Representations of measurable sets in computable measure theory

This article is a fundamental study in computable measure theory. We use the framework of TTE, the representation approach, where computability on an abstract set X is defined by representing its elements with concrete"names", possibly countably infinite, over some alphabet {\Sigma}. As a basic computability structure we consider a computable measure on a computable $\sigma$-algebra. We introduce and compare w.r.t. reducibility several natural representations of measurable sets. They are admissible and generally form four different equivalence classes. We then compare our representations with those introduced by Y. Wu and D. Ding in 2005 and 2006 and claim that one of our representations is the most useful one for studying computability on measurable functions.


Introduction
Measure theory is a fundament of modern analysis.In particular, computable measure theory is a fundament of computable analysis.In recent years a number of articles have been published on computable measure theory, for example [10,22,29,14,36,27,5,11,19,15,32,16,1,17,20,4,13,33,3,21,18].Most of these articles start with a definition of computability concepts in measure theory and then prove, or disprove, a computable version of some classical theorem.
Wu and Ding [34,35] have defined and compared various definitions of computability on measurable sets.In this article we extend these fundamental studies.We use the representation approach to computable analysis (TTE) [30,8].In this approach computability is defined directly on the set Σ ω of the infinite sequences of symbols, e.g. by Turing machines.Computability is transferred to other sets X by means of representations δ : Σ ω → X where the elements of Σ ω are considered as names and computations are performed on names.Obviously, computability on the "abstract" set X depends crucially on the choice of the representation δ.Only those representations are of interest which can relate the important structure properties of X with corresponding ones of Σ ω .
We start from a computable measure on a computable σ-algebra which has proved to be a very useful fundamental concept of computability in measure theory [34,35,36].In addition to the representations studied in these articles we introduce several new representations of the measurable sets and compare all of them w.r.t.reducibility.
In Section 2 we outline very shortly some concepts from the representation approach.In Section 3 we summarize elementary definitions and facts from measure theory which we will need for introducing the new computability concepts.
In Section 4 we define computable σ-algebras (Ω, A, R, α) where R is a countable ring which generates the σ-algebra A in Ω such that Ω = R and α : ⊆ Σ * → R is a notation of the ring such that set union and difference become computable.A measure µ is computable if µ(R) is finite for every ring element R and R → µ(R) is computable.Then we introduce and study representations ζ + , ζ − and ζ of the measurable sets which exactly allow to compute µ(R ∩ A) for for R ∈ R and A ∈ A from below, from above or from below and above, respectively.We study reducibility and characterize the degree of non-computability for the negative results.
In Section 5 for the sets of finite measure we define a computable metric space and compare its Cauchy representation with the representations defined before.
In Section 6 we partition the set Ω computably by a (majorizing) sequence (F i ) i∈N of ring elements.For each number i, the measure restricted to F i is finite and induces a computable metric space, the metric of which can be normalized to a metric d ′ i bounded by 1.The weighted sum d = i 2 −i • d ′ i is a computable metric on the whole σ-algebra the Cauchy representation of which allows to compute the measures of measurable sets from below and above and hence is equivalent to the representation ζ from Section 4.
In Section 7 we show that all the representations are admissible [30].We compare our representations with those from [34,35].It turns out that ζ + for which there is no equivalent one in [34,35] is most interesting.

