aa r X i v : . [ m a t h . D S ] D ec LOGARITHMIC CAPACITY OF RANDOM G δ SETS
FERNANDO QUINTINO
Abstract.
We study the logarithmic capacity of G δ subsets of theinterval [0 , . Let S be of the form S = \ m [ k ≥ m I k , where each I k is an interval in [0 ,
1] with length l k that decrease to0. We provide sufficient conditions for S to have full capacity, i.e.Cap( S ) = Cap([0 , ,
1] randomly with respect to somegiven distribution. The random G δ sets generated by such distributionsatisfy our sufficient conditions almost surely and hence, have full capac-ity almost surely. This study is motivated by the G δ set of exceptionalenergies in the parametric version of the Furstenberg theorem on ran-dom matrix products. We also study the family of G δ sets { S ( α ) } α> that are generated by setting the decreasing speed of the intervals to l k = e − k α . We observe a sharp transition from full capacity to zerocapacity by varying α > Introduction
The setting.
Finite signed Borel measures on C form a vector spaceover R . Given two finite singed measures Borel measures µ and ν on C wedefine their interaction by I ( ν, µ ) := Z Z ( − log | z − w | ) dν ( z ) dµ ( w ) . (1.1)The interaction is a bilinear form on the vector space of finite singed Borelmeasures on C with the following properties:(1) I ( ν, µ ) = I ( µ, ν ) , (2) I ( ν, µ ) >
0, if ν and µ are probability measures and the union oftheir support has diameter of at most 1,(3) I ( ν, µ + µ ′ ) = I ( ν, µ ) + I ( ν, µ ′ ) and I ( ν, cµ ) = cI ( ν, µ ). Date : December 4, 2020.2010
Mathematics Subject Classification.
Primary: 31A15, 31C15. Secondary: 28A12.
Key words and phrases.
Logarithmic capacity, phase transition, parametric Fursten-berg theorem.The author was supported in part by NSF grants DMS-1700143 (PI - M. Foreman)and DMS-1855541 (PI - A. Gorodetski).
This bilinear form is a generalization of the energy of measure µ , whichis defined as I ( µ ) := I ( µ, µ ) . We can think of the energy of a measure as self-interacting . In physics µ is considered as a charge distribution on C and I ( µ ) as the total energy of µ on C (see [10, pg. 56]).The logarithmic capacity of a subset E ⊂ C is then defined by minimizingthe energy: Cap( E ) = exp( − inf { I ( µ ) } ) , where the infimum is taken over the set of Borel probability measures whosesupport is a compact subset of E (we interpret e −∞ as 0). Capacity gaugeshow far away a set is from being polar . A set E is said to be polar if I ( µ ) = ∞ for every non-trivial measure µ with compact support in E .Most of the literature on capacity has been devoted to the study of com-pact sets in C , but non-compact sets has also been studied (see [10], [11,Appendix A]). Moreover, G δ sets have also been of interest (see [3], [7], [10]).In [2, Section 9], the authors discuss the applications of potential theory tospectral theory.Our focus will be on the capacity of G δ ’s of the form: S = \ m [ k ≥ m I k , (1.2)where each I k is an open interval of length l k with center at c k ∈ (0 , { l k } is taken to approach 0 as k → ∞ . It is immediate that S is a G δ subset of [0 , S has full capacity on the unit interval: Definition 1.
Let J ⊂ C . A set E ⊂ J is said to have full capacity on J ifCap( E ) = Cap( J ) . The capacity of an interval J is Cap( J ) = | J | (see, e.g. [10, pg. 135], [11,Example A.17]).Our results and methods can be extended to higher dimensions, but we donot elaborate on that here. In this paper, we will be focused on G δ subsetsof an interval of the real line. We are mostly interested in one-dimensionbecause our motivation came from the one-dimensional random G δ sets ofexceptional energies in the parametric version of the Furstenberg theorem(see Section 1.3).1.2. Main results.
Our first main result is devoted to the random setting,being a “toy model” for the exceptional energies in the parametric versionof the Furstenberg theorem (see Section 1.3). Namely, the set of excep-tional parameters in [6] is generated by exponentially small intervals, thatare asymptotically distributed with respect to some (dynamically defined)measure. However, their positions are random.
OGARITHMIC CAPACITY OF RANDOM G δ SETS 3 A random G δ set is obtained by viewing the centers { c k } as random vari-ables. As a toy model, it is reasonable to consider first the set generatedby random intervals, that are placed independently (with the same “reason-able” distribution of their centers) - instead of some complicated definitioncoming from the random dynamical systems. Theorem 1.
Let { c k } be i.i.d. with an absolutely continuous distributionon any interval J with almost everywhere positive and uniformly boundeddensity function. Take l k = e − λk for some fixed λ > and let S be thecorresponding G δ -set (1.2) . Then, almost surely S has full capacity on theunit interval: Cap( S ) = Cap([0 , > . Remark 1.
Full capacity is a property that is inherited when restrictedto subintervals (see [7, Proposition 1.6]): If E is a subset of interval J such that Cap( E ) = Cap( J ), then given any subinterval J ′ ⊂ J , one hasCap( E ∩ J ′ ) = Cap( J ′ ). Remark 2.
In Theorem 1 (and Theorem 3 below), the interval [0 ,
1] maybe replaced with any bounded interval J due to the fact that Cap( β · J ) = β · Cap([0 , β >
0. Without loss of generality, we will only beworking on the interval [0 , dx to be the restriction tothe unit interval: dx | [0 , .Now, take the centers { c k } from Theorem 1 and let us vary the lengths ofthe intervals as a function of the parameter α ∈ (0 , l k = e − λk α . It turnsout that at α = 1 the capacity undergoes a (sharp) phase transition: Theorem 2 (Random phase transition) . Let { c k } be i.i.d. with an absolutelycontinuous distribution with density function that is bounded and positivealmost everywhere with respect to the Lebesgue measure. Let S be generatedby l k := e − λk α for λ > and α > . Then(1) Cap( S ) = Cap([0 , > for < α ≤ almost surely,(2) Cap( S ) = 0 for < α . Remark 3.
