How constructive is constructing measures?

Given some set, how hard is it to construct a measure supported by it? We classify some variations of this task in the Weihrauch lattice. Particular attention is paid to Frostman measures on sets with positive Hausdorff dimension. As a side result, the Weihrauch degree of Hausdorff dimension itself is determined.


Introduction
We investigate variations on the problem to construct a measure with a given support. The variations include the available information about the set, whether the support has to be precisely the given set or merely a subset, and whether the measure is required to be non-atomic. A special case of particular importance is the Frostman lemma, which links having certain Hausdorff dimension to admitting a measure with certain properties.
Two of these variations are computable: Given an overt set A, one can construct a measure with support precisely A. In a very restricted setting, a computable version of the Frostman lemma is available. Apart from these cases, the problems generally are non-computable. Using the framework of Weihrauch reducibility, we can establish the precise degree of noncomputability of each case.
The Weihrauch degrees from the framework for the research programme to classify the computational content of mathematical theorems formulated by Brattka and Gherardi [5] (also Gherardi & Marcone [17], P. [32]). The core idea is that S is Weihrauch reducible to T , if S can be solved using a single invocation of T and otherwise computable means.

A short introduction to represented spaces
We briefly present some fundamental concepts on represented spaces following [34], to which the reader shall also be referred for a more detailed presentation. The concept behind represented spaces essentially goes back to Weihrauch and Kreitz [24], the name may have first been or κ X : X → V(X) as κ X ({x}) = {x} instead. The image of X under κ X shall be denoted by X κ . The following definition essentially goes back to Schröder [35] and provides an effective counterpart to the definition in [36]: Definition 1. A space X is called computably admissible, if X and X κ are computably isomorphic.
Note that X κ is always computably admissible, i.e. isomorphic to (X κ ) κ . The computably admissible spaces are precisely those that can be regarded as topological spaces, based on the fact that the computable map f → f −1 : C(X, Y) → C(O(Y), O(X)) becomes computably invertible iff Y is computably admissible.
As a special case of represented spaces, we define computable metric spaces following Weihrauch's [41]. The computable Polish spaces, are derived from complete computable metric spaces by forgetting the details of the metric, and just retaining the representation (or rather, the equivalence class of representations under computable translations). Definition 2. We define a computable metric space with its Cauchy representation such that: 1. A computable metric space is a tuple M = (M, d, (a n ) n∈N ) such that (M, d) is a metric space and (a n ) n∈N is a dense sequence in (M, d).

The relation
As any computable metric space is (effectively) countably based, the following well-known characterization of the open sets in such spaces is useful for us: Proposition 3 ([19, Proposition 11]). Let X have an effective countable base (U i ) i∈N and be computably separable. Then the map : O(N) → O(X) defined via (S) = i∈S U i is computable and has a computable multivalued inverse.

