hasboth bounded variation and Luzin’s (N).We observe that being absolutely continuous is a Π -property, and recall that Luzin’s (N)is Π -complete [9]. As such, restricting our attention to functions of bounded variation shouldalter the situation significantly. Proposition 45. If f : [0 , → R is computable and absolutely continuous, then f reflectsweak-2-randomness.6 Randomness reflection Proof. First, we consider how we can exploit connectedness of R to say something about theimages of open sets under computable functions. We are given open sets in the form U = (cid:83) i ∈ N I i ,where each I i is an open interval with rational endpoints. We can then compute sup f ( I i ) andinf f ( I i ) (as these are equal to max f ( I n,i ) and min f ( I n,i ), and we can compute minima andmaxima of continuous functions on compact sets). Let V = (cid:83) i ∈ N (inf f ( I i ) , sup f ( I i )). We notethat we can compute V from U , that V ⊆ f [ U ], and that f [ U ] \ V can only contain computablepoints. In particular, λ ( V ) = λ ( f [ U ]).Now we assume that f additionally is absolutely continuous, and that we are dealing witha Π -null set A = (cid:84) n ∈ N U n witnessing that some x ∈ A is not weak-2-random. We assume that U n +1 ⊆ U n . As A is null, we know that lim n →∞ λ ( U n ) = 0. Since f is absolutely continuous,we also have lim n →∞ f [ U n ] = 0. Let V n be obtained from U n as in the first paragraph, and B = (cid:84) n ∈ N V n . It follows that λ ( B ) = 0, and moreover, f [ A ] is contained in B with the potentialexception of some computable points. So we can conclude that f ( x ) is not weak-2-random,either because f ( x ) is computable, or because f ( x ) ∈ B .Note that if we had started with a Martin-L¨of test in the argument above, we would have noguarantee of ending up with one, because the modulus of absolute continuity is not computablein general. Indeed, absolute continuity does not imply MLR reflection. See Corollary 53. Lemma 46. If f : [0 , → R is computable, has bounded variation, and reflects ∅ (cid:48) -Kurtzrandomness, then f has property ( N ). Proof. Suppose that f does not have ( N ). Since f has bounded variation, it must fail absolutecontinuity. Let ε > δ > 0, there is a finite union of intervals A δ ⊆ [0 , µ ( A δ ) < δ and µ ( f ( A δ )) > ε . Computably, given δ we can find such A δ by searching. Let A = ∩ n U n , where U n = ∪ m>n A − m . Then A is Π , and µ ( A ) = 0, and µ ( ∩ n f ( U n )) ≥ ε . Weclaim that its subset f ( A ) also has µ ( f ( A )) ≥ ε . Let Var f : [0 , → R denote the cumulativevariation function of f , defined by setting Var f ( x ) to be equal to the variation of f on [0 , x ].Since f has bounded variation and U n +1 ⊆ U n , (cid:80) n Var f ( U n \ U n +1 ) is finite, so by choosing N large enough, we can make (cid:80) n>N µ ( f ( U n \ U n +1 )) as small as we like. Now observe that nomatter how large N we choose, (cid:32)(cid:92) n f ( U n ) \ f ( A ) (cid:33) ⊆ (cid:91) n>N f ( U n \ U n +1 ) . This proves the claim. We have found a Π set A = ∩ n U n which witnesses the failure of ( N ).Observe that for any c.e. open set U , µ ( f ( U )) is c.e.. Therefore, since f has boundedvariation, ∅ (cid:48) can search around to find, for each n , a closed set F n ⊆ U n such that µ ( f ( U n \ F n )) < − n − ε . The existence of such a closed set is guaranteed by f having bounded variation.Let F = ∩ n F n . Then F ⊆ A and A \ F = ∪ n ( A \ F n ). So µ ( f ( A \ F )) ≤ (cid:88) n µ ( f ( A \ F n )) ≤ (cid:88) n µ ( f ( U n \ F n )) ≤ (cid:88) n − n − ε < ε. The positive measure of f ( F ) then follows as ε ≤ µ ( f ( A )) ≤ µ ( f ( A \ F )) + µ ( f ( F )) . Therefore, F is an ∅ (cid:48) -computable closed set of measure zero whose image has positive measure.So f does not reflect ∅ (cid:48) -Kurtz randomness. . Pauly, L. Westrick & L. Yu Theorem 47. The following are equivalent