ALGORITHMIC RANDOMNESS AND CAPACITY OF CLOSED SETS

. We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m , a computable capacity T may be deﬁned by letting T ( Q ) be the measure of the family of closed sets K which have nonempty intersection with Q . We prove an eﬀective version of Choquet’s capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions on the measure m that characterize when the capacity of an m -random closed set equals zero. This includes new results in classical probability theory as well as results for algorithmic randomness. For certain computable measures, we construct eﬀectively closed sets with positive capacity and with Lebesgue measure zero. We show that for computable measures, a real q is upper semi-computable if and only if there is an eﬀectively closed set with capacity q .


Introduction
The study of algorithmic randomness has been an active area of research in recent years.The basic problem is to quantify the randomness of a single real number.Here we think of a real r ∈ [0, 1] as an infinite sequence of 0's and 1's, i.e. as an element in 2 N .There are three basic approaches to algorithmic randomness: the measure-theoretic approach of Martin-Löf tests, the incompressibility approach of Kolmogorov complexity, and the betting approach in terms of martingales.All three approaches have been shown to yield the same notion of (algorithmic) randomness.The present paper will consider only the measure-theoretic approach.A real x is Martin-Löf random if for any effective sequence S 1 , S 2 , . . . of c. e. open sets with µ(S n ) ≤ 2 −n , x / ∈ n S n .For background and history of algorithmic randomness we refer to [8,15].
The study of random sets and in particular of random closed sets is a vibrant area in probability and statistics, with many applications in science and engineering.The notion of capacity plays an important role here as a part of the analysis of imprecise or uncertain observations, for example in intelligent systems.For background on the theory of random sets see [14].
In a series of recent papers [4,2], G. Barmpalias, P. Brodhead, D. Cenzer, S. Dashti, J.B. Remmel and R. Weber have defined a notion of algorithmic randomness for closed sets and continuous functions on 2 N .Here the Polish space 2 N is equipped with usual product topology and has a basis of clopen sets.Definitions are given below in section 1.The space C of closed subsets of 2 N has the hit-or-miss or Fell topology which is also described in section 1.In general when we discuss closed sets in this paper we are refering to closed subsets of 2 N .
The study of randomness for closed sets and continuous functions has several interesting aspects concerning properties of those sets and properties of the members of such sets.The topological and measure-theoretic properties of effectively random closed sets has been studied.For example, it is shown in [4] that every effectively random closed set is perfect and has Lebesgue measure 0. The complexity of effectively random closed sets as subsets of 2 N was considered in [4], where it was shown that no effectively closed (Π 0 1 ) set is random but there is a random ∆ 0 2 closed set.The members of a closed set are reals and hence we can study the complexity of the members of an effectively random closed set.The following results were obtained in [4].Every effectively random closed set contains a random member but not every member is random.Every random real belongs to some random closed set.Every effectively random ∆ 0 2 closed set contains a random ∆ 0 2 member.Effectively random closed set contain no computable elements (in fact, no n-c.e. elements).It was shown in [2] that the set of zeroes of an effectively random continuous function is an effectively random closed set.
Just as an effectively closed set in 2 N may be viewed as the set of infinite paths through a computable tree T ⊆ {0, 1} * , an algorithmically random closed set in 2 N may be viewed as the set of infinite paths through an algorithmically random tree T .Diamondstone and Kjos-Hanssen [11,10] give an alternative definition of algorithmic randomness for closed sets according to the Galton-Watson distribution and show that this definition produces the same family of algorithmically random closed sets.
We note that in probability theory a random closed subset of a topological space X is considered a random variable which takes on the values in the space C(X) of closed subsets of X.That is, let (Ω, A, P ) be a probability space with underlying topological space Ω, σ-algebra A ⊆ P(X) and measure P such that P (Ω) = 1 and P (S) is defined for all sets S ∈ A. For example, we might have Ω = 2 N , A the family of Borel subsets of 2 N , and P the standard Lebesgue measure.The map X induces a probability measure P X on C(X) given by P (X −1 (S)).Classically, the statement that a random closed set has no computable elements means that the collection of closed sets with no computable elements has measure one.In effective randomness, there is a particular collection R of algorithmically random closed sets which has measure one.In this context, the statement that effectively random closed sets have no computable elements is to say that the closed sets in R have no computable elements.The latter result of course implies the former, but is stronger.