The phase transition in Theorem 2 (and in Theorem 4 below)is analogous to the one observed in [7] (see Equation (1.7) and Theorem 6 inSection 1.3). It is interesting to note that in the present paper we actuallyestablish the full capacity at the critical point α = 1 , while for the settingin [7] full capacity at the critical value α = 2 was only a conjecture.Theorem 1 and Theorem 2 follow from our deterministic results. Ourfirst main deterministic result provides sufficient conditions for a G δ set S ,defined by (1.2), to have full capacity on the unit interval: Theorem 3 (Sufficient conditions for full capacity) . Assume that the in-tervals { I k } from (1.2) have exponentially decreasing lengths l k = e − λk for F. QUINTINO some fixed λ > and satisfy the assumptions A.1 - A.3 below. Then the G δ set S has full capacity on the unit interval: Cap( S ) = Cap([0 , > . The assumptions that are imposed in this theorem, roughly speaking,state that these intervals are sufficiently uniformly placed and sufficientlywell-spaced (both in terms of “average” and minimal distances between theircenters).First, we will pack intervals { I k } into groups with the indices from A n := { n, . . . , n − } , (1.3)and then pack these into larger groups: A n,q ( n ) := A n ∪ A n ∪ · · · ∪ A q n , (1.4)where q ∈ N . We will assume the following: A.1 (distribution)
The centers are distributed with respect to somedensity function φ ( x ) ∈ L ([0 , , dx ) , where φ ( x ) > f ∈ C ([0 , A n ) X k ∈A n f ( c k ) → Z f ( x ) φ ( x ) dx as n → ∞ . Also, there exists a sequence q ( n ) of integer numbers, such that q ( n ) → ∞ as n → ∞ and that the following two conditions hold: A.2 ( log -average spacing)
For every ε >
0, there exists δ > n large enough for any n ′ , n ′′ ∈ { n, n, . . . , q ( n ) n } we have1 A n ′ × A n ′′ ) X ( − log | c i − c j | ) < ε, where the sum is over i ∈ A n ′ , j ∈ A n ′′ such that i = j and | c i − c j | <δ . A.3 (gap control)
For every ε >
0, for all large enough n , we have thatfor every i, j ∈ A n,q ( n ) with i = j the following holds: l i + l j | c i − c j | < ε. Applying Theorem 3, we obtain our next result, a deterministic phasetransition for the capacity. Again, take the centers { c k } that assumptions A.1 - A.3 are satisfied for the choice of lengths l k = e − λk for some fixed λ > Theorem 4 (Deterministic phase transition) . Let the centers { c k } be thesame centers from Theorem 3. Let S be generated by l k := e − λk α for α > .Then OGARITHMIC CAPACITY OF RANDOM G δ SETS 5 (1)
Cap( S ) = Cap([0 , > for < α ≤ ,(2) Cap( S ) = 0 for < α . The next theorem states that both the random theorems (Theorem 1and Theorem 2) follow from the deterministic theorems (Theorem 12 andTheorem 4):
Theorem 5.
Let { c k } be i.i.d. with an absolutely continuous distributionon any interval J with almost everywhere positive and uniformly boundeddensity function. Take l k = e − λk for some fixed λ > . Then, almost surelyassumptions A.1 - A.3 are satisfied.
Motivation and historical background.
In this section, we willdiscuss the motivation behind our project and the historical background.Gorodetski and Kleptsyn in [6, Section 1.2] studied the set of exceptionalenergies in the parametric version of the Furstenberg theorem. Consider T n,ω,a := A ω n ( a ) . . . A ω ( a )where matrices A ω k ( a ) ∈ SL (2 , R ) are i.i.d., depending on a parameter a ,taking values in some interval J ⊂ R . Furstenberg’s theorem implies thatfor every a ∈ J , for almost every ω , we getlim n →∞ n log k T n,ω,a k = λ F ( a ) > . (1.5)Questions on switching the quantifiers in the limit appear naturally inspectral theory, specifically, in Anderson localization proofs.In [6, Theorem 1.5], the authors proved that almost surely switching thequantifiers leads to the occurrence of a different kind of behavior. Namely,under some technical assumptions, it was shown that for almost every ω ,there exists some random exceptional energies subset of parameters S e ( ω ) ⊂ J such that (1.5) does not hold. Additionally, there also exists a smaller setof parameters G δ -set S ( ω ) such that for all a ∈ S ( ω ), we havelim inf n →∞ n log k T n,ω,a k = 0 . Both these sets are random G δ ’s of the form (1.2).Additionally, in [6] it was shown that the set S e ( ω ) (and thus S ( ω ))have zero Hausdorff dimension. Capacity is a finer measurement than theHausdorff dimension in the sense that any set E ⊂ C that has zero capacitymust have zero Hausdorff dimension. The question as to what is the capac-ity of both S e ( ω ) and S ( ω ) is still open. If one can show that those setssatisfy assumptions A.1 - A.3 (and this is what we conjecture), our Theo-rem 3 will imply that these sets have full capacity, that is Cap( S e ( ω )) =Cap( S ( ω )) = Cap( J ), in the same way as we get full capacity in the “toymodel” Theorem 2. F. QUINTINO
The capacity of such G δ ’s is also interesting because it showcases a phasetransition. That is, a drastic transition from zero capacity to full capacityprecisely when the series X k | log l k | (1.6)transitions from convergent to divergent. As we mentioned above, capacitygauges how far away a set is from being polar. Hence, as we change the speedof intervals so that the series (1.6) transitions from convergent to divergent, S goes from being polar to being as far away as possible from polar, thereis no middle ground. This transition was first noticed by Kleptsyn andQuintino (see [7]) in the case when the centers { c k } are equidistributed inthe following way: for every n we consider n equally spaced centers: c j,n = 2 j + 12 n for every j = 0 , . . . , n − , and with the restriction that the corresponding interval J j,n have the samelength r n for j = 0 , . . . , n −
1. The uniform G δ -set e S , corresponding to thesequence r n , is given by e S := ∞ \ m =1 ∞ [ n = m n − [ j =0 J j,n . (1.7)Any uniform G δ set e S may be written in the generic setting (1.2) by ordering J j,n and re-labeling. They noticed that there is a “phase transition” in which e S goes from having zero capacity to full capacity: Theorem 6 (Phase transition [7, Theorem 1.2]) . For r n = e − n α ,(1) if α > , then Cap( e S ) = 0 ,(2) if α < , then Cap( e S ) = Cap([0 , . We refer to α = 2 as the critical case because it is precisely when thesum (1.6) transitions from convergent to divergent. Note that there are n intervals of length e − n α in (1.7), and that is why the critical case is α = 2in Theorem 6 and not α = 1. Also, note that full capacity of e S in thecritical case in [7] was conjectured , but not proved; contrary to this, in thesetting of the present paper the analogous statement for the critical α = 1is established (see Remark 3).The zero capacity part in all the theorems above goes back to the worksin the first half of twentieth century: a 1918 paper by Lindeberg [8] and 1937by Erd¨os and Gillis [5]. They were working on connecting the notion of the h - volume of a set with the logarithmic capacity of a set. A function h thatis defined in some right neighborhood of 0 is called a measuring function provided that h is continuous, positive, increasing, concave, and h (0) = 0. OGARITHMIC CAPACITY OF RANDOM G δ SETS 7
The h - volume of a set E ⊂ R is defined as m h ( E ) := lim ε → inf { ( x j ,r j ) j ∈ N }∈I ( E,ε ) X j h ( r j ) , where the infimum is taken over the set I ( E, ε ) of covers of E by balls ofdiameter less than ε : I ( E, ε ) = ( x j , r j ) j ∈ N | [ j U r j ( x j ) ⊃ E, ∀ j r j < ε . In particular, the function h ( x ) = | log x | provided the following link: Theorem 7 (Erd¨os and Gillis [5, p. 187], generalizing Lindeberg [8, p. 27]) . If for a set E one has m h ( E ) < + ∞ , then Cap( E ) = 0 . This was later re-proved by Carleson [1], and noticed in [7, Thm. 1.3] tobe a corollary of the Cauchy-Schwarz inequality. Theorem 7 immediatelyimplies (see [7, Corollary 1.4]):
Corollary 8.