for computable functions f : [0 , → R havingbounded variation:1. f has Luzin’s (N).2. f reflects weak-2-randomness.3. f reflects ∅ (cid:48) -Kurtz randomness.4. f reflects ∆ ( O )-randomness.5. f reflects Z -Kurtz randomness for any Z ≥ ∅ (cid:48) . Proof. The implication from (1) to (2) is given by Proposition 45. To see that (2) implies (3),first observe that weak-2-randomness reflection implies that µ ( f ( A )) = 0 for any null Π set A , for if f ( A ) had positive measure then it would certainly contain weak-2-random elements.A Π ( ∅ (cid:48) ) set is in particular Π , so the image of any ∅ (cid:48) -Kurtz test has measure 0, and is thusalso an ∅ (cid:48) -Kurtz test because the continuous image of a compact set is uniformly compact. Theimplication (3) ⇒ (1) is in Lemma 46.Finally, the equivalence of (1) and (4) is just Theorem 16, the implication from (1) to (5) isProposition 22, and the implication from (5) to (3) is Proposition 23. Corollary 48. If a computable function f : [0 , → R of bounded variation reflects ML-randomness, then it has Luzin’s (N).The converse is false; see Corollary 53. Proof. The same argument works as for the implication (2) ⇒ (3) in Theorem 47.In this section we have stated all results for f : [0 , → R because this is a natural setting inwhich to consider functions of bounded variation. Of course, our pointwise results are equallytrue for any computable f : R → R which is locally of bounded variation.An often useful result about continuous functions of bounded variation is that they can beobtained as difference between two strictly increasing continuous functions. In light of our in-vestigation of Luzin’s (N) for strictly increasing functions, one could wonder why we are notexploiting this property here. There are two obstacles: One the one hand, the computablecounterpart of the decomposition result is false: There is a computable function of boundedvariation, which cannot be written as the difference between any two strictly increasing com-putable functions [26]. On the other hand, Luzin’s (N) is very badly behaved for sums. Forexample, for every continuous function f having Luzin’s (N) there exists another continuousfunction g having Luzin’s (N) such that f + g fails (N) [20]. For increasing functions we see a connection to absolute continuity of measures. Recall that ameasure µ is absolutely continuous w.r.t. a measure ν (in symbols µ (cid:28) ν ), if ν ( A ) = 0 impliesthat µ ( A ) = 0. The notions are related through the following observations: Observation 49. If continuous surjective f : [0 , → [0 , 1] is increasing, then the probabilitymeasure µ defined as µ ( A ) = λ ( f ( A )) is non-atomic, and µ (cid:28) λ iff f has Luzin’s (N).8 Randomness reflection Observation 50. If µ is a non-atomic measure on [0 , µ ( x ) := µ ([0 , x ]) is a continuous increasing function which has Luzin’s (N) iff µ (cid:28) λ .In [2], Bienvenu and Merkle have done an extensive survey of the conditions under whichtwo computable measures µ and ν share the same randoms for a variety of notions of random-ness (Kurtz, computable, Schnorr, MLR, and weak-2-random). Two trivial situations where µ -randomness and λ -randomness fail to coincide is if µ has an atom or if µ ( J ) = 0 for someopen interval J . When discussing the connections among Luzin’s (N), randomness reflection,and coincidence of randomness notions, we will restrict our attention to computable measures µ which avoid these two degenerate situations. When µ is atomless, cdf µ is continuous andcomputable. To say µ ( J ) > J , it is equivalent to say that cdf µ is strictlyincreasing. When the degenerate situations are avoided, cdf µ is a computable homeomorphismof [0 , − µ is also a computable homeomorphism. In this situation, randomness reflectionfor cdf µ is exactly randomness preservation for cdf − µ . Proposition 51. Let µ be a