Concepts from classical measure theory
In this Section we summarize elementary definitions and facts from measure theory which we will need for introducing the new computability concepts.
Let Ω be a set.
The elements of A are called the measurable sets.Every σ-algebra is a ring.
-For a set T ⊆2 Ω , R(T ) denotes the smallest ring containing T and A(T ) denotes the smallest σ-algebra containing T .
-A measure on a ring R is a function µ The sets can be assumed to be pairwise disjoint: for For two sets A, B let A ∆ B := (A \ B) ∪ (B \ A) be their symmetric difference.Some useful rules for the symmetric difference are listed in the appendix Section 9.
Two sets A and B with µ(A ∆ B) = 0 are essentially identical in measure theory.Notice that the following are equivalent: Therefore the following operations are well-defined on equivalence classes: Proof.Straightforward, using in particular (9.6) and (9.7).
Our computability concepts in measure theory are based on the following theorem.
Theorem 3.3 (Carathéory extension theorem [2,12]).Every σ-finite measure on a ring R has a unique extension to a measure on the σ-algebra A(R).
Therefore, for specifying a measure µ on the σ-algebra A(R), it suffices to define µ(E) for every E ∈ R.
Let µ be a σ-finite measure on a ring R and let (F i ) i∈N be a sequence of ring elements which satisfy (3.1).For any set R ∈ R, we have µ(R) = i∈N µ(R ∩ F i ).This implies that the measure µ on the ring R is completely determined by its restriction to the subring R f which consists of all ring elements with finite measure.
In our computable measure theory we will consider only σ-algebras A(R) spanned by a finite or countable ring R (which is non-empty since ∅ ∈ R) and measures µ such that µ(R) < ∞ for all R ∈ R and µ is σ-finite on R. Lemma 3.4.Let µ be a measure on a countable ring (2) Since R⊆2 Ω ′ , µ is σ-finite in Ω ′ by (1).
Therefore, if R = Ω, we obtain a σ-finite measure by ignoring Ω \ R. We will use the next two theorems for defining representations of the measurable sets.For a measure µ on a σ-algebra A and a subset E⊆A let E f := {A ∈ E | µ(A) < ∞} be the set of elements of E of finite measure.
Special cases of the following theorem are proved in most introductory texts.A complete proof is added in the appendix Section 10.Theorem 3.5.Let µ be a measure on a σ-algebra A. On A f the Fréchet metric is defined by d(A, B) A set A ∈ A is determined uniquely up to a set of measure 0 by the values µ(A ∩ E) for ring elements E of finite measure.We will use this fact for defining various representations of the set [A]. Lemma 3.6.Let R be a ring and let µ be a measure on (9.9).⇐=: Suppose µ(A ∆ B) > 0. We may assume, without loss of generality, µ(A \ B) > 0. We want to find some

The basic representations
In computable analysis computability on an uncountable structure is usually introduced by selecting a countable substructure which "generates" it and defining the meaning of "computable" on this substructure (example: computability on the field Q, completion to R).The results from the last section suggest that a countable ring with a σ-finite measure should be a good substructure.Then ring operations should become computable as well as the measure restricted to the ring.
For a computable σ-algebra the intersection operation on the ring is also computable because From the notation α of the ring a representation δ of the σ-algebra A(R) can be defined inductively as follows: In this case, if δ(p) = B then p encodes a finite-path tree (a term) which protocols the generation of the set B from ring elements by repeated application of the unary operation "complement" and the ω-ary operation "countable union".The tremendous amount of information contained in a δ-name is not really necessary if we are only interested in computing the measure of the set.Instead, for given measure µ the σ-algebra A is factorized by the equivalence relation In the following let µ be a computable measure on the computable σ-algebra (Ω, A, R, α).