Let S be defined by (1.2) . If the series P k | log l k | converges,then the set S is of zero capacity. In the same 1937 paper, Erd¨os and Gillis [5, (C), p. 186] have mentioned aconjecture, going back to Nevanlinna’s paper [9], that aimed at generalizingTheorem 7 to other h -volume settings. This conjecture was disproved byUrsell [12]; the re-distribution construction that was used in [7] and that weare using in the present paper can be seen as an extension of his technique.1.4. Sketch of the proof and plan of the paper.
In this section, wewill give a sketch of the proofs and end with the plan of the paper.The statement in the phase transition theorems 2 and 4 for α >
A.1 - A.3 (see Theorem 7above).Due to monotonocity of capacity, the statement in the phase transitiontheorems 2 and 4 for 0 < α < α = 1. For the deterministic phase transition this is Theorem 3. For therandom phase transition the result follows from Theorem 3 by showing thatassumptions A.1 - A.3 hold almost surely, that is Theorem 5.In Section 2, we will show that the centers from the random phase tran-sition satisfy assumptions
A.1 - A.3 for α = 1 (Theorem 5). Hence, therandom phase transition holds.Thus, the main task is to show that S has full capacity for α = 1 (The-orem 3). The method that we will employ to show full capacity is the re-distribution technique under assumptions A.1 - A.3 for α = 1.We introduce this technique in Section 3.1. Namely, we will begin withthe equilibrium measure ν J on the interval J , then we will construct a prob-ability measure ν such that the energy I ( ν ) approximates the energy I ( ν J ) F. QUINTINO and whose support is a subset of supp ν J and is a finite union of intervals { I k } . Then we will construct another probability measure ν such that theenergy I ( ν ) approximates the energy I ( ν ) and whose support is a subsetof supp ν and is a finite union of intervals { I k } . Inductively, repeating thisprocedure, we get a sequence of probability measures that have their ener-gies that are arbitrarily close to I ( ν J ) and such that their supports create adecreasing sequence of compact subsets. After passing to the weak-limit weobtain a measure supported on S (Proposition 12), thus proving the desiredfull capacity for the set S (see Section 3.2 for the proof). Proposition 13states that the above technique is applicable when assumptions A.1 - A.3 aresatisfied. Hence, Theorem 3 follows from Proposition 12 and Proposition 13.Finally, in Section 4 we develop the tools to prove Proposition 13. InSection 4.5 we conclude with the proof of Proposition 13.2.
In the random setting
A.1 - A.3 are a.s. satisfied
This section is devoted to the proof of Theorem 5: we assume that thecenters { c k } are i.i.d. random variables and show that if their distributionsare nice, then assumptions A.1 - A.3 are satisfied.The distribution immediately follows from the law of large numbers:
Lemma 9.
Under the assumptions of Theorem 5, assumption
A.1 is almostsurely satisfied.
Now, take q ( n ) = [log (log n )]. The uniform gap control can be obtainedby a straightforward estimate of the probability of two random centers beingclose to each other: Lemma 10.
Under the assumptions of Theorem 5, for q ( n ) = [log (log n )] ,assumption A.3 is almost surely satisfied.Proof.
Let K be the upper bound for the density of the distribution, andlet ε > i = j , i, j ∈ A n,q ( n ) , if l i + l j | c i − c j | < ε does not hold, it implies that | c i − c j | ≤ l i + l j ε < ε e − λn , and the probability of such an event (for any given i and j ) does not ex-ceed Kε e − λn . As there are less than 2 q ( n )+1 n < n possible indices i and j , the total probability that the condition is violated for a given n does notexceed 4 n · Kε e − λn . The series X n n · Kε e − λn OGARITHMIC CAPACITY OF RANDOM G δ SETS 9 converges, and the application of the Borel-Cantelli Lemma concludes theproof. (cid:3)
Finally, the log-averages of spaces also can be controlled quite directly:
Lemma 11.
Under the assumptions of Theorem 5, for q ( n ) = [log (log n )] ,assumption A.2 is almost surely satisfied.Proof.