non-atomic computable probability measure on [0 , 1] with cdf µ strictly increasing. Then x is µ -MLR ( µ -Schnorr random, µ -Kurtz random, µ -∆ -random) iffcdf µ ( x ) is Martin-L¨of random (Schnorr random, Kurtz random, ∆ -random) w.r.t. the Lebesguemeasure. Proof. For any set A , we have µ ( A ) = λ (cdf µ ( A )), and cdf µ and cdf − µ are both computablehomeomorphisms. We can thus move any relevant test from domain to codomain and viceversa.Therefore, cdf µ reflects a given notion of randomness exactly when the µ -randoms are con-tained in the λ -randoms for that notion of randomness. Similarly, cdf − µ reflects a given notionof randomness exactly when the λ -randoms are contained in the µ -randoms.Using our previous results, we obtain the following corollary. The equivalence of (1) and (4)was proved in ([2, Proposition 58]), but the others are new. Corollary 52. The following are equivalent for a computable probability measure µ .1. µ is mutually absolutely continuous with the Lebesgue measure.2. cdf µ is a homeomorphism and both cdf µ and cdf − µ have Luzin’s (N).3. µ -∆ ( O )-randomness and ∆ ( O )-randomness coincide.4. µ -weak-2-randomness and weak-2-randomness coincide.5. µ -Kurtz( ∅ (cid:48) )-randomness and Kurtz( ∅ (cid:48) )-randomness coincide. Proof. First observe that in all cases above, cdf µ is a homeomorphism. That is because none ofthe cases is compatible with µ having an atom or assigning measure 0 to an interval.Then (1) ⇐⇒ (2) follows from Observation 50 for the case of cdf µ , and by similar reasoningfor the case of cdf − µ .Since cdf µ and cdf − µ are computable functions of bounded variation, by Theorems 16, and47, they have Luzin’s (N) if and only if they reflect each kind of randomness mentioned in (3)-(6).So the implications (2) ⇐⇒ (3) , (2) ⇐⇒ (4) , and (2) ⇐⇒ (5) now follow from Proposition51. . Pauly, L. Westrick & L. Yu µ which is mutually absolutely con-tinuous with Lebesgue measure, but µ -MLR does not coincide with λ -MLR, µ -Schnorr randomdoes not coincide with with λ -Schnorr random, and µ -computably random does not coincidewith λ -computably random. Essentially, µ is obtained by thinning out the Lebesgue measurearound Chaitin’s Ω in a way that derandomizes Ω without introducing new null sets. Corollary 53. Luzin’s (N) does not imply any of Martin-L¨of randomness reflection, Schnorrrandomness reflection nor computable-randomness reflection; even for strictly increasing com-putable functions. Proof. If Luzin’s (N) were to imply reflection for any of these kinds of randomness, they couldbe included in the list in Corollary 52 by the same reasoning, but this would contradict Bienvenuand Merkle’s result above.We still need to discuss reflection of (unrelativized) Kurtz randomness. In [2, Proposi-tion 56], Bienvenu and Merkle construct a non-atomic computable probability measure µ suchthat µ -Kurtz random and Kurtz random coincide, yet makes the Lebesgue measure not abso-lutely continuous relative to µ . The construction is based on an involved characterization of2-randomness in terms of Kolmogorov complexity obtained by Nies, Stephan and Terwijn [18].We could already conclude that Kurtz randomness reflection does not imply Luzin’s (N) fromthis, but instead we will provide a direct, more elementary construction in the following. Ourseparation works “the other way around”, that is we obtain a probability measure µ which is notabsolutely continuous w.r.t. the Lebesgue measure. This shows that the Lebesgue measure hasno extremal position for relative absolutely continuity inside the class of measures having thesame Kurtz randoms. For comparison, a measure satisfies Steinhaus theorem iff it is absolutelycontinuous w.r.t. Lebesgue measure [16]. Theorem 54. There is an increasing surjective computable function f : [0 , → [0 , 1] which isnot absolutely continuous, yet for any Π set A