We define various representations of the class [A]. By Lemma 3.6 and Definition 4.1, [A] is defined uniquely by the set of all
A ζ + -name of a set A consists of all rational lower bounds of the µ(R∩A) (A ∈ R).Since the numbers µ(R) are ρ-computable, a ζ − -name of A, yields a list of all rational lower bounds of µ(R \ A) (A ∈ R) (Definition 4.6, Lemma 4.7).In [35] rational lower bounds of µ(A \ R) instead of µ(R \ A) are used for defining representations which then differ significantly from the ones defined here.We must show that the definitions do not depend on the representative A of the class [A].Then for all r ∈ Q and for all R ∈ R, hence The argument is the same for ζ.For the case ζ − replace "<" in (4.1) by ">".
The representation Proof.The statements can be derived from a general theorem [31,Theorem 13.1].We give a direct proof here.
(1) There is a Type-2 machine M that on input (p, v) On the other hand, suppose that the function Then there is a Type-2 machine M which on input (p, v) ∈ dom(γ) × dom(α) writes a list of all u ∈ dom(ν Q ), such that ν Q (u) < µ(α(v)∩γ(p)).From M we can construct a Type-2 machine N which on input p writes a list of all (u, v) ∈ dom(ν Q ) × dom(α) such that the machine M on input (p, v) writes u in finitely many steps of computation.Therefore, (2) and ( 3) can be proved accordingly.
For representations γ and δ, γ ∧ δ is the greatest lower bound of γ and δ for the reducibility ≤, where (γ ∧ δ) p, q = x ⇐⇒ γ(p) = δ(q) = x [30, Section 3.3].Remember that for the well-known representations of the real numbers, and vice versa.Therefore, we can define representations such that names allow to compute all µ(α(v) \ A) which are equivalent to the former ones. ( Notice that There is a computable measure on a computable σ-algebra such that ζ + ≤ ζ (see the proof of Theorem 4.8 (2) below).As usual already translation by a continuous function is impossible, ζ + ≤ t ζ.We determine the degree of unsolvability of the translations from ζ + to ζ and the other similar ones.
Let X 1 , Y 1 , X 2 , Y 2 be represented sets and let 7,6], where ≤ W is called Weihrauch reducibility).This means that composition with G and H in this manner transforms every realization of f 2 to a realization of f 1 .The multi-functions It is known that , where ρ n is the representation of the real numbers by (not necessarily fast) converging sequences of rational numbers [30].These five translation problems are of the same sW-degree of unsolvability.Furthermore, the identity ECf : (2 N , En) → (2 N , Cf) and complementation of enumeration CE : (2 N , En) → (2 N , En), K → N \ K, are in this sW-degree [28], where Cf is the characteristic function representation and En is the enumeration representation of the subsets of N [30].
Let En * : Σ ω → 2 Σ * be the canonical enumeration representation of the set of subsets of Σ * , that is, ι(w) is a subword of p ∈ Σ ω iff w ∈ En * (p).Then also complementation for all functions f on represented sets.(1) For every computable measure on a computable σ-algebra, (2) There is a computable probability measure on a computable σ-algebra such that Proof.
(1) We prove id +− ≤ sW id +0 .By Lemma 4.5 there is a computable function We prove id +0 ≤ W id +− .By Lemma 4.5 there is a computable function h such that (ζ . Define H and G by H(p, q) := h( p, q )) and G(p) There is a Type-2 machine M that on input q ∈ Σ ω writes a list of all v, w such that v ∈ dom(α) and for some u Combining the two cases we obtain, In summary id +− ≤ sW id +0 ≤ W id +− ≤ sW CE.Applying (4.3) we obtain (4.4).
First, we prove CE ≤ sW id +− .We show that the function h : There is a Type-2 machine M that on input p ∈ dom(En) produces a list of all u, v such that ν Q (u) < µ(α(v) ∩ A k ) for some k, where A k is the set of all n such that 01 n+1 0 is a subword of the first k symbols of p.
Let µ ′ := 2/3 • µ.Then µ ′ (Ω) = 1, hence µ ′ is a probability measure and the results hold as well for µ ′ .Lemma 4.9.The function There is a machine that on input (p, u) writes a list of all (v, w) such that (h(v, u), w) is listed in p.Therefore, The other two statements can be proved accordingly.
Let ρ < and ρ > be the lower and upper representation of R := R∪{−∞, ∞}, respectively, and let ρ = ρ < ∧ ρ > [30, Secton 4.1].Informally, ρ < (p) = x iff p is a list of all a ∈ Q such that a < x, and ρ > (p) = x iff p is a list of all a ∈ Q such that a > x.
(1) µ : Proof.(1) Since Ω = R and R is countable and closed under union, µ(A) = µ(A ∩ Ω) = sup R∈R µ(R ∩ A).There is a Type-2 machine M which on input p writes a list q of all u such that for some v, (u, v) is listed in p.

