The interplay of classes of algorithmically random objects

We study algorithmically random closed subsets of $2^\omega$, algorithmically random continuous functions from $2^\omega$ to $2^\omega$, and algorithmically random Borel probability measures on $2^\omega$, especially the interplay between these three classes of objects. Our main tools are preservation of randomness and its converse, the no randomness ex nihilo principle, which say together that given an almost-everywhere defined computable map between an effectively compact probability space and an effective Polish space, a real is Martin-L\"of random for the pushforward measure if and only if its preimage is random with respect to the measure on the domain. These tools allow us to prove new facts, some of which answer previously open questions, and reprove some known results more simply. Our main results are the following. First we answer an open question of Barmapalias, Brodhead, Cenzer, Remmel, and Weber by showing that $\mathcal{X}\subseteq2^\omega$ is a random closed set if and only if it is the set of zeros of a random continuous function on $2^\omega$. As a corollary we obtain the result that the collection of random continuous functions on $2^\omega$ is not closed under composition. Next, we construct a computable measure $Q$ on the space of measures on $2^\omega$ such that $\mathcal{X}\subseteq2^\omega$ is a random closed set if and only if $\mathcal{X}$ is the support of a $Q$-random measure. We also establish a correspondence between random closed sets and the random measures studied by Culver in previous work. Lastly, we study the ranges of random continuous functions, showing that the Lebesgue measure of the range of a random continuous function is always contained in $(0,1)$.


Introduction
In this paper, we have two primary goals: (1) to study the interplay between algorithmically random closed sets on 2 ω , algorithmically random continuous functions on 2 ω , and algorithmically random measures on 2 ω ; and (2) to apply two central results, namely the preservation of randomness principle and the no randomness ex nihilo principle, to the study of the algorithmically random objects listed above. Barmpalias, Brodhead, Cenzer, Dashti and Weber initiated the study of algorithmically random closed subsets of 2 ω in [BBC + 07]. Algorithmically random closed sets were further studied in, for instance, [Axo10], [DKH12], and [CW13]. In the spirit of their definition of algorithmically random closed set, Barmpalias, Brodhead, Cenzer, Dashti and Weber also defined a notion of algorithmically random continuous function on 2 ω in [BBC + 08]. The connection between random closed sets and effective capacities was explored in [BCTW11]. More recently, Culver has studied algorithmically random measures on 2 ω in [Cul14]. One of the central results [BBC + 08] is that the set of zeros of a random continuous function of 2 ω is a random closed subset of 2 ω . Inspired by this result, we here investigate similar bridge results, which allow us to transfer information about one class of algorithmically random objects to another.
Two tools that are central to our investigation, mentioned in (2) above, are the preservation of randomness principle and the no randomness ex nihilo principle. In 2 ω , the space of infinite binary sequences, the preservation of randomness principle tells us that if Φ : 2 ω → 2 ω is an effective map and µ is a computable measure on 2 ω , then Φ maps µ-random members of 2 ω to members of 2 ω that are random with respect to the measure ν obtained by pushing µ forward via Φ. Furthermore, the no randomness ex nihilo principle tells us that any sequence that is random with respect to ν is the image of some µ-random sequence under Φ. Used in tandem, these two principles allow us to conclude that the image of the µ-random sequences under Φ are precisely the ν-random sequences.
With the exception of [Cul14], the studies listed above do not make use of these two tools used in tandem. As we will show, they not only allow for the simplification of a number of proofs in the above-listed studies, but they also allow us to answer a number of questions that were left open in these studies.