Computable measure theory
Computable measure theory requires a further special represented space for its development. We can introduce the space R < by identifying a real number x with the set {y ∈ R | y < x} ∈ O(R). Equivalently, using {y ∈ Q | y < x} ∈ O(Q) provides the same result. A third way is to use a monotone growing sequence (q n ) n∈N ∈ C(N, Q) as a stand-in for sup n∈N q n ∈ R. We will make use of the following: Proof. Using type conversion, we show instead that given U ∈ O([0, 1] N ) and y ∈ R, it is recognizable if ∃(x 0 , x 1 , . . .) ∈ U i∈N x i ≤ y . Given y, we can simultaneously try (q 0 , q 1 , . . . , q n , 0, 0, . . .) ∈ U ? for all rational vectors (q 0 , . . . , q n ) such that y > n i=0 q i . If we find such a vector, then clearly the answer is yes. On the other hand, if any (x 0 , x 1 , . . .) ∈ U with i∈N x i ≤ y exists, then there must a rational eventually-zero such vector since U is open.
Given some represented space X, we direct our attention to the space C(O(X), R < ) of continuous functions from the open subsets of X to R < . Note that for any µ ∈ C(O(X), R < ) we find that if U ⊆ V for some U, V ∈ O(X), then µ(U ) ≤ µ(V ). We introduce the space M(X) as a subspace of C(O(X), R < ) by: The space P(X) of probability measures is obtained in the straightforward way as P(X) := {µ ∈ M(X) | µ(X) = 1}. Given a point x ∈ X, we can define the point-measure π x by π x (U ) = 1 iff x ∈ U and π x (U ) = 0 otherwise. Then x → π x : X → P(X) is computable. Also, the usual push-forward operation (f, µ) → f * µ : C(X, Y) × M(X) → M(Y) is computable. However, this works only for the continuous functions, not for any larger class of measurable functions: Definition 5. Let P(X) be the space of probability measures on X. To define the space M C (X, Y) of measurable functions from X to Y, identify a measurable function f : X → Y with its lifted version f * : P(Y) → P(X), and define M C (X, Y) as the according subspace of C(P(Y), P(X)).
Proof. Given a continuous function f : X → Y, we can get f −1 : O(Y) → O(X), and compose this with a measure ν ∈ P(Y) to obtain f * ν. This establishes one direction.
For the other direction, note that x → π x : The left hand side is recognizable by the definition of P(Y), and admissibility of Y means that the recognizability of the right hand side for arbitrary U implies continuity of f . As a representation always is a continuous function (by definition of continuity), we see that we can push a measure on Baire space out to the represented space. As shown by Schröder, in many cases the converse is also true: Theorem 7 (Schröder [37]). Let X be a complete computably admissible space. The map µ → δ * X µ : P(N N ) → P(X) is computable and computably invertible. In a computable Polish space it is sufficient to know the value a measure takes on the basic open balls, as the following (simple) generalization of a result by Weihrauch [42] shows: Proposition 8. Let X be a computable Polish space with dense sequence (a n ) n∈N . Then the map µ → (µ(B(a n , 2 −k ))) n,k ∈N : M(X) → C(N, R < ) is computable and computably invertible.
Proof. That this map is computable is straight-forward element-wise function application. For the inverse direction, note that given some open set U we can effectively approximate it from the inside by finite disjoint unions of basic open balls. Both finite sums and countable suprema are computable on R < , and this is all that is required.
We will frequently use the preceding proposition, and construct measures simply by providing their values on the basic open balls without further notice.
In the (albeit very restricted) case of probability measures on N there is further interesting characterization available. Essentially, one may use typical sequences as names for measures: Theorem 9 ( [31]). Uniformly in ε > 0 there is a computable and computably invertible function S ε :⊆ N N → P(N) such that for all µ ∈ P(N) we find that µ(S −1 ε (µ)) ≥ 1 − ε, and if ν = µ, then ν(S −1 ε (µ)) = 0. Here µ denotes the induced product measure on N N .