A random closed set is a specific type of random recursive construction, as studied by Graf, Mauldin and Williams [9].McLinden and Mauldin [13] showed that the Hausdorff dimension of a random closed set is log 2 (4/3), that is, almost every closed subset of 2 N has Hausdorff dimension log 2 (4/3).It was shown in [4] that every effectively random closed set has box dimension log 2 (4/3).The effective Hausdorff dimension of members of effectively random closed sets is studied in [11].It is shown that every member of an effectively random closed set has effective Hausdorff dimension ≥ log 2 (3/2) and that any real with effective Hausdorff dimension > log 2 (3/2) is a member of some effectively random closed set.
In the present paper we will examine the notion of computable capacity and its relation to computable measures on the space C of nonempty closed sets.Given a domain U , a capacity T is a real-valued function defined on some σ-field of subsets of U , which is closely related to measure.T may be thought of as a belief function in the context of reasoning with uncertainty.(See [14, p. 71] and also [18]. ) The capacity T (A) for a set A is the probability that a randomly chosen set S is a subset of A.
Choquet [6] developed the Choquet capacity for the space C of closed subsets of an infinite set X. A probability measure µ * on C induces a capacity T on C by defining the capacity T (C) of a closed set C to be µ * ({K ∈ C : K ⊆ C}).Choquet's capacity theorem states that every capacity T on C arises in this way from some measure µ * .
In section one, we give some basic definitions including the definition of the space of C(X) of closed subsets of a computable Polish space X.We present a family of computable measures on C which will lead to different notions of effective randomness for closed sets.
In section two, we define the notion of computable capacity and show how a measure on the space of closed sets induces a capacity.An effective version of Choquet's capacity theorem is proved.
The main theorem of section three gives conditions under which the capacity T (Q) of a µ * -random closed set Q is either equal to 0 or > 0. In particular, suppose that the measure µ b on {0, 1, 2} N is defined so that, for all σ ∈ {0, 1, 2} * , µ b (I(σ ⌢ i)) = b • µ b (I(σ)) for i = 0, 1 and define the corresponding probability measure µ * b and capacity T b on the space C of closed sets and the corresponding capacity T b .This means that for any node σ in the tree T Q , σ has unique extension σ ⌢ 0 in T Q with probability b, and similarly σ has unique extension σ ⌢ 1 with probability b.Then we show the following.
It is important to note that, since the random closed sets have measure one in the space C of closed sets, this result implies that almost all closed sets have capacity zero.This is a new result about the classical measure and capacity of closed sets in general and not only about algorithmic randomness or computability.
On the other hand 2 , then every effectively µ * b -random closed set Q has capacity T b (Q) > 0, and hence almost every closed set has positive capacity.A more general result is given.
In section four, we consider the capacity of effectively closed sets.Fix computable reals b 0 and b 1 such that 0 < b 1 ≤ b 0 and b 0 + b 1 < 1 and define the measure µ on {0, 1, 2} N so that for any σ ∈ {0, 1, 2} * and for i ∈ {0, 1}, µ(I(σ ⌢ i)) = b i • µ(I(σ)).Let µ * be the corresponding measure on C and let T be the corresponding capacity.It is easy to see that for any effectively closed set Q, T (Q) is an upper-semi-computable real.Conversely, for any upper-semi-computable real q, there exists an effectively closed set Q with capacity T (Q) = q.We also show that if b 0 = b 1 , there exists an effectively closed set Q with Lebesgue measure zero and with positive capacity.
A preliminary version [3] of this paper appeared in the electronic proceedings of the conference CCA 2010.The current paper contains several improvements and new results, including Theorems 3.1, 3.4, 3.8 and 4.3.We thank the referees for very helpful comments.

Computable Measures on the Space of Closed Sets
We present an effective version of Choquet's theorem connecting measure and capacity.