The outline of the remainder of this paper is as follows. In Section 2, we provide the requisite background for the rest of the paper. In Section 3, we review the basics of algorithmic randomness, including preservation and the no randomness ex nihilo principle. We also provide the definitions of algorithmic randomness for closed sets of 2 ω , random continuous functions on 2 ω , and measures on 2 ω and list some basic properties of these objects. Section 4 contains simplified proofs of some previously obtained results from [BBC + 07] and [BBC + 08], as well as a proof of a conjecture in [BBC + 08] that every random closed subset of 2 ω is the set of zeros of a random continuous function on 2 ω . We study the support of a certain class of random measures in Section 5 and establish a correspondence between between random closed sets and the random measures studied in [Cul14]. Lastly, in Section 6, we prove that the Lebesgue measure of the range of a random continuous function on 2 ω is always non-zero, from which it follows that no random continuous function is injective (which had not been previously established). We also strengthen a result in [BBC + 08] (namely, that not every random continuous function is surjective) by proving that no random continuous function is surjective, from which it follows that the Lebesgue measure of the range of a random continuous function is never equal to one.
n ω is the space of all infinite sequences over the alphabet n. The elements of n ω are also called reals. The product topology on n ω is generated by the clopen sets where σ ∈ n <ω and x ≻ σ means that σ is an initial segment of x. When x is a real and k ∈ ω, x ↾ k denotes the initial segment of x of length k.
A tree is a subset of n <ω that is closed under initial segments; i.e. T ⊆ n <ω is a tree if σ ∈ T whenever τ ∈ T and σ τ . A path through a tree T ⊆ n <ω is a real x ∈ n ω satisfying x ↾ k ∈ T for every k. The set of all paths through a tree T is denoted [T ]. Recall the correspondence between closed sets and trees.
Proposition 2.1. A set C ⊆ n ω is closed if and only if C = [T ] for some tree T ⊆ n <ω . Moreover, C is non-empty if and only if T is infinite.
Given a measure µ on n ω and σ, τ ∈ n <ω , µ(στ | σ) is defined to be 2.2. Some computability theory. We assume the reader is familiar with the basic concepts of computability theory as found, for instance, in the early chapters of [Soa87]. A Σ 0 1 class S ⊆ n ω is an effectively open set, i.e., an effective union of basic clopen subsets of n ω . P ⊆ n ω is a Π 0 1 class if 2 ω \ P is a Σ 0 1 class. A partial function Φ : ⊆ n ω → m ω is computable if the preimage of a Σ 0 1 subset of m ω is a Σ 0 1 subset of the domain of Φ, uniformly; that is, if for every Σ 0 1 class U ⊆ m ω , there is a Σ 0 1 class V ⊆ n ω such that Φ −1 (U) = V ∩ dom(Φ), and an index for V can be uniformly computed from an index for U. Equivalently, Φ : ⊆ n ω → m ω is computable if there is an oracle Turing machine that when given x ∈ n ω (as an oracle) and k ∈ ω outputs Φ(x)(k). We can relativize the notion of a computable function Φ : ⊆ n ω → m ω to any oracle z ∈ 2 ω to obtain a z-computable function.
A measure µ on n ω is computable if µ(σ) is a computable real number, uniformly in σ ∈ n <ω . Clearly, the Lebesgue measure λ is computable.
If µ is a computable measure on n ω and Φ : ⊆ n ω → m ω is a computable function defined on a set of µ-measure one, then the pushforward measure µ Φ defined by for each σ ∈ m <ω is a computable measure.

Algorithmically random objects
In this section, we lay out the definitions of the various algorithmically random objects that are the subject of this study. For more details, see [Nie09] or [DH10].
Definition 3.1. Let µ be a computable measure on n ω and let z ∈ m ω .
(i) A µ-Martin-Löf test relative to z (or simply a µ-test relative to z) is a uniformly Σ 0,z 1 sequence (U i ) i∈ω of subsets of n ω with µ(U n ) ≤ 2 −n .
We will often write "random" instead of "Martin-Löf random," since we are only working with one notion of randomness in this paper (although there are other reasonable notions of algorithmic randomness that one might consider). The collection of µ-random sequences relative to z will be denoted MLR z µ . When z is computable we simply write MLR µ , say that x is µ-random, call (U i ) i∈ω a µ-test, etc.
The following is well-known and straightforward.
Proposition 3.2. Let µ be a computable measure on n ω and z ∈ m ω . If C ⊆ n ω is Π 0,z 1 and µ(C) = 0, then C ∩ MLR z µ = ∅. The following is likely folklore, but was at least observed in [BP12].
1 , uniformly. One of the central tools that we will use in this study is the following.