Weihrauch reducibility
Definition 10 (Weihrauch reducibility). Let f, g be multi-valued functions on represented spaces. Then f is said to be Weihrauch reducible to g, in symbols f ≤ W g, if there are computable functions K, H : The relation ≤ W is reflexive and transitive. We use ≡ W to denote equivalence regarding ≤ W , and by < W we denote strict reducibility. By W we refer to the partially ordered set of equivalence classes. As shown in [33,6], W is a distributive lattice, and also the usual product operation on multivalued function induces an operation × on W. The algebraic structure on W has been investigated in further detail in [20,11].
We will make use of an operation ⋆ defined on W that captures aspects of function decomposition. Following [8,10] We understand that the quantification is running over all suitable functions f 0 , g 0 with matching types for the function decomposition. It is not obvious that this maximum always exists, this is shown in [11] using an explicit construction for f ⋆ g. Like function composition, ⋆ is associative but generally not commutative.
An important source for examples of Weihrauch degrees relevant in order to classify theorems are the closed choice principles studied in e.g. [5,4]: Definition 11. Given a represented space X, the associated closed choice principle C X is the partial multivalued function C X :⊆ A(X) ⇒ X mapping a non-empty closed set to an arbitrary point in it.
For any uncountable compact metric space X we find that C X ≡ W C [0,1] . For well-behaved spaces, using closed choice iteratively does not increase its power, in particular 1] is closely linked to WKL in reverse mathematics, while C N is Weihrauch-complete for functions computable with finitely many mindchanges.
Further variations of closed principle providing a fruitful area of study are obtained by restriction to certain subclasses of the closed sets. In [9,10] choice for connected closed subsets of [0, 1] k was studied (and related to Brouwer's Fixed Point theorem). Convex and finite sets were compared in [26,25]. Most related to the present investigation, choice for sets of positive Lebesgue measure was studied in [12,7]. This yields a Weihrauch degree Once more, replacing [0, 1] with another uncountable compact metric space X does not change the Weihrauch degree.
Another typical degree is obtained from the limit operator lim :⊆ N N → N N defined via lim(p)(n) = lim i→∞ p( n, i ). This degree was studied by von Stein [40], Mylatz [28] and Brattka [2,3], with the latter noting in [3] that it is closely connected to the Borel hierarchy. Hoyrup, Rojas and Weihrauch have shown that lim is equivalent the Radon-Nikodym derivative in [23]. It also appears in the context of model of hypercomputation as shown by Ziegler [44,43], and captures precisely the additional computational power certain solutions to general relativity could provide beyond computability [21]. It is related to the examples above via Important further representatives of the degree of lim are found in the following: This can be generalized further: Proposition 13. Let X be a computable metric space admitting a computable sequence a ∈ C(N, X) together with a computable sequence r ∈ N N such that ∀n, m ∈ N (n = m ⇒ d(a n , a m ) > 2 −rn ). Then: Proof. We show the reduction (id : Given some closed set A ∈ A(N), we construct a closed set B ∈ A(X) as follows: When n ∈ N is removed from A, we remove B(a n , 2 −rn ) from B. By choice of the sequence, we then find that n ∈ A ⇔ B(x n , 2 −rn ) ∩ B = ∅, hence knowing B as an overt set implies knowing A as an overt set.
Whether any infinite computable metric space admits a computable sequence as above seems to be an open problem. Certainly such sequences exist without the computability requirement, thus we obtain: Corollary 14. Let X be an infinite computable metric space. Then lim ≤ W (id : A(X) → V(X)) relative to some oracle.
Proposition 15. Let X be a locally compact computable metric space. Then: Proof. We may assume a basis of basic open balls with compact closures in this case. Then, given A ∈ A(X), we can enumerate all basic open balls B n such that B n ∩ A = ∅. Using (id : A(N) → V(N)) ≡ W lim, we can transform such an enumeration to its characteristic function then there is some B n with B n ⊆ U and B n ∩ A = ∅. Given χ, we can effectively search for such a candidate, thus, A ∈ V(X) is computable from χ.
Corollary 16. Let X be an infinite locally compact computable metric space. Then relative to some oracle: Proof. As the (closure of the) image of an overt set under a continuous function is overt, the By assumption on (a n ) n∈N , we now find that n / ∈ A ⇔ a n / ∈ {a i | i ∈ A}. Thus, we have a reduction: By v Stein's result (Theorem 12), this is equivalent to our claim.
Proposition 18. Let X be computably countably based. Then: Proof. Let (U n ) n∈N be a computable countable basis. Given A ∈ V(X), we may compute , and will find this to be equivalent to A ∈ A(X).

Hausdorff dimension and the Frostman lemma
The Frostman lemma essentially states that having positive Hausdorff dimension is equivalent to admitting a measure that is far from being atomic -and being far from atomic is given a quantitative interpretation and exactly tied to the Hausdorff dimension. We introduce the Hausdorff dimension only as a property of closed subsets of [0, 1] here: , dim H (A) = 0 for any countable A and dim H (A) = 1 whenever λ(A) > 0 where λ denotes the Lebesgue measure. A way to construction sets of given Hausdorff dimension is provided in [15]: Given a sequence (d i ) i∈N of non-negative reals, we define a family ( The claim for ϑ now follows from dim H ((θ • γ) −1 ({p})) = 1 for any p ∈ {0, 1} N . This can be shown using the mass distribution principle (e.g. [14]). For this, we define a measure µ p on C using the intervals [a w , b w ] occurring in the construction of C. Start with µ p ([0, 1] . This yields indeed a measure, and we find µ p ((θ • γ) −1 ({p})) = 1. To invoke the mass distribution principle, we further need that lim r→0 log µp(B(x,r)) log r = 1 for all x ∈ supp(µ p ).
We can now introduce Frostman measures, and state the Frostman lemma: {∃µ | supp(µ) ⊆ A ∧ µ is an s-Frostman measure} We will in particular investigate the Weihrauch degree of the following maps:

Measures and support
We shall begin by investigating how a measure and its support are related. We show that the support is fundamentally an overt set, rather than a closed set; and that both obtaining the support of a given measure as a closed set, and constructing a measure with support as a given closed set are equivalent to the lim-operator. The measures constructed here will generally fail to be non-atomic.
Theorem 25. supp : M(X) → V(X) is computable. If X is a computable metric space, it has a computable multivalued inverse. Proof.
Note that an open set U intersects supp(µ) iff µ(U ) > 0. It is easy to see that that > 0 : R < → S is computable. Taking into consideration the definitions of M and V, we see that f → (U → > 0(f (U ))) is a computable realizer of supp. For the reverse we shall construct a probability measure µ given a non-empty overt set A ∈ V(X) such that supp(µ) = A. Let (x i ) i∈N be a computable dense sequence in X. We begin by associating numbers c i,k ∈ R < to the basic open balls B(x i , k2 −k ). First, we test for any In the next round, we test for all B(x i , 2 −2 ) if they intersect A, and again obtain an infinite sequence B(x l ′ 1 , 2 −2 ), B(x l ′ 2 , 2 −2 ), . . . of balls doing so, potentially with repetitions. For any i ∈ N, let l i be the least j such that B(x l ′ j , 2 −2 ) ⊆ B(x i , 2 −1 ), provided that this exists, and i otherwise. Now we set c i,2 = {j|l j =i} c i,1 + {j|l ′ j =i} 2 −j+2 . We proceed to construct the remaining numbers in this pattern. Note that we find i∈N c i,k = 1 2 + . . .   If we just demand that the (non-zero) measure to be constructed is supported by the given (non-empty) closed set, the resulting operator ConstructMeasure X :⊆ A(X) ⇒ M(X) is strictly simpler for many spaces: Theorem 27. Let X be a computable Polish space. Then ConstructMeasure X ≡ W C X .
Proof. For ConstructMeasure X ≤ W C X , use C X to pick a point x in A, and then compute the point measure µ x , which is non-zero and satisfies supp(µ x ) = {x} ⊆ A.
For the other direction, we need to show that given a non-zero measure µ supported by A we can compute some point x ∈ A. We do this by searching for a basic open ball B(x 1 , 2 −1 ) with µ(B(x 1 , 2 −1 )) > 0, which we will find eventually. Then we search for some x 2 ∈ B(x 1 , 2 −1 ) such that µ(B(x 2 , 2 −2 )) > 0, which also will eventually be detected. We continue to produce a fast Cauchy sequence, of which we can compute the limit x. Now x ∈ A is easy to see.

Non-atomic measures
The picture painted in Section 3 of the constructivity (or lack thereof) of constructing measures crucially depends on the option of resulting measures having atoms, i.e. single points carrying positive measure. In the present section we first introduce the notion of flows on infinite trees as a technical tool (which could be of some interest in its own right). We then proceed to investigate the role of overtness, which drastically differs from the results above. Considering some Weihrauch degrees related to Hausdorff dimension then leads up to the Frostman lemma.

Non-atomic measures on [0, 1] and flows
We consider assignments of non-negative real numbers to the edges of a full infinite binary tree, i.e. the space R + For the other direction, we inductively define a decreasing sequence of real numbers for each edge of the tree by setting a v 0 = f (v) and a v n+1 = min{a v n , a v0 n + a v 1 n }. Then (v → lim i→∞ a v i ) ∈ MaxFlow(f ), so using lim countably times in parallel suffices to find a valid max flow. As stated above, this is equivalent to using lim just once. is defined via g ∈ NonZeroFlow(f ) iff g ≤ f , g is a flow and g(ǫ) > 0.
Proof. To see that NonZeroFlow ≤ W C N ×C {0,1} N we use a non-deterministic algorithm following [4]. We guess some number k ∈ N together with an assignment g : {0, 1} * → [0; 2 −k ] where g(ε) = 2 −k (note that the latter is an element of a computably compact computable metric space). If g is not a flow with g ≤ f where f is the input to NonZeroFlow, we will detect this eventually.
For the other direction, we may prove WKL×UpperBound ≤ W NonZeroFlow instead. Thus, we are given an infinite binary tree T and a monotone and bounded sequence of natural numbers (n i ) i∈N . Let λ T n := |{v ∈ T | |v| = n}| − 1. We define an assignment f : if v ∈ T and f (v) = 0 otherwise. Any non-zero flow g smaller than f then computes both an infinite path through T (just go down some path carrying positive flow) and an upper bound for (n i ) i∈N (in form of N s.t. g(ε) > 2 −N ).
Let the multivalued map ConcentrateFlow map a non-zero flow f to any non-zero concentrated flow g with f (ε)g ≤ f .