Representations of the sets of finite measure
In this section we introduce and study representations of the set [A f ]) for the set A f of measurable sets of finite measure.µ(Ω) may be finite or infinite.By Theorem 3.5, (A f , d) with d(A, B) = µ(A ∆ B) is a complete pseudometric space with R f as a dense subset.Remember that for our computable measure µ on the computable σ-algebra, (As usual, we use the same symbol d for the pseudometric and its factorization.) A computable metric space is a quadruple (M, d, A, ν) such that (M, d) is a metric space, A⊆M is dense and ν : ⊆ Σ * → A is a notation of A such that dom(ν) is recursive and the metric d restricted to A is (ν, ν, ρ)-computable (equivalently, the set of all (t, u, v, w) such that ν

e.). The Cauchy representation of a computable metric space is defined by δ
We introduce two further representations of the set [A f ] of measurable sets of finite measure by adding the measure of Proof.(1), (2) Obvious. ( There is a machine N which on input p writes a sequence v 0 , v 1 , . . . of words where v i is computed as follows: N runs M as a subprogram and searches some (v i , k) such that M on input (p, v i ) writes the rational number (5.1) by (9.10).Since intersection on R is (α, α, α)-computable, from an α-name of R and r ∈ dom(ξ C ) encoding the sequence R 0 , R 1 , . . .we can compute a sequence s ∈ Σ ω encoding the sequence is the limit of an increasing computable sequence of rational numbers which may be finite or ∞.By Example 4.11 there is a computable finite measure with finite non-computable µ(Ω).