We will also use a relativization of Theorem 3.5.
Lastly, the following result, known as van Lambalgen's theorem, will be useful to us.
Theorem 3.7 ( [VL90]). Let µ and ν be computable measures on m ω and n ω , respectively. Then for (x, y) ∈ m ω × n ω , (x, y) ∈ MLR µ⊗ν if and only if x ∈ MLR y µ and y ∈ MLR ν . 3.2. Algorithmically random closed subsets of 2 ω . Let C(2 ω ) denote the collection of all non-empty closed subsets of 2 ω . As noted in Proposition 2.1, these are the sets of paths through infinite binary trees. Thus to randomly generate a non-empty closed set, it suffices to randomly generate an infinite tree. Following [BBC + 07], we will code infinite trees by reals in 3 ω , thereby reducing the process of randomly generating infinite trees to the process of randomly generating reals. Given x ∈ 3 ω , define a tree T x ⊆ 2 <ω inductively as follows. First ǫ, the empty string is Under this coding T x has no dead ends and hence is always infinite. Note that every tree without dead ends can be coded by some x ∈ 2 ω .
The main facts about random closed sets that we will use in the sequel are as follows.  3.3. Algorithmically random continuous functions on 2 ω . Let F (2 ω ) denote the collection of all continuous F : ⊆ 2 ω → 2 ω . To define a random continuous function, we code each element of F (2 ω ) by a real x ∈ 3 ω (as carried out in [BBC + 08]). The coding is a labeling of the edges of 2 ω (or equivalently, all nodes in 2 <ω except ǫ) by the digits of x. Having labeled the edges according to x, the function F x coded by x is defined by F x (y) = z if z is the element of 2 ω left over after following y through the labeled tree and removing the 2's. (In the case where only finitely many 0's and 1's remain after removing the 2's, F x (y) is undefined.) Formally, define a labeling function ℓ . . . Note that every F ∈ F (2 ω ) has infinitely many codes.
Remark 3.1. F x is continuous (on its domain) because it is computable relative to some oracle, namely x. Since 2 ω is compact and Hausdorff, it follows that F x is a closed map and hence that ran(F ) is Π 0,F 1 . We will make use of the following facts about random continuous functions.
3.4. Algorithmically random measures on 2 ω . Let P(2 ω ) be the space of probability measures on 2 ω . Given x ∈ 2 ω , the n-th column x n of x is defined by x n (k) = 1 if and only if x( n, k ) = 1, where n, k is some fixed computable bijection between ω 2 and ω. We write x = ⊕ n∈ω x n . We define a map Ψ : 2 ω → P(2 ω ) that sends a real x to the measure µ x satisfying (i) µ x (ǫ) = 1 and (ii) µ x (σ n 0) = x n · µ x (σ n ), where x n is the real number corresponding to the n-th column of x and σ n is the n-th element in the standard enumeration of 2 ω . This coding was first given in [Cul14].
Let P be the pushforward measure on P(2 ω ) induced by λ and Ψ. Then we have the following.
The support of a measure µ on 2 ω is defined to be It is not hard to see that Supp(µ) = 2 ω for every random measure.
In [Cul14], it was shown, among other results, that random measures are atomless (that is, µ({x}) = 0 for every x ∈ 2 ω ) and that the reals that are random with respect to some random measure are precisely the reals in MLR λ .

Applications of Randomness Preservation and No Randomness Ex Nihilo
In this section, we demonstrate the usefulness of preservation of randomness and the no randomness ex nihilo principle in the study of algorithmically random objects such as closed sets, continuous functions, and so on.
As a warm-up, we provide a new, simpler proof of a known result from [BBC + 07].
Theorem 4.1. Every random closed set contains an element of MLR λ and every element of MLR λ is contained in some random closed set.
Proof. We define a computable map Φ : C(2 ω ) × 2 ω → 2 ω that pushes forward the product measure λ C ⊗ λ to λ and satisfies Φ(C, x) ∈ C for every pair (C, x) ∈ C(2 ω ) × 2 ω . Having done this, preservation of randomness and no randomness ex nihilo imply that the image of a (λ C ⊗ λ)-random pair is λ-random and any λ-random is the image of some (λ C ⊗ λ)-random pair. The result then follows because by Van Lambalgen's theorem (Theorem 3.7), a pair (C, x) is (λ C ⊗ λ)-random if and only if C is λ C -random and x is λ-random relative to C.