Given ε > G ( X, Y ) = ( − log | X − Y | ) (0 ,δ ) ( | X − Y | ) , where δ > G ( X, X ) = 0. We have that1 A n ′ × A n ′′ ) X < | c i − c j | <δ ( − log | C i − C j | ) = 1 A n ′ × A n ′′ ) X i ∈A n ′ ,j ∈A n ′′ G ( C i , C j ) . Suppose the law of large numbers holds for G ( X, Y ): as n → ∞ we have1 A n ′ × A n ′′ ) X i ∈A n ′ ,j ∈A n ′′ G ( C i , C j ) → E G ( C , C ) , where for n ′ , n ′′ ∈ { n, n, . . . , q ( n ) n } . Then we may find a δ such that A.2 holds.The law of large numbers holds by considering the difference: H ( x, y ) := G ( x, y ) − c − E [ G ( x, y ) − c | y ] − E [ G ( x, y ) − c | x ] , where c = E G ( x, y ) and their average1 n ′ n ′′ S n ′ ,n ′′ = 1 n ′ n ′′ X i = j H ( x, y ) . (2.1)Now, consider the fourth power of S n ′ ,n ′′ and take its expectation: E ( S n ′ ,n ′′ ) = X E [ H ( c i , c i ) H ( c j , c j ) H ( c k , c k ) H ( c l , c l )] . (2.2)The function H ( x, y ) has the property that E [ H ( x, y ) | y ] = E [ H ( x, y ) | x ] = 0.Hence, if a term has an independent random variable, say c i , then E [ H ( c i , c i ) H ( c j , c j ) H ( c k , c k ) H ( c l , c l )] = 0 . On the other hand, when every random variable is depended on anotherrandom variable, we can count the non-vanishing terms. There are ( n ′ n ′′ )terms of the form E [ H ( c i , c i ) ] . There are ( n ′ n ′′ ) terms of the form E [ H ( c i , c i ) H ( c j , c j ) ] . There are at most ( n ′ n ′′ )( n ′ + n ′′ )4 terms of the form E [ H ( c i , c i ) H ( c j , x ) H ( c k , y )] , where x, y ∈ { c i c i } . Lastly, there are at most n ′ n ′′ ( n ′ + n ′′ ) terms of theform E [ H ( c i , c i ) H ( c i , c j ) H ( c k , c i ) H ( c k , c j )] . Since E [ H ( x, y ) ] < ∞ , then E S n ′ ,n ′′ ≤ C ′ max { ( n ′ n ′′ ) , n ′ n ′′ , n ′ n ′′ } , where C ′ > P (cid:0) | S n ′ ,n ′′ | > ε ( n ′ n ′′ ) (cid:1) ≤ E ( S n ) / ( ε ( n ′ n ′′ )) ≤ C ′ ε max (cid:26) n ′ n ′′ ) , n ′ n ′′ , n ′ n ′′ (cid:27) . Since n ≤ n ′ , n ′′ , then P (cid:0) | S n ′ ,n ′′ | > ε ( n ′ n ′′ ) (cid:1) ≤ C ′ ε n . We have that ∞ X n =1 2 q ( n ) n X n ′ = n q ( n ) n X n ′′ = n P (cid:0) | S n ′ ,n ′′ | > ε ( n ′ n ′′ ) (cid:1) ≤ C ′ ε ∞ X n =1 q ( n ) n ≤ C ′ ε ∞ X n =1 (log n ) n , which is finite. By Borel-Cantelli lemma, | S n ′ ,n ′′ | > ε ( n ′ n ′′ ) does not occurinfinitely often with probability 1. Let ε k be a sequence of positive numbersthat decreases to 0 as k → ∞ . For each ε k , | S n ′ ,n ′′ | > ε k ( n ′ n ′′ ) does notoccur infinitely often with probability 1. Since the countable intersection ofsets of full measure has full measure, then for all ε >
0, there exists n ∈ N such that for any n ≥ n , for every n ′ , n ′′ ∈ { n, n, . . . , q ( n ) n } , we have | S n ′ ,n ′′ | < ε ( n ′ n ′′ ) with probability 1. That is, the average (2.1) goes to 0 as n → ∞ . (cid:3) Together, lemmas 9, 10, 11 imply Theorem 5.3.
The re-distribution technique
Introducing the technique.
Our main tool for establishing full ca-pacity for a set S (what is needed for the proof of Theorem 3) will bethe re-distrubtion technique that was introduced in [7]. We will recall themethod in this section. The main property that allows it to work will bethe following one: Definition 2.
We say that S (in the generic setting (1.2)) is re-distributable if the following holds: for every probability measure ν with piecewise contin-uous density that is supported on a finite collection of intervals in [0 ,
1] andfor every ε > m ∈ N , there exists another probability measure ν ′ with piecewise continuous density such that(1) I ( ν ′ ) < I ( ν ) + ε , OGARITHMIC CAPACITY OF RANDOM G δ SETS 11 (2) ν ′ is supported on supp ν ∩ V n for some n ≥ m ,where V n is a finite union of I k ’s with k ≥ n .The following proposition then allows us to establish full capacity: Proposition 12. If S is re-distributable, then S has full capacity on theunit interval: Cap( S ) = Cap([0 , . Proposition 13.
Assume
A.1 - A.3 for interval lengths l k = e − λk forsome λ > . Then the set S is re-distributable. Section 4 is devoted to the proof of Proposition 13.3.2.
Proof of Proposition 12.
In this section, we will prove that S hasfull capacity when S is re-distributable.As we have mentioned in Section 1.4, the proof of Proposition 3 is obtainedby inductively constructing a sequence of measures with smaller and smallersupport. Let us make these arguments formal: Proof of Proposition 12.
The density function for the equilibrium measurefor the unit interval is f [0 , ( x ) = 1 π p x (1 − x ) , for x ∈ (0 ,
1) and 0 otherwise (see e.g. [11, Eq. (A.53)]). Given ε >
0, thereexists a continuous density function f such that I ( f ( x ) dx ) < I ( f [0 , ( x ) dx ) + ε. Let dν ( x ) := f ( x ) dx with support [0 , ν , thereexists ν with support V n and I ( ν ) < I ( ν ) + ε/ . Apply Definition 2 to ν , there exists ν with support V n ∩ V n and I ( ν ) < I ( ν ) + ε/ . and n < n . By induction and applying Definition 2, for each m ∈ N thereexists a Borel probability measure ν m that is supported on C m := V n ∩ · · · ∩ V n m . We consider the telescoping sum: I ( ν m ) − I ( ν ) = m X i =1 (cid:0) I ( ν i ) − I ( ν i − ) (cid:1) < m X i =1 ε i +1 . It follows that I ( ν m ) < I ( ν ) + ε. As in [7], any weak* limit will work. Assume that ν ∞ is a weak* limit of { ν m } . Passing to a weak* limit can only decrease the energy (see [10, Lemma3.3.3]): I ( ν ∞ ) ≤ lim inf m →∞ I ( ν m ) < I ( ν ) + ε < I ( f [0 , ( x ) dx ) + 2 ε. Since ε > I ( ν ∞ ) ≤ I ( f [0 , ( x ) dx ) . If ν ∞ has compact support contained in S , then we are done. The weak*limit only allows us to conclude that ν ∞ has compact support contained in¯ C ∞ := \ m Cl( C m ) . However, ¯ C ∞ differs from C ∞ := \ m C m ⊂ S, by at most a countable set P (the collection of boundary points of each C m ). Since I ( ν ∞ ) < ∞ , then ν ∞ ( P ) = 0 (see [10, Theorem 3.2.3]). Byregularity of Borel measures, we may find a Borel probability measure withcompact support contained in C ∞ that differs from I ( ν ∞ ) as small as wewant. Hence, I ( f [0 , ( x ) dx ) = inf { I ( ν ) : ν ∈ P ( C ∞ ) } . Since C ∞ ⊂ S , then S has full capacity:Cap( S ) = Cap([0 , . (cid:3) Proving Proposition 13:
A.1 - A.3 imply re-distribution
Properties of assumptions A.1-A.3.