In this section, we describe the hit-or-miss topology on the space C of closed sets, we define certain probability measures µ d on the space {0, 1, 2} N and the corresponding measures µ * d on the homeomorphic space C.These give rise to notions of algorithmic randomness for closed sets.
Some definitions are needed.For a finite string σ ∈ {0, 1} n , let |σ| = n.Let λ denote the empty string so that |λ| = 0.For two strings σ, τ , say that σ is an initial segment of τ and write , . . .).Let x⌈n = (x(0), . . ., x(n − 1)).Two reals x and y may be coded together into z = x ⊕ y, where z(2n) = x(n) and z(2n + 1) = y(n) for all n.For a finite string σ, let I(σ) denote {x ∈ 2 N : σ ❁ x}.We shall call I(σ) the interval determined by σ.Each such interval is a clopen set and the clopen sets are just finite unions of intervals.We let B denote the computable Boolean algebra of clopen sets.Note that this is a countable atomless Boolean algebra.
A set T ⊆ {0, 1} * is a tree if it is closed under initial segments.For an arbitrary tree T ⊆ {0, 1} * , let [T ] denote the set of infinite paths through T .It is well-known that P ⊆ 2 N is a closed set if and only if P = [T ] for some tree T .P is a Π 0 1 class, or an effectively closed set, if P = [T ] for some computable tree T .
A closed set P may be identified with a tree T P ⊆ {0, 1} * where T P = {σ : P ∩I(σ) = ∅}.Note that T P has no dead ends.That is, if σ ∈ T P , then either σ ⌢ 0 ∈ T P or σ ⌢ 1 ∈ T P .The complexity of the closed set P is generally identified with that of T P .Thus P is said to be a Π 0 2 closed set if T P is Π 0 2 ; in this case P = [T ] for some ∆ 0 2 tree T .The complement of an effectively closed set is sometimes called a c. e. open set.We remark that if P is an effectively closed set, then T P is a Π 0 1 set, but it is not, in general, computable.For any σ ∈ {0, 1} * and any There is a natural effective enumeration P 0 , P 1 , . . . of the effectively closed sets and thus an enumeration of the c. e. open sets.Thus we can say that a sequence S 0 , S 1 , . . . of c. e. open sets is effective if there is a computable function, f , such that S n = 2 N − P f (n) for all n.For a detailed development of effectively closed sets, see [5].
It was observed in [4] that there is a natural isomorphism between the space C of nonempty closed subsets of {0, 1} N and the space {0, 1, 2} N (with the product topology) defined as follows.Given a nonempty closed Q ⊆ 2 N , let T = T Q be the tree without dead ends such that Q = [T ].Let σ 0 , σ 1 , . . .enumerate the elements of T in order, first by length and then lexicographically.We then define the code For a finite tree T ⊆ {0, 1} ≤n , the finite code ρ T is similarly defined, ending with ρ T (k) where σ k is the lexicographically last element of T ∩ {0, 1} ≤n .We defined in [4] a measure µ * on the space C of closed subsets of 2 N as follows.
for any X ⊆ C and µ is the standard measure on {0, 1, 2} N .Informally this means that given σ ∈ T Q , there is probability 1 3 that both σ ⌢ 0 ∈ T Q and σ ⌢ 1 ∈ T Q and, for i = 0, 1, there is probability 1  3 that only Note that the equal probability of 1  3 for the three cases of branching allows the application of Schnorr's theorem that Martin-Löf randomness is equivalent to prefix-free Kolmogorov randomness.
The standard (hit-or-miss) topology [7, p. 45] on the space C of closed sets is given by a sub-basis of sets of two types, where U is any open set in 2 N .
so that ∅ is an isolated element of C under this topology.Thus we may omit ∅ from C without complications.
A basis for the hit-or-miss topology may be formed by taking finite intersections of the basic open sets.We want to work with the following simpler basis.For each n and each finite tree A ⊆ {0, 1} ≤n , let Note that the sets U A are in fact clopen.That is, for any tree A ⊆ {0, 1} ≤n , define the tree For any finite n and any tree T ⊆ {0, 1} ≤n , define the clopen set [T ] = σ∈T I(σ).Then K ∩ [T ] = ∅ if and only if there exists some A ⊆ {0, 1} ≤n such that K ∈ U A and

Similarly, K ⊆ [T ] if and only if there exists some
The following lemma can now be easily verified.