The map Φ provides a path through C (when viewed as the paths through a tree) by using x to tell us which way to go whenever we encounter a branching node. Specifically, having Φ(C, x) ↾ n = σ such that σ ∩ C = ∅, we define Φ(C, x)(n) = 0 if σ1 ∩ C = ∅ and Φ(C, x)(n) = 1 if σ0 ∩ C = ∅. If neither σ0 ∩ C = ∅ nor σ1 ∩ C = ∅, then we define Φ(C, x)(n) = x(n).

6
The map Φ is clearly computable. It pushes λ C ⊗ λ forward to λ because if Φ has output σ ∈ 2 n after n steps, then Φ outputs a next bit of 0 if and only if either σ1 ∩ C = ∅ or both σ1 ∩ C and σ0 ∩ C are non-empty and x(n) = 0. The former happens with probability 1 3 , and the latter happens with probability 1 6 = 1 3 · 1 2 by independence. The proof is now complete since 1 3 + 1 6 = 1 2 . Let F ∈ F (2 ω ). We define the zeros of F to be Z F = {x : F (x) = 0 ω }, which is clearly a closed subset of 2 ω . In [BBC + 08], the following was shown.
Theorem 4.2 ([BBC + 08]). Let F ∈ F (2 ω ) be random. Then Z F is a random closed set provided it is non-empty.
In [BBC + 08] it was conjectured that the converse also holds, but this was left open. We show that this is the case. To do so, we provide a new proof of Theorem 4.2, from which the converse follows immediately. We also make use of an alternative characterization of random closed sets due to Diamondstone and Kjøs-Hanssen [DKH12].
Just as a binary tree with no dead ends is coded by a sequence in 3 ω (see the paragraph preceding Definition 3.8), an arbitrary binary tree is coded by a sequence in 4 ω , except now a 3 at a node indicates that the tree is dead above that node. That is, given x ∈ 4 ω , we define a tree S x ⊆ 2 <ω inductively as follows. First ǫ, the empty string, is included in S x by default. Now suppose that σ ∈ S x is the (i + 1)-st extendible node in S x . Then . This coding can be thought of as a labeling of the nodes of 2 ω by the digits of x; a 0 at a node means that only the left branch is included, a 1 means that only the right branch is included, a 2 means that both branches are included, and a 3 means that neither branch is included. Note that every tree except 2 <ω itself has infinitely many codes.
Via this coding we can also think of µ GW as a measure on Tree, the space of binary trees. Then the probability of extending a string in a tree by only 0 is 2/9, by only 1 is 2/9, by both 0 and 1 is 4/9, and by neither is 1/9. We call a tree T GW-random if it has a random code; i.e. if there is x ∈ MLR µ GW such that T = S x (here GW stands for Galton-Watson, since GW -trees are obtained by a Galton-Watson process).

Lemma 4.3 (Diamondstone and Kjøs-Hanssen [DKH12]).
A closed set C is random if and only if C is the set of paths through an infinite GW-random tree.

By means of Lemma 4.3 we prove:
Theorem 4.4. (i) For every random F ∈ F (2 ω ), Z F is a random closed set provided that it is non-empty. (ii) For every random C ∈ C(2 ω ), there is some random F ∈ F (2 ω ) such that C = Z F . Proof. We define a computable map Ψ : F (2 ω ) → Tree that pushes forward λ F to µ GW such that the set of paths through Ψ(F ) ∩ dom(F ) is exactly Z F . Given our representation of functions as members of 3 ω and binary trees as members of 4 ω , we are really defining a computable map Ψ : 3 ω → 4 ω that pushes forward λ to µ GW .