In this section, we will discusssome of the properties of assumptions
A.1 - A.3 that will be needed in theproofs.The gap control property (assumption
A.3 ) is aimed at controlling thegaps between two distinct intervals in a collection of intervals in a uniformway. Let
I, I ′ ⊂ (0 ,
1) be two disjoint intervals with centers c, c ′ , then thegap between I and I ′ isdist( I, I ′ ) = | c − c ′ | −
12 ( | I | + | I ′ | ) > . (4.1)We will control the gaps by controlling the ratio of the average of the lengthsand the distance between their centers (see A.3 ). OGARITHMIC CAPACITY OF RANDOM G δ SETS 13
Remark 4.
Notice that by letting ε <
A.3 , we get (4.1). Hence, gapcontrol implies that the intervals in { I k : k ∈ n, . . . , q ( n ) n − } , are pairwise disjoint. Each measure that we construct in Section 4.2 will besupported on pair-wise disjoint collection: { I k : k ∈ A n } = { I n , I n +1 , . . . , I n − } . In Section 4.5, we will construct a measure that is an average of measuresfrom Section 4.2. Hence, the average measure will be supported on: { I n , I n +1 , . . . , I n − }{ I n , I n +1 , . . . , I n − } ... { I q ( n ) − n , I q ( n ) − n +1 , . . . , I q ( n ) n − } . Assumption
A.3 allows the collection of intervals above to be disjoint.The distribution property (assumption
A.1 ) requires the centers to be distributed with respect to some function φ ( x ) ∈ L ([0 , , dx ) , where φ ( x ) > A n ) X k ∈A n f ( c k ) → Z f ( x ) φ ( x ) dx as n → ∞ , (4.2)for every continuous function f ∈ C ([0 , Remark 5.
Note that (4.2) will hold for piecewise continuous functions f since we may approximate such functions from above and below by contin-uous functions in L . Equation (4.2) extends to 2-dimensions: Let f be anypiecewise continuous function. Then1 A n × A n ′ ) X i ∈A n ,j ∈A n ′ f ( c i ) f ( c j ) → Z Z f ( x ) f ( y ) φ ( x ) φ ( y ) dxdy, (4.3)as m → ∞ and n, n ′ ≥ m .To show that S is re-distributable (see Definition 2), we will show thatthe following statement holds: P.1
For each positive continuous function f on the interval [0 , { µ n } so that each µ n has a piecewise continuous density with support contained in a finite unionof disjoint I k ’s with k ≥ n and with asymptotic behavior: I ( µ n ) = I ( f ( x ) dx )( R f ( x ) dx ) + o (1) . With the distribution assumption
A.1 and log-average spacing assump-tion
A.2 , we can see that the centers have the asymptotic behavior that isneeded in
P.1 : Lemma 14.
Under assumptions
A.1 and
A.2 , for every f ∈ C ([0 , , as n → ∞ and n ≤ n ′ , n ′′ , we have that A n ′ × A n ′′ ) X i = j ( − log | c i − c j | ) f ( c i ) f ( c j ) → I ( f ( x ) φ ( x ) dx ) , (4.4) where the sum is taken over ( i, j ) ∈ A n ′ × A n ′′ and i = j . Moreover, I ( f ( x ) φ ( x ) dx ) < ∞ . Remark 6.
The density function φ ( x ) in assumption A.1 is not to beconfused with the continuous density function f in Definition 2 and in P.1 .Once the centers are distributed with respect to φ ( x ), the function φ ( x )is fixed. The continuous density function f in Definition 2 and in P.1 isarbitrary.
Proof of Lemma 14.
Given ε >
0, let δ >
A.2 . For s >
0, define f s ( x ) = − log x for x ≥ s and 0 otherwise. Using Fatou’slemma, we get Z Z | x − y | <δ ( − log | x − y | ) φ ( x ) φ ( y ) dx dy ≤ lim inf s → + Z Z | x − y | <δ f s ( | x − y | ) φ ( x ) φ ( y ) dx dy. Using assumption
A.1 for n ′ , n ′′ ∈ { n, n, . . . , q ( n ) n } , we havelim inf s → + Z Z | x − y | <δ f s ( | x − y | ) φ ( x ) φ ( y ) dx dy = lim inf s → + lim n →∞ X < | c i − c j | <δ f s ( | c i − c j | ) A n ′ × A n ′′ ) ≤ lim n →∞ X < | c i − c j | <δ ( − log | c i − c j | ) A n ′ × A n ′′ ) ≤ ε, where the last holds by assumption A.2 for some δ >
0. Hence, for everycontinuous function f , we have finite energy: I ( f ( x ) φ ( x ) dx ) = Z Z ( − log | x − y | ) f ( x ) φ ( x ) f ( y ) φ ( y ) dx dy < ∞ . OGARITHMIC CAPACITY OF RANDOM G δ SETS 15
Note that X i = j ( − log | c i − c j | ) f ( c i ) f ( c j ) A n ′ × A n ′′ ) − X i = j ( − log | c i − c j | ) f δ ( c i ) f δ ( c j ) A n ′ × A n ′′ )= X < | c i − c j | <δ ( − log | c i − c j | ) f ( c i ) f ( c j ) A n ′ × A n ′′ ) . Let n → inf, then by assumptions A.1 and
A.2 , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim n →∞ X i = j ( − log | c i − c j | ) f ( c i ) f ( c j ) A n ′ × A n ′′ ) − Z Z f δ ( | x − y | ) f ( x ) f ( y ) φ ( x ) φ ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε · (max | f | ) . By letting δ →
0, we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim n →∞ X i = j ( − log | c i − c j | ) f ( c i ) f ( c j ) A n ′ × A n ′′ ) − I ( f ( x ) φ ( x ) dx ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε · max | f | . Since ε > (cid:3)
Construction of a single-level re-distribution.