Lemma 1.1.The family of sets {U A : A ⊆ {0, 1} ≤n A is a tree} is a basis of clopen sets for the hit-or-miss topology on C.
Recall the mapping from C to {0, 1, 2} N taking Q to x Q .It can be shown that this is in fact a homeomorphism.(See Axon [1] for details.) Let B * be the family of clopen subsets of C; each set is a finite union of basic sets of the form U A and thus B * is a computable atomless Boolean algebra.Note that elements U of B * are collections of closed sets and are closed and open in the hit-or-miss topology on the space C of closed subsets of {0, 1} N .Recall that B denotes the family of clopen subsets of {0, 1} N .Proposition 1.2.The space C of nonempty closed subsets of 2 N is computably homeomorphic to the space {0, 1, 2} N .Furthermore, the corresponding map from B to B * is a computable isomorphism of these computable Boolean algebras.
Next we consider probability measures µ on the space {0, 1, 2} N and the corresponding measures µ * on C induced by µ.
A probability measure on {0, 1, 2} N may be defined as in [16] from a function d : The corresponding measure µ d on {0, 1, 2} N is then defined by letting µ d (I(σ)) = d(σ).Since the intervals I(σ) form a basis for the standard product topology on {0, 1, 2} N , this will extend to a measure on all Borel sets.If d is computable, then µ d is said to be computable.The measure µ d is said to be nonatomic or continuous if µ d ({x}) = 0 for all x ∈ {0, 1, 2} N .We will say that µ d is bounded if there exist bounds b, c ∈ (0, 1) such that, for any σ ∈ {0, 1, 2} * and i It is easy to see that any bounded measure must be continuous.We will say that the measure Let us say that a measure µ * on C is computable if the restriction of µ * to the family B * of clopen sets is computable.That is, if there is a computable function F mapping B * to [0, 1] such that F (B) = µ * (B) for all B ∈ B * .Proposition 1.3.For any computable d, the measure µ * d is a computable measure on C. Proof.For any tree A ⊆ {0, 1} ≤n , it is easy to see that

Computable Capacity and Choquet's Theorem
In this section, we define the notion of capacity and of computable capacity.We present an effective version of Choquet's theorem connecting measure and capacity.For details on capacity and random set variables, see Nguyen [14] and also Matheron [12].
(2) T has the alternating of infinite order property, that is, for n ≥ 2 and any Q ( We will also assume, unless otherwise specified, that the capacity T (2 N ) = 1.
We will say that a capacity T is computable if it is computable on the family of clopen sets, that is, if there is a computable function F from the Boolean algebra B of clopen sets into [0, 1] such that F (B) = T (B) for any B ∈ B. Define is the probability that a randomly chosen closed set meets Q.Here is the first result connecting effective measure and effective capacity.This follows easily from the classical proof of Choquet.
Proof.Certainly T d (∅) = 0.The alternating property follows by basic probability.For (iii), suppose that If d is computable, then T d may be computed as follows.For any clopen set where each σ i ∈ {0, 1} n , we compute the probability distribution for all trees of height n and add the probabilities of those trees which contain one of the σ i .
Choquet's Capacity Theorem states that any capacity T is determined by a measure, that is T = T d for some d.See [14] for details.We now give an effective version of Choquet's theorem.It is not so easy, but this does follow from the classical proof of Choquet [6].See also [12] and Axon [1].

Theorem 2.3 (Effective Choquet Capacity Theorem).
If T is a computable capacity, then there is a computable measure µ * d on the space of closed sets such that T = T d .Proof.Given the values T (U ) for all clopen sets I(σ 1 ) ∪ • • • ∪ I(σ k ) where each σ i ∈ {0, 1} n , there is in fact a unique probability measure µ d on these clopen sets such that T = T d and this can be computed as follows.