Given F ∈ F (2 ω ), which we think of as a {0, 1, 2}-labeling of the edges of the full binary tree, we build the desired tree by declaring that σ ∈ Ψ(F ) if and only if the labels by F of the edges of σ consists only of 0's and 2's. More formally, as in the paragraph preceding Definition 3.11, F comes with a labeling function ℓ F : 2 <ω \ {ǫ} → 3 defined by ℓ F (σ i ) = j if and only if x(i) = j where x is the given code for F . So σ ∈ Ψ(F ) if and only if ℓ F (σ↾k) ∈ {0, 2} <ω for every 0 < k ≤ |σ|. Clearly this map is computable. Now we show that the map Ψ pushes λ F forward to µ GW . Suppose σ ∈ Ψ(F ), which, as stated above, means that ℓ F (σ↾k) ∈ {0, 2} <ω for every 0 < k ≤ |σ|. Then The right-hand side of the equivalence occurs with probability (2/3)(1/3) = 2/9. Similarly, where this latter event also occurs with probability 2/9. Next, with the latter event occurring with probability (2/3)(2/3) = 4/9. Lastly, where the event on the right-hand side occurs with probability (1/3)(1/3) = 1/9. Now by construction, it follows immediately that any path through the tree Ψ(F ) is a sequence X such that either F (X) = 0 ω (in the case that ℓ F (X↾n) = 0 for infinitely many n) or F (X)↑ (in the case that ℓ F (X↾n) = 0 for only finitely many n). By preservation of randomness and no randomness ex nihilo, a tree is GW-random if and only if it is the image of some random continuous function F . The conclusion then follows by Lemma 4.3.
One consequence of Theorem 3.12 and Theorem 4.4(ii), not noted in [BBC + 08], is that the composition of two random continuous functions need not be random.
Corollary 4.5. For every random F ∈ F (2 ω ), there is some random G ∈ F (2 ω ) such that G • F is not random.
Another consequence of Theorem 4.4 is that we can answer another open question from [BBC + 08] involving random pseudo-distance functions. Given a closed set C ∈ C(2 ω ), a function δ : 2 ω → 2 ω is a pseudo-distance function for C if C is the set of zeros of δ. In [BBC + 08] it was shown that if δ is a random pseudo-distance function for some C ∈ C(2 ω ), then C is a random closed set, but the converse was left open. By Theorem 4.4, the converse immediately follows.
Corollary 4.6. Let C ∈ C(2 ω ). C has a random pseudo-distance function if and only if C is a random closed set.

The support of a random measure
In the previous section, we established a correspondence between random closed sets and and random continuous functions: a closed set C is random if and only if it is the set of zeros of some random continuous function. In this section, we establish similar correspondences between random closed sets and random measures.
Since the support of a measure µ, i.e., the set Supp(µ) = {x ∈ 2 ω : ∀n µ(x↾n) > 0} is a closed set, one might hope to establish such a correspondence by considering the supports of random measures. However, it is not hard to see that for each random measure µ, Supp(µ) = 2 ω .
If we consider a different computable measure on P(2 ω ) than the measure P defined above in Section 3.4, then such a correspondence can be given. In the first place, we want a measure Q on P(2 ω ) with the property that no Q-random measure has full support. In fact, we can choose a measure Q such that each Q-random measure is supported on a random closed set.
Theorem 5.1. There is a computable measure Q on P(2 ω ) such that (i) every Q-random measure is supported on a random closed set, and (ii) for every random closed set C ⊆ 2 ω , there is a Q-random measure µ such that Supp(µ) = C.
Proof. We will define the measure Q so that each Q-random measure is obtained by restricting the Lebesgue measure to a random closed set. That is, each Q-random measure will be uniform on all of the branching nodes of its support. We define Q in terms of an almost total functional Φ : 3 ω → 2 ω . On input x ∈ 3 ω , Φ will treat x as the code for a closed set and will output the sequence y = ⊕ i∈ω y i defined as follows. For each i ∈ ω, we set If we think of the columns of y as encoding the conditional probabilities of a measure µ y , then if (σ i ) i∈ω is the standard enumeration of 2 <ω , these conditional probabilities are given by That is, Φ(x) = y, where y represents the unique measure µ y such that µ y (σ0 | σ) = p σ for each σ ∈ 2 <ω . Let Q be the measure on P(2 ω ) induced by the composition of Φ and the representation map Ψ : 2 ω → P(2 ω ) defined in Section 3.4.