Proposition 33. ConcentrateFlow is computable.
Proof. We may normalize the input flow to f (ε) = 1. We define g iteratively, starting with g(ε) = 1 2 . We want to ensure the invariant g throughout the process -any flow constructed this way clearly is a valid answer. The invariant holds at the initial step.
For the continuation step, we can computably select a true case among In the first case, set g(v0) = g(v) and g(v1) = 0. In the second case, g(v0) = 0 and g(v1) = g(v). In the third case,

Overtness and non-atomic measures
It is clear that isolated points cannot be part of the support of an isolated measure. Thus, when constructing measures supported by given sets, we either need to consider only perfect sets, or be satisfied if the support is included in the set, rather than demanding equality. In the latter situation, overtness becomes useless as demonstrated next: Given some closed set A, let A * be the largest perfect set contained in A. Alternatively, let A * be the set resulting from A after removing the isolated points α times for some countable ordinal α, after which no isolated points remain. Proof. We assume that we have an enumeration of all basic open intervals whose closure disjoint is with the input A, and that we have to decide for each basic open interval whether it intersects the constructed output B. Basic open intervals are assumed to be finite, hence have compact closure. We will decide on these answers in some order, and in particular deal with smaller intervals only after those containing it. In the following, let I always the interval we currently need to decide on. We further maintain a set X of rational numbers, which at any stage of the construction will be finite, and initially is empty.
2. If we have already learned that I ∩ A = ∅, and additionally I ∩ X = ∅, then we decide that I ∩ B = ∅.
3. If I ∩ X = ∅, and it is consistent with the information we have read about A so far that I ∩ A = ∅, then we decide that I ∩ B = ∅ and keep monitoring I in the following.
4. If I is an interval we monitor due to (3.), and we do learn that I ∩ A = ∅, then at this particular time there is still some basic open interval L ⊆ I which is disjoint with any interval J for which we already had decided that J ∩ B = ∅, due to compactness. We can effectively find such an L. Let x be the midpoint of L and add it to the set X. Let L 1 and L 2 be the remaining open halves of L. We also decide that L 1 ∩ B = ∅ and L 2 ∩ ∅. These rules ensure that the decisions made for the individual intervals are actually consistent, and thus the construction actually yields some closed and overt set B, which incidentally is of the form B = A ∪ X with X ∩ A = ∅. The removal of L 1 and L 2 in Item (4.) ensures that any x ∈ X is isolated in B, hence B ∈ PerfectCore(A).
Proof. Let µ be the input. We search for some k ∈ N, such that µ([0, 1]) ≥ 2 −k . Now we can construct a flow f from µ such that f (ε) = 2 −k and f (v) ≤ µ(D • v ) for any v ∈ {0, 1} * . From this flow we obtain a concentrated flow by Proposition 33, which then yields the desired concentrated measure.
Proposition 37. Let X be a computably compact computable metric space. Then ConcentratedSupport is computable.
Proof. The V-name is computed as in Theorem 25. As we work in computably compact space, and are dealing with non-atomic measures, we can actually compute the measure of open balls as real numbers. Then to find the A-name, note that given a concentrated measure ν for any dyadic open ball B we have that ν(B) < Cr 2 ⇔ B ∩ supp(ν). The left hand side is recognizable, and then the right hand side produces the desired result.

Some Weihrauch degrees related to Hausdorff dimension
Before continuing with our investigation of measures, we shall classify the Hausdorff dimension itself. Note that our result does make use of the fact that we have defined Hausdorff dimension only for subsets of [0, 1] -the result generalizes directly to compact subsets of a metric space though.
Proof. First, we show dim H ≤ W lim ⋆ lim, split over the following two lemmata.
as follows from arguments in [30,28]. Since LPO ⋆ lim ≡ W lim ⋆ lim, we can decide whether inf U d = 0 for all rational d in parallel, and this suffices to obtain inf{d ∈ [0, 1] | inf U d = 0} ∈ R.
For the other direction, first note that a standard argument establishes: Proof. For the ≤ W -direction, note that inf ≡ W sup ≡ W lim ≡ W lim, e.g. [40,6].
Given some closed set A ∈ A([0, 1]) and some interval [a, b], let A ≺ [a, b] be the rescaling of A into [a, b]. Not only is this a computable operation, but even ( is computable. Moreover, we have: Combining this with the construction of sets of given Hausdorff dimension in Subsection 2.4 (which takes care of the inner inf for free) and the preceding lemma, we obtain the claim.