Suppose first that T (I(i)) = a i for i < 2 and note that each a i ≤ 1 and a 0 + Now suppose that we have defined d(τ ) and that τ is the code for a finite tree with elements σ 0 , . . ., σ n = σ and thus d(τ ⌢ i) is giving the probability that σ will have one or both immediate successors.We proceed as above.Let T (I(σ ⌢ i)) = a i • T (I(σ)) for i < 2. Then as above d(τ ⌢ 2) = d(τ ) • (a 0 + a 1 − 1) and d(τ ⌢ i) = d(τ ) • (1 − a i ) for each i.

Zero Capacity
In this section, we compute the capacity of a random closed set under certain computable probability measures.In particular, suppose that µ d is a symmetric measure, that is, let d(σ ⌢ 0) = d(σ ⌢ 1) for all σ.We show the following.If d(σ ⌢ 2) ≤ We say that K ∈ C is µ * d -random if x K is Martin-Löf random with respect to the measure µ d .(See [16] for details.)Our results show that the T d capacity of a µ * d -random closed set depends on the particular measure.
In the following proofs, the key idea is that an arbitrary closed set Q can be given as the intersection of a sequence Q n n∈ω of clopen sets, so that the capacity Thus we want to compute the capacity q n of Q n when Q is a random closed set, or at least to compute bounds on this capacity.Now the capacity of Q is the probability that Q ∩ K = ∅ for a random closed set K, that is to say Thus we first compute the probability that Q n ∩ K n = ∅ for randomly chosen closed sets Q and K and use this to determine T (Q) for a random closed set.In the first two theorems, these computations can be converted into Martin-Löf tests, so that the capacity of an effectively µ * d -random closed set can be determined.Proof.We first present the proof for a uniform measure µ d and then give the modifications necessary for non-uniform measure.
Fix b with 1 2 and suppose that, for all σ, d(σ We will compute the probability, given two closed sets Q and K, that Q ∩ K is nonempty.Here we define the usual product measure on the product space C × C of pairs (Q, K) of nonempty closed sets by letting Let p n be the probability that Q n ∩ K n = ∅ for two arbitrary closed sets K and Q, relative to our measure µ * .It is immediate that Next we will determine the quadratic function f such that p n+1 = f (p n ).There are 9 possible cases for Q 1 and K 1 , which break down into 4 distinct cases in the computation of p n+1 .so that for Q ∈ A m,n , we have K∈C F (Q, K)dK ≥ 2 −n .It follows that Multiplying both sides by 2 n completes the proof of Claim 3.2.
Since the computable sequence p n n∈ω converges to 0, there must be a computable subsequence m 0 , m 1 , . . .such that p mn < 2 −2n−1 for all n.We can now define our Martin-Löf test.Let S r = A mr,r and let Now suppose that R is a random closed set.The sequence V n n∈ω is a computable sequence of c. e. open sets with measure ≤ 2 −n , so that there is some n such that R / ∈ S n .Thus for all r > n, µ * ({K : K mr ∩ R mr = ∅}) < 2 −r and it follows that Thus T d (R) = 0, as desired.
This completes the proof when the function d is independent of σ.
Next suppose that the value b such that d( 2 )p 2 as above and let f b(p) = f (p).Let p n be the probability computed above corresponding to b σ = b for all σ, so that p n+1 = f (p n ).Define p d n to be the probability, under µ * d , that K n ∩ Q n = ∅, for closed sets K and Q.We will argue by induction on n that p d n ≤ p n .
Claim 3.3.For any reals b, c, p Let p i be the probability under d i that Q n ∩ K n = ∅.Then the probability under d i+1 that Q n+1 ∩ K n+1 = ∅ can be computed in the four cases as above to equal By induction, both of p 0 and p 1 are ≤ p n and it follows easily that Finally, suppose that we only have that b σ ≥ b for σ with |σ| ≥ n.Let R be µ * d -random and for each σ of length n, let d σ be defined so that The appropriate Martin-Löf test can now be given as before to show that any µ * d -random closed set will have capacity zero.
Next we consider the case where random closed sets will have positive capacity.
2 is fixed and that the measure µ d is defined by d such that, for all sufficiently long σ, d(σ As in the proof of Claim 3.2, we have which completes the proof of Claim 3.5.