Our first step inconstructing the probability measures in
P.1 is to construct a single-level re-distribution probability measure. This section is devoted to the constructionof such probability measures.We begin with a “re-distribution” type of measure f ( x ) dx | [0 , onto asingle interval: µ k = f ( x ) dx | I k | I k | . We do not call this a re-distribution as in [7] because the measure is notnecessarily a probability measure. We consider the average of µ k ’s: µ A n := 1 A n X k ∈A n µ k , where A n are defined in (1.3) and satisfy the gap control property A.3 .Recall that gap control implies that for large enough n , our collection ofintervals are disjoint (see Remark 4). Notice that each measure µ A n is notnecessarily a probability measure. To correct that, let us define V n = [ k ∈A n I k . Then, we consider the single-level re-distribution probability measure:ˆ µ n := µ A n µ A n ( V n ) , (4.5)which is supported on V n and it’s energy is I (ˆ µ n ) = 1( µ A n ( V n )) I ( µ A n ) . Thus, we are interested in the asymptotic behavior of µ A n ( V n ) and theasymptotic behavior of: I ( µ A n ) = 1( A n ) X k ∈A n I ( µ k ) + 1( A n ) X i = j I ( µ i , µ j ) . (4.6)The first sum is referred to as the self-interaction sum because in I ( µ k ) = I ( µ k , µ k ) the same measure is interacting with itself. The second sum isreferred to as outer-interaction sum because we have two measures withdisjoint supports interacting with each other in I ( µ k , µ j ).In Section 4.3, we will discuss the asymptotic behavior of the outer-interaction and in Section 4.4 we will work on controlling the asymptoticbehavior of the self-interaction sum. In Section 4.5, we will put the two to-gether. We will finish the section with the asymptotic behavior of µ A n ( V n ) : Lemma 15. If A.1 and
A.3 hold, then µ A n ( V n ) = Z f ( x ) φ ( x ) dx + o (1) . Proof.
Since for large n the intervals in { I k : k ∈ A n } are disjoint (see Remark 4), then µ k ( V n ) = | I k | R I k f ( x ) dx. Hence, µ A n ( V n ) = 1 A n X k ∈A n | I k | Z I k f ( x ) dx. Due to the uniform continuity of f on the interval [0 , ε > δ > | f ( x ) − f ( y ) | < ε if | x − y | < δ. Since the lengths of the intervals I k approach 0, then there exists N ∈ N such that for every k ≥ N we have | f ( x ) − f ( c i ) | < ε if x ∈ I k . Therefore, for every n ≥ N and every k ∈ A n , we have | f ( x ) − f ( c k ) | < ε if x ∈ I k . OGARITHMIC CAPACITY OF RANDOM G δ SETS 17
It follows that for large enough n , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ A n ( V n ) − A n X i ∈A n f ( c i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε. As the centers are distributed with respect to φ ( x ) A.1 , it follows that (cid:12)(cid:12)(cid:12)(cid:12) µ A n ( V n ) − (cid:18)Z f ( x ) φ ( x ) dx + o (1) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < ε. Since ε > (cid:3)
Asymptotic behavior of outer-interaction.
In this section, we areinterested in the asymptotic behavior of the outer-interaction sum:1( A n ) X i = j I ( µ i , µ j ) , where the sum is taken over i, j ∈ A n and i = j . It is the outer-interactionsum that gives the limit point in P.1 : Lemma 16.
We have A n ) X i = j I ( µ i , µ j ) = I ( f ( x ) φ ( x ) dx ) + o (1) , (4.7) where i, j ∈ A n . To show Lemma 16, we want to estimate ( − log | x − y | ) by ( − log | c − c ′ | )where x and y are in intervals with centers c and c ′ , respectively. The nextlemma allows us to do that. Lemma 17.
Let f be any continuous function on [0 , and let J, J ′ be twodisjoint intervals in [0 , with lengths r, r ′ and centers c, c ′ , respectively.Define µ := 1 r f ( x ) dx | J and µ ′ := 1 r ′ f ( x ) dx | J ′ . Let ε > . If r + r ′ | c − c ′ | ≤ (1 − e − ε ) , (4.8) then (cid:12)(cid:12) I ( µ, µ ′ ) − ( − log | c − c ′ | ) f ( c ) f ( c ′ ) (cid:12)(cid:12) ≤ (2 K ( − log | c − c ′ | ) + K ) ε, where K = k f k ∞ .Proof. Let us first prove that for a, b >
0, we have | log a − log b | ≤ ε, if | a − b | ≤ b (1 − e − ε ) . We have that | log a − log b | ≤ ε holds if and only if − ε ≤ log( a/b ) ≤ ε, if and only if be − ε ≤ a ≤ be ε , if and only if − b (1 − e − ε ) ≤ a − b ≤ b ( e ε − . Since (1 − e − ε ) ≤ ( e ε − | log a − log b | ≤ ε holds when | a − b | ≤ b (1 − e − ε ) . For any two disjoint intervals
J, J ′ ∈ [0 ,
1] with centers c, c ′ and withlengths r, r ′ respectively, we have || x − y | − | c − c ′ || ≤ | ( x − c ) + ( c ′ − y ) | ≤ r + r ′ . Since assumption (4.8) is equivalent to r + r ′ ≤ | c − c ′ | (1 − e − ε ) , then | ( − log | x − y | ) − ( − log | c − c ′ | ) | < ε. Since f is uniformly continuous on [0 , δ > | f ( a ) − f ( b ) | < ε if | a − b | < δ . If 0 < r, r ′ < δ , we have that | f ( x ) − f ( c ) | < ε and | f ( y ) − f ( c ′ ) | < ε. We would like to combine the three inequalities.Suppose
A, a, B, b ∈ R and ε a , ε b > | A − a | < ε a and | B − b | < ε b . We have that | AB − ab | ≤ | AB − Ab | + | Ab − ab | ≤ ( | A | ε b + | b | ε a ) . One application of the above gives | f ( x ) f ( y ) − f ( c ) f ( c ′ ) | ≤ Kε, where K = k f k ∞ . A third application yields | ( − log | x − y | ) f ( x ) f ( y ) − ( − log | c − c ′ | ) f ( c ) f ( c ′ ) | < (2 Kε ( − log | c − c ′ | ) + K ε ) . OGARITHMIC CAPACITY OF RANDOM G δ SETS 19
Integrating by 1 r dx | J and 1 r ′ dy | J ′ finishes the proof. (cid:3) Now that we can estimate ( − log | x − y | ) by ( − log | c − c ′ | ) where x and y are in intervals with centers c and c ′ , respectively, we are ready to estimate1( A n ) X i = j I ( µ i , µ j ) , by 1( A n ) X i = j ( − log | c i − c j | ) f ( c i ) f ( c j ) . Let us go back to Lemma 16 and prove the asymptotic behavior of theouter-interaction:
Proof of Lemma 16.