We now verify (i) by showing that Φ maps each x ∈ MLR to a Q-random measure supported on a random closed set. Let x ∈ MLR and set Φ(x) = y. By preservation of randomness, Ψ(Φ(x)) = µ y is Q-random.
Next, since x ∈ MLR, [T x ] is a random closed set. We claim that Supp(µ y ) = [T x ]. Suppose that σ ∈ 2 <ω is the (n + 1)-st extendible node of T x . Then one of the following holds: (a) σ0 ∈ T x and σ1 / ∈ T x ; (b) σ0 / ∈ T x and σ1 ∈ T x ; or (c) σ0 ∈ T x and σ1 ∈ T x .
To show (ii), let C ⊆ 2 ω be a random closed set. By definition there is some Martin-Löf random z ∈ 3 ω such that C = [T z ]. Hence Ψ(Φ(z)) is a Q-random measure ν. By the definition of Φ, ν has support [T z ] = C, which establishes the claim.
Instead of changing the measure on P(2 ω ) we can also establish a correspondence between random closed sets and random measures by considering not the support of a random measure but what we refer to as its 1/3-support.
Definition 5.2. Let µ ∈ P(2 ω ) and set Then the 1/3-support of the measure µ is the closed set [T µ ].
Theorem 5.3. A closed set C ∈ C(2 ω ) is random if and only it is the 1/3-support of some random measure µ ∈ P(2 ω ).

The range of a random continuous function
In [BBC + 08], it was shown that for each y ∈ 2 ω λ({x ∈ 2 ω : y ∈ ran(F x )}) = 3/4. from which it follows that every y ∈ 2 ω is in the range of some random F ∈ F (2 ω ). In this section, we prove that λ(ran(F )) ∈ (0, 1) for every random function F . First we will prove that λ(ran(F )) > 0 for each random function, from which it follows that no random function is injective and that the range of a random function is never a random closed set. These improve two results of [BBC + 08] according to which (i) not every random function is injective and (ii) the range of a random function is not necessarily a random closed set. Our proof requires us to prove some auxiliary facts about the measure induced by a random function.
To prove that λ(ran(F )) < 1 for every F ∈ F (2 ω ), we will show that no random function is surjective, from which the result immediately follows. Our result on surjectivity also improves a result of [BBC + 08] according to which not every random function is surjective.
We begin by proving the following, which is similar to a result in [Cul14] for random measures.
Proof. By change of variables, it suffices to show that for each σ ∈ 2 <ω . Without loss of generality, we assume σ = 0 n . We proceed then by induction on n. Equation (1) holds when σ = ǫ since each random F is total by Theorem 3.13. Now supposing that equation (1) holds for 0 n , we show it also holds for 0 n+1 . Suppose then that F (2 ω ) λ(F −1 0 n ) dλ F = 2 −n . To compute F (2 ω ) λ(F −1 0 n+1 ) dλ F , we note that by symmetry Recall that any F ∈ F (2 ω ) can be viewed as a labeling by 0's, 1's, and 2's of the nodes of full binary branching tree (where the root node is unlabeled). We compute F (2 ω ) λ( 0 ∩ F −1 0 n+1 ) dλ F by considering the three equiprobable cases for the label of the node 0 for an arbitrary F ∈ F (2 ω ). The point is that the label 0 contributes to producing an output beginning with 0 n+1 , the label 1 rules out the possibility of producing an output beginning with 0 n+1 , and the label 2 neither contributes to nor rules out the possibility of producing an output beginning with 0 n+1 .
Case 1: If the node 0 is labeled with a 0, then the measure of all sequences extending the node 0 that (after removing 2's) yield an output extending 0 n+1 is equal to the measure of all sequences that yield an output extending 0 n times 1/2 (the measure determined by the initial label 0), i.e., 1/2 · 2 −n .