with λ ( A ) = 0, it holds that λ ( f ( A )) = 0. Corollary 55. There is a non-atomic probability measure µ such that µ -Kurtz random andKurtz random coincide, yet µ (cid:54)(cid:28) λ . Proof. Let ˆ µ be the probability measure whose cumulative distribution function is f , equivalentlyˆ µ ( B ) := λ ( f ( B )). Since f does not have Luzin’s (N), there is some set B with λ ( B ) = 0 andˆ µ ( B ) > 0. Let µ = ˆ µ + λ . Then using the same B , we see that µ (cid:54)(cid:28) λ . On the other hand, if A is a Π set, then λ ( A ) = 0 implies ˆ µ ( A ) = 0, and thus λ ( A ) = 0 if and only if µ ( A ) = 0. Corollary 56. For increasing computable functions f : [0 , → [0 , h : [0 , → [0 , 1] is a piecewise linear increasingfunction, B ⊆ [0 , 1] is a finite union of intervals with rational endpoints, and δ > 0. We definea new function Concentrate( h, B, δ ) : [0 , → [0 , λ ( B )-much measure onto a set of Lebesgue measure at most δ , as follows.0 Randomness reflection Definition 57 (Definition of Concentrate) . Given h, B, δ as above, write B = ∪ k Suppose that ( B n ) n ∈ N is a computable sequence of finite unions of intervals in[0 , f n ) n ∈ N inductively by setting f ( x ) = x and f n +1 = Concentrate( f n , B n , − n ) . Then ( f n ) n ∈ N converges uniformly to a computable increasing function f . Furthermore, if thereis some ε > λ ( B n ) > ε for all n , then f fails Lusin’s (N). Proof. The uniform convergence to a computable f follows from the third property in the defi-nition of Concentrate, and f is increasing because each f n is. Observe that Concentrate neverchanges the value of h at a break point of h . Therefore, the second property in the definition ofConcentrate, which tells us that f n ( F ) = B for some F with λ ( F ) < − n , implies that f ( F ) = B as well (here we also used the fact that f is continuous and increasing). It follows that f is notabsolutely continuous, and thus fails Lusin’s (N). Proof of Theorem 54. We construct a computable sequence ( B n ) n ∈ N such that λ ( B n ) > / n , and argue that the function f constructed as in Lemma 58 satisfies λ ( f ( P )) = 0 whenever P ∈ Π and λ ( P ) = 0.The strategy for a single Π class P e is as follows. Let C e, be some interval of length ε e .Let B s = [0 , \ C e,s . As long as f s ( P e,s ) ∩ C e,s has measure at least ε e / 2, define C e,s +1 = C e,s .If f ( P e,s ) ∩ C e,s has measure less than ε e / 2, define C e,s +1 = ( f s ( P e,s ) ∩ C e,s ) ∪ C , where C is anew interval or finite union of intervals almost disjoint from ∪ t ≤ s C e,t . Choose C so that that C e,s +1 has measure ε e , if possible; if this is not possible, choose C so that ∪ t ≤ s +1 C e,t = [0 , C e,s +1 may be less than ε e and this is also fine. If we reachthis degenerate situation, we also stop checking the measures and simply let C e,t = C e,s +1 forall t > s .We claim that if λ ( P e ) = 0, then λ ( f ( P e )) = 0. Suppose at some stage s we have that themeasure of f s ( P e,s ) ∩ C e,s is greater than ε e / 2. If this continues for all t > s , then f and f s coincide on the set J := f − ( C e,s ). It follows that f is piecewise linear on J , but f ( P e ∩ J )has positive measure; this is impossible since P e has measure 0. We conclude that nothing lastsforever; eventually we do reach a stage s where ∪ t ≤ s C e,s = [0 , C e,s never changes again, f and f s again coincide on J := f − ( C e,s ). Observe also that P e ⊆ J . Since f s is piecewiselinear and λ ( P e ) = 0, we also have λ ( f s ( P e )) = 0, and thus λ ( f ( P e )) = 0.The above strategy works purely with negative requirements, specifically freezing f on f − s ( C e,s ). If other requirements also freeze f on other places, it has no effect on the proof . Pauly, L. Westrick & L. Yu λ ( B s ) > / s , where we now define B s = [0 , \ (cid:91) e Let K, L ⊆ [0 , be non-empty sets containing only Kurtz randoms. Then“whenever