Representations by means of a partition
We still assume that µ is a computable measure on the computable σ-algebra (Ω, A, R, α).As we have mentioned there are ring elements and it is majorising if there is a computable function Lemma 6.2.There is a majorising partition for α.
Proof.There is a bijective computable function h : N → dom(α).For the numbering Then F satisfies (6.1).Since union and set difference are (E, E, E)-computable, there is some computable function i≤g ′ (w) F i .Then g ′ is computable and satisfies (6.2).
For i ∈ N and A ∈ A let µ i (A) := µ(F i ∩ A).Then µ i is a computable measure on (Ω, A, R, α) such that µ i (Ω) = µ(F i ) is (finite and) ρ-computable (see Lemma 4.10 and Theorem 5.3).For every i, d i defined by ) is a computable pseudometric on A (not only on A f ).Notice that µ i is the restricton of the measure µ to F i and d i (A, B) is the finite distance of A and B restricted to  [23].The statements hold accordingly for metrics.
Proof.By the above remarks d is a pseudometric on A, and since Since union, intersection and difference on R are (α, α, α)-computable, the restriction of d to R is (α, α, ρ)-computable.Below, we show that R is dense in (A, d).Let k ∈ N. Since the metric d is (ξ, α, ρ)-computable and R is dense in (A f , d), for every i we can find some u i such that for and hence This implies that R is dense in (A, d).
. Some u such that d(A, S) ≤ 2 −k−i−1 for S = α(u) can be computed from p, i and k.
Since µ(F i ∩ R ∩ (A ∆ S)) ≤ µ(F i ∩ (A ∆ S)) and the function e : x → x/(1 + x) is increasing, hence by (6.3), µ( Proof.This follows from Theorems 6.4 and 6.5. In the proof of Lemma 6.2 we have constructed a majorizing partition F for α.Although the metric d on [A] and the representation ξ C introduced in Definition 6.4 depend on F , the equivalence class of ξ C is the same for all such partitions.In [34,35] Wu and Ding have introduced several other representations of the measurable sets.First, we consider [35].The representation δ T 1 [35,Theorem 4.1] can be expressed informally as follows: δ T 1 (p) = [A] iff p consists of a list of all pairs (E, r) such that µ(E \ A) < r and a list of all pairs (E, r) such that µ(A \ E) < r (where E ∈ R and r ∈ Q).Since µ(E) = µ(E \ A) + µ(E ∩ A) and µ(E) can be computed, the first list can be replaced by a list of all pairs (E, r) such that r < µ(E ∩ A).
Define δ 1 p, q = [A] iff ζ + (p) = [A] and ρ > (q) = µ(A).Then δ 1 ≡ δ T 1 (without proof).Therefore, the restriction of δ T 1 to the sets of infinite measure is equivalent to ζ + and its restriction to the sets of finite measure is equivalent to ξ + , hence also equivalent to ξ − , ξ and ξ C by Theorem 5.3.
Accordingly, the representation δ T 2 from Section 4.2 is equivalent to the following representation δ 2 defined by δ 2 p, q, r = The third representation δ T 3 from [35, Section 4.3] uses a computable sequence (C i ) i∈N where C n = i<n D i for some partition (D i ) i∈N for α such that µ(D i ) > 0. The condition µ(D i ) > 0 excludes some spaces from consideration.It is irrelevant for the representaion δ T 3 but important for the representaton δ D 1 below.The representation δ T 3 can be defined informally as follows: From p we can compute a list of all (E, k, r) (r rational) such that µ((A ∆ E) ∩ D k ) <r.Using arguments similar to those in the proof of Theorem 6.5 we can prove δ T 3 ≡ ζ.The additional condition µ(D i ) > 0 in [35,Theorem 3.3] is not used in this proof.If the partition D is majorising then δ T 3 ≡ ζ (without proof).
In [34,Definiton 5.1] a metric on [A] is defined by This definition is only meaningful if µ(D i ) > 0 for all i.Therefore, for the metric d in (6.5) we use the denominators 1 + µ(D i ∩ (A ∆ B)) instead of µ(D i ).The Cauchy representation for the computable metric space A function f : Ω → X to a topological space X is measurable, if f −1 (U ) is measurable for every open set U .Since intersection and countable union are computable on the open subsets of a computable topological space [31] these operations should also be computable on the measurable sets (since, for example, f −1 ( U i ) = i f −1 (U i )).From all the representations of measurable sets mentioned in this article only for the representation ζ + intersection and countable union are computable.Therefore, we claim that ζ + is the most useful one for studying computability of measurable functions.
In [35,Sections 4.1 and 4.2] proper supersets of σ := {↑ (E, r) | R ∈ R, r ∈ Q + } where ↑ (E, r) := {A ∈ A | µ(R \ A) < r} have been used as subbases of topologies for defining the representations δ T 1 and δ T 2 of the measurable sets.The set σ itself would yield a representation which is equivalent to ζ + .The authors have not taken this case into consideration.
A representation δ : ⊆ Σ ω → X of a topological T 0 -space (X, τ ) is admissible, iff it is continuous and γ ≤ δ for every other continuous representation γ of X [30,25,24,26,8].For admissible representations, a function on the represented sets is continuous, iff it can be realized by a continuous function on the names.
Let λ : Σ * → σ be a notation of a set of subsets of X such that σ is a subbase of a T 0topology (X, τ ).Define a representation δ : ⊆ Σ ω → X as follows: δ(p) = x iff p is a list of all w such that x ∈ λ(w).Then δ is an admissible representation of the space (X, τ ) where τ is the final topology of δ [31].All the other representations of measurable sets defined in this article can be written in this way and hence are admissible.In each case a subbase of the final topology can be directly extracted from the definition.From (2) or (3) it follows that (A f , d) is a complete pseudometric space.
We prove (4), i.e. density of R f .For C⊆Ω let U (C) be the set of all sequences (R i ) i∈N of ring elements such that C⊆ i∈N R i .In the Carathéodory proof of the extension theorem [2] the measure µ is defined on A by its values on the ring as follows: Let C ∈ A f and let ε > 0. There is some sequence (R i ) i∈N ∈ U (C) such that C⊆ i∈N R i and 0 ≤ i∈N µ(R i ) − µ(C) < ε/2.Then (∀i) R i ∈ R f .Let S 0 := R 0 and S i := R i \ (R 0 ∪

Definition 3 . 1 .
Let µ be a measure on a ring R. Define an equivalence relation on R by A ∼ B ⇐⇒ µ(A ∆ B) = 0. Let [A] := {B ∈ A | A ∼ B} be the equivalence class containing A. For E⊆R, let [E] := {[A] | A ∈ E}.