Let ǫ > K = k f k ∞ . The gap control A.3 guarantees that there exists N ∈ N such that for every n ≥ N andevery i = j , where i, j ∈ A n , we have l i + l j | c i − c j | ≤ (1 − e − ε ) , which is the condition (4.8) in Lemma 17. Since l k decrease to 0 as k → ∞ ,then for all large enough n we may apply Lemma 17 to get | I ( µ i , µ j ) − ( − log | c i − c j | ) f ( c i ) f ( c j ) | ≤ (2 K ( − log | c i − c j | ) + K ) ε, for every n ≥ N and every i = j , where i, j ∈ A n . Adding this up for i = j where i, j ∈ A n and then dividing by ( A n ) , gives us: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A n ) X i = j I ( µ i , µ j ) − A n ) X i = j ( − log | c i − c j | ) f ( c i ) f ( c j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Kε ( A n ) X i = j ( − log | c i − c j | ) + K ε. We will apply Lemma 14 twice to the last two sums. We apply the lemmato the last sum by taking f = 1 in Lemma 14, and then we apply the lemmaagain for arbitrary f in Lemma 14 to get: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim n →∞ A n ) X i = j I ( µ i , µ j ) − I ( f ( x ) φ ( x ) dx ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (2 KI ( φ ( x ) dx ) + K ) ε. Lemma 14 also informs us that the energy of φ ( x ) dx is finite, hence 0 < (2 KI ( φ ( x ) dx ) + K ) ε < ∞ . As ε > A n ) X i = j ( − log | c i − c j | ) f ( c i ) f ( c j ) → I ( f ( x ) φ ( x ) dx ) . Lemma 16 holds. (cid:3)
Asymptotic behavior of self-interaction.
In this section, we willto control the self-interaction:
Lemma 18. If l k = e − λk where λ > , then A n ) X k ∈A n I ( µ k ) = O A n ) X k ∈A n k + o (1) . Proof.
By shifting and a change of variables, we get I ( 1 l k dx | I i ) = − log l k + I ( dx | [0 , ) = − log l k · (1 + o (1)) . If l k = e − λk , then adding the above over k ∈ A n gives us our result. (cid:3) If the self-interaction sum vanishes in the limit, then we will be able tofinish the proof with a single-level re-distribution. Let us see what the self-interaction tells us:
Lemma 19. If A n := { n, . . . , p ( n ) − } where p ( n ) is an integer-valuedfunction such that p ( n ) ≥ n , then A n ) X k ∈A n k = p ( n ) + n − p ( n ) − n ) . If p ( n ) >> n , then the right-hand side is close to .Proof. Let S = X k ∈A n k. Then the arithmetic sum becomes S = ( p ( n ) + n − A n )2 . Since A n = p ( n ) − n , then dividing by ( A n ) we get what we want. (cid:3) Remark 7.
Lemma 19 tells us that no matter how many intervals areincluded in our single-level of re-distribution, the self-interaction sum willnever vanish.
OGARITHMIC CAPACITY OF RANDOM G δ SETS 21
But since we can bound the self-interaction sum uniformly for all n , wewill be able to apply a multi-level re-distribution in Section 4.5. That is, wewill take the average of measures ˆ µ n to handle the self-interaction sum.4.5. The proof of Proposition 13.
In this section, we will use a multi-level re-distribution to show
P.1 holds and prove Proposition 13.Let us first see where the asymptotic behavior of a single-level re-distributionleads:
Proposition 20 (Single-level re-distribution) . Let l k := e − λk and λ > . Ifassumptions A.1 - A.3 are satisfied, then I (ˆ µ n ) = I ( f ( x ) φ ( x ) dx )( R f ( x ) φ ( x ) dx ) + O (cid:18) R f ( x ) φ ( x ) dx ) (cid:19) + o (1) = O (1) , where each ˆ µ n is the corresponding measure defined in (4.5) .Proof. We have that I (ˆ µ n ) = 1( µ A n ( V n )) I ( µ A n ) . Breaking down the last, we get I ( µ A n ) = 1( A n ) X k ∈A n I ( µ k ) + 1( A n ) X i = j I ( µ i , µ j ) . Applying Lemma 16 to the outer-interaction sum and Lemma 19 to theself-interaction sum, and lastly, applying Lemma 15 to the normalization µ A n ( V n ) completes the proof. (cid:3) Remark 7 tells us that using a single level re-distribution will not render S to be re-distributable no matter how many intervals are included in { I k : k ∈ A n } . We will need to take the average of q ( n ) single-level re-distributionmeasures, where q ( n ) is an integer-valued function such that q ( n ) → ∞ as n → ∞ .Let ˆ µ n be a single-level re-distribution as defined in (4.5). For each n , weconsider a multi-level re-distribution probability measure: µ m := 1 B m X s ∈B m ˆ µ s m , where B m := { , . . . , q ( m ) − } . Since each ˆ µ n is supported on V n := [ k ∈A n I k = n − [ k = n I k , then µ m is supported on V n , V n , V n , . . . , V q ( n ) − n . See Remark 4 for details.Our convex measure can now be partitioned into a new self-interactionsum and a new outer-interaction sum: I ( µ m ) = 1( B m ) X s ∈B m I (ˆ µ s m ) + 1( B m ) X s,t ∈B m s = t I (ˆ µ s m , ˆ µ t m ) . Proposition 20 tells us that I (ˆ µ n ) = O (1) . Hence, 1( B m ) X s ∈B m I (ˆ µ s m ) = 1( B m ) X s ∈B m O (1) ≤ O (1)( B m ) → m → ∞ . That is, the self-interaction sum vanishes. The outer-interaction sum giveswhat we aim:
Lemma 21.
Assume
A.1 - A.3 . As m → ∞ we have that B m ) X s = t I (ˆ µ s m , ˆ µ t m ) → I ( f ( x ) φ ( x ) dx )( R f ( x ) φ ( x ) dx ) , where the sum is over s = t and s, t ∈ B m . We will leave the proof of Lemma 21 to the end of the section. Note thatthe vanishing of the self-interaction sum and Lemma 21 gives us:
Proposition 22.
For every f ∈ C ([0 , , we have that I ( µ m ) = I ( f ( x ) φ ( x ) dx )( R f ( x ) φ ( x ) dx ) + o (1) . Notice that with Proposition 22 we can show that S is re-distributablewhen φ ( x ) ≡
1. In order to remove φ ( x ), we will need to apply Proposi-tion 22 to a continuous approximation of 1 /φ ( x ) and take an appropriatesubsequence: Proposition 23.
Suppose for each continuous function f , there exists asequence of probability measures { µ n } so that each µ n has a piecewise con-tinuous density with support V n , where V n is a finite union of disjoint I k ’swith k ≥ n with asymptotic behavior: I ( µ n ) = I ( f ( x ) φ ( x ) dx )( R f ( x ) φ ( x ) dx ) + o (1) . (4.9) Then property
P.1 holds.