Claim 3.6.{Q : and suppose that µ * (A) = 0.As in the proof of Claim 3.2, we have Since µ * (A) = 0, it follows that for any B ⊆ C, we have Furthermore, T d (Q) < m b for almost all Q, so there exists some P with T d (P ) < m b − ǫ for some positive ǫ.This means that for some n, µ * ({K : Then we have This contradiction demonstrates Claim 3.6. It is now easy to see that T d (R) > 0 with probability one.That is, let p be the probability that T d (R) = 0. Then by considering the first level of R, we can see that p = 2bp + (1 − 2b)p 2 and hence either p = 0 or p = 1.Since we know that p < 1, it follows that p = 0.
Since the set of µ * -random closed sets has measure one, there must be a random closed set R such that T d (R) ≥ m b and furthermore, almost every µ * -random closed set has positive capacity.
Furthermore, we can construct a Martin-Löf test as follows.First observe that for any computable q, {Q : T d (Q) < q} is a c. e. open set.This is because T d (Q) < q ⇐⇒ (∃n)T d (Q n ) < q and T d (Q n ) can be uniformly computed from Q. Now let h(p) be the probability that T d (Q) < p.Note that if T d (Q i ) ≥ p for i = 0 or for i = 1, then T d (Q) ≥ bp.It follows that h(bp) ≤ h(p) 2 .Since T d (Q) = 0 with probability zero, it follows that lim p→0 h(p) = 0. Take a rational q small enough so that h(q) < 1 2 .Then h(b n q) ≤ ( 12 ) Due to he inequalities needed for |b 0 − b 1 | in the theorem, it seems that the proof given above does not easily extend to provide a result for non-uniform measures or to prove that, in the second case above, every random closed set has positive capacity.

Effectively Closed Sets
In this section, we consider the capacity of effectively closed sets.A random closed set can never be effectively closed.But we can still construct an effectively closed set with measure zero and with positive capacity.
We begin by characterizing the possible capacity of effectively closed sets.For the following results we will take T = T d where µ d is the computable measure defined by Q is the effective intersection of the decreasing sequence [T n ] of clopen sets, where T n = T ∩ {0, 1} ≤n .Thus for a computable measure T d , the capacity T d (Q) is the limit of a computable, decreasing sequence and is therefore an upper semi-computable real.We will show that for every upper semi-computable real q ∈ [0, 1], there exists an effectively closed set Proof.For a closed set K, K ∩ Q = ∅ if and only if one of the following holds: , where the last inequality holds by the induction hypothesis.Theorem 4.3.Let the real number q ∈ [0, 1] be upper semi-computable, i.e. there is a computable, decreasing sequence {q n : n ∈ N} such that lim q n = q.Then there exists an effectively closed set P such that T (P ) = q.Moreover, P can be written as n P n where {P n : n ∈ N is a computable sequence of clopen sets with q n+1 ≤ T (P n ) ≤ q n .Proof.We may assume without loss of generality that q 0 = 1.We will construct P n by recursion beginning with P 0 = 2 N .Now suppose we have constructed the clopen set Q n−1 = m k=0 I(σ k ) such that q n ≤ T d (Q n−1 ) ≤ q n−1 .
Let δ = q n − q n−1 and compute s large enough so that ( In this way, we have constructed a computable, decreasing sequence Q n of clopen sets with q n ≤ T d (Q n ) ≤ q n−1 , so that, for Proof.First let us compute the capacity of X n = {x : x(n) = 0}.For n = 0, we have T d (X 0 ) = 1− b.That is, Q meets X 0 if and only if Q 0 = I(0) (which occurs with probability b), or Q 0 = 2 N (which occurs with probability 1 − 2b).Now the probability T d (X n+1 ) that an arbitrary closed set K meets X n+1 may be calculated in two distinct cases.As in the proof of Theorem 3.7, let Case II: If K 0 = I((i)) for some i < 2, then T d (X n+1 ) = T d (X n ).