f ( x ) ∈ K , then already x ∈ L ” is a Π -hard property of continuous functions f : ([0 , \ Q ) × [0 , → [0 , . Proof. It is well-known that [0 , \ Q and N N are homeomorphic, and even computably so. Weidentify the spaces in such a way that the Lebesgue measure induced on N N satisfies λ ( { p ∈ N N |∀ n p n = p n +1 } ) = 0.We construct a function f T : N N × [0 , → [0 , from a countably-branching tree T . First, wemodify T to obtain ˆ T = { w w w w . . . w n − w n − w n | w ∈ T }∪{ w w w w . . . w n − w n − w n w n | w ∈ T } . Clearly, T is well-founded iff ˆ T is, and [ ˆ T ] contains no Kurz-randomns (so inparticular,[ ˆ T ] × [0 , ∩ L = ∅ ). For any p ∈ N N , let | T, p | = n iff n is minimal such that p (cid:22) n / ∈ ˆ T , and | T, p | = ∞ if p ∈ [ ˆ T ].Let s ∞ : [0 , → [0 , be a computable space-filling curve, and let ( s n ) n ∈ N be a computablefast Cauchy sequence converging to s ∞ such that any s n ([0 , f T ( p, x ) = s | T,p | ( x ). This construction is computable in T . We claim that f T has our reflection property iff T is well-founded.If T is well-founded, then the range of f T is (cid:83) n ∈ N s n ([0 , s n ([0 , -set, we see that f T never takes any Kurtz random values (in particular, none in K ), andthus vacuously, if f ( x ) ∈ K then x ∈ L . The argument in fact establishes that for arbitrary T ,whenever p / ∈ ˆ T then f T ( p, x ) is not Kurtz random regardless of x .Now assume that T is ill-founded and that y ∈ K . We find that f − T ( { y } ) = [ ˆ T ] × s − ∞ ( { y } ).Since ˆ T is illfounded and s ∞ is space-filling, this set is non-empty. But by construction of ˆ T , itcannot contain any elements of L . Hence, f T does not have our reflection property. As a slight digression, we have a look at related properties of functions, namely those wherethe image of null sets are required to belong to some other ideals of small sets, such as beingcountable or being meager. These properties were investigated by Sierpinski [24] and Erd¨os [7],amongst others. Our results are formulated in some generality, but as a consequence, we do seethat we do not get any “regular” functions with these properties. In contrast, Erd¨os showed that2 Randomness reflection under CH there is a bijection f : R → R mapping meager sets to null sets with f − mappingnull sets to meager sets. Theorem 60. (1) If A is a nonnull Σ set and f is a continuous function mapping any nullsubset of A to a countable set, then the range of f restricted to A is countable. Inparticular, if A is an interval, then range of f restricted to A is a constant function.(2) Assume CH, there is a function f mapping any null set to a countable set such that therange of f is R , and f ( A ) is uncountable for any nonnull set A but for every y , f − ( y ) isan uncountable Borel null set.(3) If A is a nonnull set and f is a continuous function mapping any null subset of A to ameager set, then the range of f restricted to A is meager. In particular, if A is an interval,then range of f restricted to A is a constant function.(4) If f is a measurable function and maps a null set to a meager set, then the range of f ismeager. In particular, if f is continuous with the property, then f is constant.(5) If f has the Baire property and maps a meager set to a null set, then the range of f isnull. In particular, if f is continuous with the property, then f is constant. Proof. (1). Fix a real x so that f is computable in x and A is Σ ( x ). Now for any real z ∈ A ,let g be a ∆ ( O x ⊕ z )-generic real. Then z cannot be Martin-L¨of random relative to g and sothere there must be a ∆ ( g )-null set G so that z ∈ G ∩ A . By the assumption, f ( G ∩ A ) is aΣ ( x ⊕ g )-countable set and so every real in f ( G ∩ A ) is hyperarithmetically below x ⊕ g . Inparticular, f ( z ) ≤ h x ⊕ g . Since f is computable in x , we also have that f ( z ) ≤ h x ⊕ z . Then f ( z ) ≤ h x since g is ∆ ( O x ⊕ z )-generic real. By the arbitrarility of z , the range of f restrictedto A is countable. So if A is an interval, then range of f restricted to A is a constant function.