OGARITHMIC CAPACITY OF RANDOM G δ SETS 23
Let us go back to show that S is re-distributable using a multi-level re-distribution before we prove Proposition 23. Proof of Proposition 13 .
Since φ ( x ) > P.1 holds.Given any probability measure ν with piecewise continuous density thatis supported on a finite collection of intervals in [0 ,
1] and given any ε >
P.1 to a continuous L approximation of the density functionof ν to show that there exists a probability measure ν ′ satisfying properties(1) and (2) in Definition 2. Thus, S is re-distributable. (cid:3) Now, let us analyze the asymptotic behavior of the new outer-interactionsum:
Proof of Lemma 21 .
The goal is to show that I (ˆ µ n , ˆ µ n ′ ) = I ( µ A n , µ A n ′ ) µ A n ( V n ) · µ A n ′ ( V n ′ ) → I ( f ( x ) φ ( x ) dx )( R f ( x ) φ ( x ) dx ) , as m → ∞ and independently of our choice of n, n ′ ∈ { s m : s ∈ B m } . Oncewe accomplish this, then1( B m ) X s = t I (ˆ µ s m , ˆ µ t m ) → I ( f ( x ) φ ( x ) dx )( R f ( x ) φ ( x ) dx ) , as m → ∞ .Lemma 15 shows that 1 µ A n ( V n ) · µ A n ′ ( V n ′ ) → R f ( x ) φ ( x ) dx ) , as m → ∞ and independently of our choice of n, n ′ ∈ { s m : s ∈ B m } .Let us focus on I ( µ A n , µ A n ′ ). Given n = n ′ where n, n ′ ∈ { s m : s ∈ B m } ,we have that I ( µ A n , µ A n ′ ) = 1( A n )( A n ′ ) X i = j I ( µ i , µ j ) , (4.10)where ( i, j ) ∈ A n × A n ′ . By the gap control assumption A.3 , we know thatfor all large enough m and every i = j , where i, j ∈ { m, . . . , q ( m ) m − } ,we have l i + l j | c i − c j | ≤ (1 − e − ε ) , which is the needed condition (4.8) to apply Lemma 17 for all large enough m . Lemma 17 gives us | I ( µ i , µ j ) − ( − log | c i − c j | ) f ( c i ) f ( c j ) | ≤ (2 K ( − log | c i − c j | ) + K ) ε, for every i = j where i, j ∈ { m, . . . , q ( m ) m − } and for all large enough m .Adding this up over ( i, j ) ∈ A n × A n ′ and then dividing by ( A n )( A n ′ )gives us: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A n )( A n ′ ) X i = j I ( µ i , µ j ) − A n )( A n ′ ) X i = j ( − log | c i − c j | ) f ( c i ) f ( c j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Kε ( A n )( A n ′ ) X i = j ( − log | c i − c j | )+ K ε. We remark that the inequality holds independently of our choice of n, n ′ ∈{ s m : s ∈ B m } for all large enough m . By the distribution assumption A.1 and the log-average spacing assumption
A.2 , we can apply Lemma 14 twiceto the last two sums above. One application yields:1( A n )( A n ′ ) X i = j ( − log | c i − c j | ) f ( c i ) f ( c j ) → I ( f ( x ) φ ( x ) dx ) , as m → ∞ with n, n ′ ≥ m . For the second application we take f = 1 inLemma 14 to get 1( A n ) X i = j ( − log | c i − c j | ) → I ( φ ( x ) dx ) < ∞ , as m → ∞ with n, n ′ ≥ m . Therefore, for n, n ′ ∈ { s m : s ∈ B m } , we have (cid:12)(cid:12)(cid:12) lim m →∞ I ( µ A n , µ A n ′ ) − I ( f ( x ) φ ( x ) dx ) (cid:12)(cid:12)(cid:12) ≤ KεI ( φ ( x ) dx ) + K ε. Since ε > I ( φ ( x ) dx ) < ∞ , then for n, n ′ ∈ { s m : s ∈B m } , we have (cid:12)(cid:12)(cid:12) lim m →∞ I ( µ A n , µ A n ′ ) − I ( f ( x ) φ ( x ) dx ) (cid:12)(cid:12)(cid:12) = 0 . Therefore, as m → ∞ , then I (ˆ µ n , ˆ µ n ′ ) → I ( f ( x ) φ ( x ) dx )( R f ( x ) φ ( x ) dx ) , where n, n ′ ∈ { s m : s ∈ B m } , which completes the proof. (cid:3) Proof of Proposition 23 .
For every continuous function h , set f ( x ) := h ( x ) φ ( x ) , OGARITHMIC CAPACITY OF RANDOM G δ SETS 25 when φ = 0 and 0 otherwise. For each ε >
0, there exists a continuousfunction f ′ such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I ( f ( x ) φ ( x ) dx )( R f ( x ) φ ( x ) dx ) − I ( f ( x ) φ ( x ) dx )( R f ( x ) φ ( x ) dx ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε/ . Applying (4.9) to f , gives us that for each ε >
0, there exists N such thatfor every n ≥ N we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I ( f ( x ) φ ( x ))( R f ( x ) φ ( x ) dx ) − I ( µ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε/ , where each µ n is a probability measure with a piecewise continuous densitywith support in V n , where V n is a finite unions of disjoint I k ’s with k ≥ n .Since f ( x ) φ ( x ) = h ( x ) a.e., then for every ε >
0, there exists n ∈ N as largeas we need such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I ( µ n ) − I ( h ( x ) dx )( R h ( x ) dx ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I ( µ n ) − I ( f ( x ) φ ( x ))( R f ( x ) φ ( x ) dx ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I ( f ( x ) φ ( x ))( R f ( x ) φ ( x ) dx ) − I ( f ( x ) φ ( x ))( R f ( x ) φ ( x ) dx ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) <ε. Hence, there exists a subsequence n k and probability measures µ n k withpiecewise continuous density supported in V n k such that I ( µ n k ) = I ( h ( x ) dx )( R h ( x ) dx ) + o (1) . (4.11)Each µ n k is a probability measure with a piecewise continuous density withsupport contained in V n k , that is a finite union of disjoint I j ’s with j ≥ k .Hence, these { µ n k } satisfy P.1 . (cid:3) Acknowledgements
I would like to thank Victor Kleptsyn for numerous discussions and forthe idea of the proof of Lemma 11, as well as for reading multiple drafts ofthis paper and offering many helpful suggestions, and Anton Gorodetski forhis remarks.
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Department of Mathematics, University of California, Irvine
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