Conclusions
In this paper, we have established a connection between measure and capacity for the space C of closed subsets of 2 N .We showed that for a computable measure µ * , a computable capacity may be defined by letting T (Q) be the measure of the family of closed sets K which have nonempty intersection with Q.We have proved an effective version of the

√ 2 2 2 2
d(σ) for all σ, then T d (R) = 0 for any µ * d -random closed set R. Thus for the uniform measure with d(σ ⌢ 0) = d(σ ⌢ 1) = 1 3 • d(σ) for all σ, effectively random closed sets have capacity zero.Thus for almost all closed sets R, T d (R) = 0.If d(σ ⌢ 2) ≥ b • d(σ) for all σ, where b > √ is a constant, then T d (R) > 0 for any µ * d -random closed set R. Thus for almost all closed sets R, T d (R) > 0. This result, and others in this section are new results about classical measure and capacity as well as results about algorithmic randomness.For non-symmetric measures, where d(σ ⌢ i) = b i •d(σ) for i < 2, the question of whether a random closed set has zero capacity depends on the sum b 0 +b 1 and also on their difference.If b 0 + b 1 ≥ 2 − √ 2 and |b 0 − b 1 | is sufficiently small, then every µ * d -random closed set will have capacity zero (so that for almost all closed sets R, T d (R) = 0) and otherwise there is a µ * d -random closed set with positive capacity.

Proof.
Fixing p and taking the derivative of f b (p) with respect to b, we obtain ∂f ∂b (b, p) = (4b − 4)p − (8b − 4)p 2 ≤ −4bp ≤ 0, with the inequality due to the fact that p 2 ≤ p on [0, 1].Now suppose that for all σ ∈ {0, 1} * and for i < 2, d(σ ⌢ i) ≥ bd(σ) and again let p d n be the µ d -probability that K n ∩ Q n = ∅.Clearly p d 0 = 1 = p 0 .Now assume that p d n ≤ p n for any d as above.Let d be given as above with d((0)) = d((1)) = b ≥ b and define d i for i = 0, 1 as follows.
one and furthermore every µ * d -random closed set has positive capacity.Thus for almost all closed sets R, T d (R) > 0. Proof.First fix b < b and fix d so that d(σ ⌢ i) = d(σ) • b for all σ and for i < 2, and let µ * = µ * d .Since 0 < 2b 2 − 4b + 1 < 1, the function f = f b defined above has a positive fixed point m b = 2b 2 −4b+1 (1−2b) 2 .It is clear that f (p) > p for 0 < p < m b and f (p) < p for m b < p.Furthermore, the function f has its maximum at p = [ 1−b 1−2b ] 2 > 1, so that f is monotone increasing on [0, 1] and hence f (p) > f (m b ) = m b whenever p > m b .As in the proof of Theorem 3.1 let p n be the probability that Q n ∩ K n = ∅ for arbitrary closed sets Q and K. Observe that p 0 = 1 > m b and hence the sequence {p n : n ∈ N} is decreasing with lower bound m b .It follows that lim n p n = m b > 0. Now B = {(Q, K) : Q ∩ K = ∅} = n B n is the intersection of a decreasing sequence of sets and hence µ 2 (B) = lim n p n = m b > 0. Claim 3.5.µ * ({Q : µ * ({K : n and the sequence (S n ) is effectively c. e. open, so that no random closed set can be belong to all S n .But if T d (Q) = 0, then of course Q ∈ S n for all n.Thus every µ * d random closed set must have positive capacity.This completes the proof when d is independent of σ.Next suppose that b < b and that, for all σ, d(σ ⌢ 0) = d(σ ⌢ 1) ≤ b • d(σ).Let p d n now be the µ * d probability that Q n ∩ K n = ∅.It follows from the monotonicity of f (Claim 3.3) that p d n ≥ p n for all d as above and thus lim n p d n ≥ m b .The same argument as above now shows that {Q : T d (Q) ≥ m b } has positive measure and thus T d (Q) has positive capacity with probability one.The argument that every µ * d -random closed set has positive capacity follows as above.

Theorem 4 . 4 .
For the uniform measure µ d defined by d(σ ⌢ i) = b • d(σ) for all σ, there is an effectively closed set Q with Lebesgue measure zero and positive capacity T d (Q).