(2). Fix an enumeration of nonempty G δ -null sets { G α } α< ℵ and all the reals { y α } α< ℵ . Wedefine f and { β α } α< ℵ by induction on α .At stage 0, define f ( x ) = y for any x ∈ G . Define β = 0.At stage α < ℵ , let β α be the least ordinal γ so that G γ \ (cid:83) α (cid:48) <α ( (cid:83) γ (cid:48) ≤ β α (cid:48) G γ (cid:48) ) is uncountable.Define f ( x ) = y α for any x ∈ G γ \ (cid:83) α (cid:48) <α ( (cid:83) γ (cid:48) ≤ β α (cid:48) G γ (cid:48) ).Clearly the range of f is R . Moreover, for any α < ℵ , f − ( y α ) = G β α \ (cid:83) α (cid:48) <α ( (cid:83) γ (cid:48) ≤ β α (cid:48) G γ (cid:48) )is an uncountable Borel null set. Now for any null set A , there must be some α < ℵ so that A ⊆ G α . By the construction, f ( A ) ⊆ f ( G α ) ⊆ { y β | β ≤ α } is a countable set.(3). Fix a real x so that f is computable in x restricted to A . Fix a 2- x -random real r ∈ A .Then f ( r ) ≤ T x ⊕ r . But f ( r ) cannot be 2- x -generic (see [18]). So the range f restricted to A ∩ { r | r is a 2 − x -random } is meager. But A \ { r | r is a 2 − x -random } is a null set. So, bythe assumption on f , the range of f restricted to A is meager.(4). Suppose that f is measurable function and maps a null set to a meager set. Withoutloss of generality, we may assume that the domain of f is [0 , { F n } n ∈ ω so that [0 , \ (cid:83) n ∈ ω F n is null and f restricted to F n is continuous for every n . By(3), the range of f restricted to F n is a meager set. So the range of f restricted to (cid:83) n ∈ ω F n isalso a meager set. Note that [0 , \ (cid:83) n ∈ ω F n is null. So the range of f is meager. In particular,if f is continuous with the property, then f is constant.(5). This is dual to (4). . Pauly, L. Westrick & L. Yu The most prominent avenue of future research seems to be the resolution of Question 28, askingfor a separation (or equivalence proof) of ∆ -randomness reflection and ∆ ( O )-randomnessreflection. There are a few further aspects that merit further investigation, though. Topological properties While we have not been systematic in exploring the impact of topo-logical properties of the domain (and maybe codomain) of the functions we explore, we observethat our proofs differ in the requirements they put on the spaces involved. For example, themajority of the arguments presented in Sections 3.1 and 3.2 are relying just on the theory ofrandomness, and are thus applicable to any space where randomness works as usual (see [10]). InSection 3.3, (local) compactness of the domain is a core ingredient in our arguments. In Section5 we do use particular properties of the reals, in particular connectedness. Further investigationof how topological properties of spaces relate to how randomness reflection behaves for functionson them seems warranted. Formalizing randomness reflection With the exception of Theorem 59, we have only con-sidered symmetric notions of randomness reflection: Whenever f ( x ) is random in some sense, wedemand that x is random in the very same sense. While this seems natural, a downside is thatwe do not get trivial implications between different notions of Π -type randomness reflection.We could consider the full square of reflection notions, ( K, L )-randomness reflection being thatwhenever f ( x ) is K -random, then x is L -random for randomness notions K , L . An extremalversion also makes sense, where we just ask for when the image of all non-randoms under f has positive measure. Whenever the latter property holds for some randomness notion L , then f cannot have ( K, L )-randomness reflection for any randomness notion K at all. 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