Proof.
By Theorem 3.5, (A f , d) with d(A, B) = µ(A ∆ B) is a complete pseudometric space with R f = R as a dense set.Since by Definition 3.1, d([A], [B]) = 0 ⇐⇒ [A] = [B], ([A f ], d) is a complete metric space with [R] as a dense subset.Obviously β is a notation of [R] with recursive domain.Since d([A], [B]) = µ(A ∆ B) and by Definition 4.1, the symmetric difference on R is computable and µ

Definition 5 . 2 .
For the space A f := ([A f ], d, [R], β) let ξ C be the Cauchy representation and define representations ξ + , ξ − and ξ by Then every µ i is a finite measure and µ(A) = i∈N µ i (A).By Lemma 3.6 for every A ∈ A, [A] is defined by the family (µ(A ∩ R)) R∈R .The representations ζ + , ζ − and ζ from Definition 4.2 are defined by means of this family ("a ζ + -name of [A] is a list of all . .." etc.).Correspondingly, for every i and A, [A ∩ F i ] is defined by the family (µ i (A ∩ R)) R∈R .Therefore [A] is defined also by the family (µ(A ∩ F i ∩ R)) (i∈N, R∈R) which is a subfamily of (µ(A ∩ R)) R∈R .We introduce representations ζ + , ζ − and ζ of [A] by means of this smaller family and compare them with ζ + , ζ − and ζ.Definition 6.1.A numbering F i .Define e : [0; ∞) → [0; 1) by e(x) := x/(1 + x).Then e −1 (y) = y/(1 − y) and e and e −1 are (ρ, ρ)-computable increasing functions such that e(x) ≤ x and e −1 (y) ≤ 2 • y for y ≤ 1/2 .(6.3)It is known that for a pseudometric d, d ′ := e • d = d/(1 + d) is a pseudometric bounded by 1 with the same induced topology.Furthermore, for a sequence (d i ) i∈N of pseudometrics bounded by 1, d(x, y) by Lemma 4.10 we can compute a ρ < -name of µ(F i ∩ A), hence a ρ < -name of µ(F i ∩ A) since µ(F i ∩ A) is finite.Since ζ ≤ ζ − ,by Definition 5.2 we can compute a ξ − -name q of [F i ∩ A].Therefore, by Theorem 5.3 from (p, i) we can compute a ξ C -name r of [F i ∩ A].Then r is (encodes) a sequence v 0 , v 1 , . . .such that d(α(v k ), F i ∩ A) ≤ 2 −k .
Let ζ(p) = [A].By Definition 6.1, for any i, an α-name of F i can be computed.So an α-name of R can be computed from p. Hence a sequence (v 0 , v 1 , . ..) can be computed such that d(A, α(v k )) ≤ 2 −k−1 , which by definition constitutes a ξ C -name of [A].Therefore ζ ≤ ξ C .By density of R, (A, d, R, α) is a computable pseudometric space and ([A], d, [R], β) is a computable metric space.ξ C ≤ ζ: We apply the following characterization which is similar to Lemma 4.4(3):

Corollary 6 . 6 .
Define ([A], d) and the Cauchy representation ξ C as in Theorem 6.5 by a majorising partition F for the notation α of the ring R. Then ζ ≡ ξ C .

7 .
Summary and final remarks Up to equivalence we have the four new representations ζ + , ζ − , ζ and ξ C .The representations ζ + , ζ − and ζ are equivalent to the first three ones if they are defined by means of a majorising partition which always exists.For the Cauchy representation ξ C of the sets of finite measure, ξ C ≡ ζ, if µ(Ω) is (finite and) ρ-computable.If the Cauchy representation ξ C is defined by means of a majorising partition, then ξ C ≡ ζ.
computability of complementation can be proved accordingly.For the representations ζ + , ζ − and ζ a name of a class [A] allows to compute µ(α(v)∩A) w.r.t.ρ < , ρ > and ρ, respectively.Since (α By a proof similar to that of Theorem 6.5 it can be shown that δ D 1 ≡ ζ.By Lemma 6.2 there is a majorising partition In this case, δ D 1 ≡ ζ by Theorem 6.4.Also for another metric a Cauchy representation δ D 2 is introduced.Only for the representations δ T 3 and δ D 1 , which are equivalent (without proof) union and intersection on the measurable sets are computable.It can be shown that union and intersection are computable also for ζ + and ζ − and that countable union is computable for ζ + but not for ζ.