Case 2: If the node 0 is labeled with a 1, then the measure of all sequences extending the node 0 that (after removing 2's) yield an output extending 0 n+1 is equal to 0.
Case 3: If the node 0 is labeled with a 2, then the measure of all sequences extending the node 0 that (after removing 2's) yield an output extending 0 n+1 is equal to the measure of all sequences that yield an output extending 0 n+1 times 1/2 (the measure determined by the initial label 2), i.e., 1/2 · s n+1 .
Putting this all together gives which yields s n+1 = 2 −n /4, as desired.
Proof. For any y ∈ 2 ω , a relativization of Theorem 4.4(i) shows that F −1 ({y}), if non-empty, is a random closed set relative to y provided that F is random relative to y. Since ran(F ) has positive Lebesgue measure, there is y ∈ ran(F ) that is random relative to F . But then by Van Lambalgen's theorem, F is also relative to y. So F −1 ({y}) is a non-empty random closed set and hence has size continuum by Theorem 3.10. Thus F is not injective.
Corollary 6.5. If F ∈ F (2 ω ) is random, then ran(F ) is not a random closed set.
Proof. By Theorem 3.9, every random closed set has Lebesgue measure 0. But by Theorem 6.3, the range of a random F ∈ F (2 ω ) has positive Lebesgue measure, and thus the conclusion follows.
From the proof of Corollary 6.4 we can also obtain the following.
The proof of following result is essentially the proof of the effective Choquet capacity theorem in [BCTW11]. We reproduce the proof here for the sake of completeness.
Lemma 6.11. The probability that a random continuous function F hits both 0 and 1 is 2q 1 − 1, and the probability that F hits both σ0 and σ1 for a fixed σ ∈ 2 <ω of length n ≥ 1, given that F hits σ , is equal to 2 q n+1 q n − 1.
Proof. First, let us write the probability that F hits σ for some fixed σ as P(F ∈ H σ ). Now since P(F ∈ H 0 ) = q 1 , it follows that P(F ∈ H 1 \ H 0 ) = 1 − q 1 (here we use the fact that every random function is total). By symmetry, we have P(F ∈ H 0 \ H 1 ) = 1 − q 1 . Since F is total with probability one, it follows that P(F ∈ H 0 ∩ H 1 ) = 1 − P(F ∈ H 0 \ H 1 ) + P(F ∈ H 1 \ H 0 ) and thus P(F ∈ H 0 ∩ H 1 ) = 1 − ((1 − q 1 ) + (1 − q 1 )) = 2q 1 − 1. Next, let σ be a string of length n ≥ 1 and let i ∈ {0, 1}. Since P(F ∈ H σ ) = q n and P(F ∈ H σ ⌢ i ) = q n+1 it follows that P(F ∈ H σ ⌢ i | F ∈ H σ ) = P(F ∈ H σ ⌢ i & F ∈ H σ ) P(F ∈ H σ ) = P(F ∈ H σ ⌢ i ) P(F ∈ H σ ) = q n+1 q n . 16 Consequently, P(F ∈ H σ1 \ H σ0 | F ∈ H σ ) = P(F ∈ H σ0 \ H σ1 | F ∈ H σ ) = 1 − q n+1 q n Thus, To complete the proof of Theorem 6.7, we now define a Martin-Löf test on F (2 ω ) that covers all surjective functions. Let us say that a function F ∈ F (2 ω ) is onto up to level n if F ∈ H σ for every σ ∈ 2 n . By Lemma 6.11, the probability of a function being onto up to level n is Thus, if we set U n = {F ∈ F (2 ω ) : F is onto up to level n}, and f (n) = min{k : ( 32 √ e) −k ≤ 2 −n }, which is clearly computable, then (U f (n) ) n∈ω is a Martin-Löf test with the property that F ∈ F (2 ω ) is onto if and only if F ∈ n∈ω U f (n) . This completes the proof.
Proof. Suppose λ(ran(F )) = 1. Then since ran(F ) is closed, it follows that ran(F ) = 2 ω . But then F is onto, so it cannot be random.
We also have the following corollary.
Theorem 6.13. No measure induced by a random function is a random measure in the sense of Definition 3.14.