The Frostman Lemma
We now finally direct our attention to the maps Frost and StrictFrost introduced in Definition 24. One can prove the Frostman lemma via the min-cut/max-flow theorem, and the construction of the flows involved (if done in the right way) yields Weihrauch reductions to NonZeroFlow and MaxFlow respectively.
Given a closed set A ∈ A([0, 1]) and some s ∈ [0, 1] we construct an infinite binary tree with capacities. As in Subsection 4.1, we assume that the vertices of the tree are labeled with (closed) dyadic intervals in the canonic way. As [0, 1] is compact, we can semidecide if A ∩ I = ∅ for a closed interval, and we run all these tests simultaneously. When it comes to assigning a weight to a vertex of depths n, we choose 2 −sn if no interval containing the label of the vertex has been identified as disjoint with A yet. If we have proof that some superset (and thus the label itself) is disjoint from A, we assign the weight 0. Any cut through this tree gives an upper bound for the s-dimensional Hausdorff content of A, and so the min-cut/max-flow theorem implies that there is a non-zero flow compatible with the tree. Any such flow then induces an s-Frostman measure using the construction from Subsection 4.1. This proves: Lemma 47. Frost ≤ W NonZeroFlow and StrictFrost ≤ W MaxFlow.
The converse directions are obtained from the following in conjunction with preceding results: Proof. As before, we use the rescaling operation 3 A ≺ [a, b]. Given a non-empty closed set A ∈ A(N) and another closed subset B ∈ A([0, 1]) with dim H (B) = 1, we may compute the set , and note that this set again has Hausdorff dimension 1. Moreover, we make use of C N ≡ W UC N , i.e. for closed choice on N we may safely assume that the closed set is a singleton. In this case, the support of any Frostman-measure on the set is inside B ≺ [2 −2n−2 , 2 −2n−1 ] for the unique n ∈ N with A = {n}. From this, we can compute a point in B ≺ [2 −2n−2 , 2 −2n−1 ], which in turn allows us to find both a point in B as well as identify n. Proof. Given A ∈ A(N), we can compute {0} ∪ n∈A [2 −2n−2 , 2 −2n−1 ] . We use StrictFrost to find a measure with this set as support, and then Theorem 25 to obtain the set as overt set. To complete the reduction, note n ∈ A ⇔ (2 −2n−2 , 2 −2n−1 ) ∩ {0} ∪ i∈A [2 −2i−2 , 2 −2i−1 ] = ∅.
A direct consequence of Corollary 50 is a computable closed set with Hausdorff dimension > 2 −n may still fail to admit a computable 2 −n -Frostman measure. Proposition 35 then shows that requiring the set to be a computable closed and overt set does not change this. On the other hand, in [15] a construction of computable Frostman measures on very special sets is given. We can make the relevant properties of the sets explicit in the following: Lemma 52. Given s ∈ [0, 1], a set A ∈ A([0, 1]) ∧ V([0, 1]) and p ∈ N N such that for any dyadic interval I we find I ∩A = ∅ or A∩I admits an s-Frostman measure µ with µ(I ∩A) ≥ 2 −p(− log |I|) we can compute an s-Frostman measure on A.
By Propositions 36, 37 the converse is true, too: Corollary 53. Let A admit a computable s-Frostman measure. Then there is a computable B ∈ A([0, 1]) ∧ V([0, 1]) with B ⊆ A and a computable sequence p ∈ N N such that for any dyadic interval I we find I ∩A = ∅ or A∩I admits an s-Frostman measure µ with µ(I ∩A) ≥ 2 −p(− log |I|) . Moreover, if an s-Frostman measure ν on A is given, we can effectively find B and p.

On computably universally measure 0 sets
Recall that a set A is called universally measure 0, if there is no non-atomic non-zero Radon measure supported by A. Clearly every countable set is universally measure 0. There are universally measure 0 sets with cardinality 2 ℵ 0 , however, such sets cannot be Borel. An example on the real line was constructed by Sierpiński and Szpilrajn-Marczeswski [39]. Later, Zindulka even found an example of a universally measure 0 set with positive Hausdorff dimension [45] -thus, the requirement of compactness of A in the Frostman lemma cannot be completely relaxed.
Our results show that in the computable world, the picture is very different: If we call a set A computably universally measure 0, if there is no non-atomic non-zero computable Radon measure supported by A, we find that there is a computable closed and computable overt set with Hausdorff dimension 1 that is computably universally measure 0.
A much stronger effective notion is arithmetically universally measure 0 -a set is this, if it does not support any non-atomic non-zero Radon measure computable relative to some arithmetic degree. This notion (under a different moniker) was employed by Gregoriades in [18] in order to study computable Polish spaces up to ∆ 1 1 -isomorphisms: A Polish space X is ∆ 1 1 -isomorphic to N N iff it is not arithmetically universally measure 0. Gregoriades then constructed computable closed and computable overt sets with cardinality 2 ℵ 0 that are arithmetically universally measure 0.