On U- and M-sets for series with respect to characters of compact zero-dimensional groups

J. Math. Anal. Appl. 446 (2017) 383–394

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

On U - and M -sets for series with respect to characters of compact zero-dimensional groups Natalia Kholshchevnikova b , Valentin Skvortsov a,∗ a b

Department of Mathematics, Moscow State University, 119991 Moscow, Russia Department of Mathematics, MSTU “STANKIN”, Moscow, Russia

a r t i c l e

i n f o

Article history: Received 13 June 2016 Available online 3 September 2016 Submitted by S. Tikhonov Keywords: Compact zero-dimensional abelian group Characters of a group U -set M -set Category problem Hausdorff h-measure

a b s t r a c t Generalizing our previous results on Walsh system, we consider the system of characters of zero-dimensional compact abelian group with the second axiom of countability and give for the sets of uniqueness with respect to this system a solution of category problem. We also obtain a generalization of a theorem on existence for this system a perfect M0 -set whose Hausdorff h-measure equals zero. © 2016 Elsevier Inc. All rights reserved.

1. Introduction This paper is related to the problem concerning the uniqueness of an expansion of a function in an orthogonal series. One of the most fascinating part of this topic is the theory of sets of uniqueness which started with Cantor’s research into the nature of exceptional sets for which the uniqueness holds in the case of the trigonometric system. That research led Cantor to the creation of set theory. Later investigations revealed deep relationship between this theory of uniqueness sets and other areas of mathematics such as descriptive set theory, functional analysis, measure theory, number theory. We recall that a set E in the domain of a system of functions is called a set of uniqueness or U -set for this system if the convergence of a series with respect to the system to zero outside the set E implies that all coefficients of the series are zero. Otherwise the set is called a set of multiplicity or M -set, i.e., a set E is said to be M -set for a system of functions if there exists a non-trivial series with respect to this system which converges to zero outside the set E. * Corresponding author. E-mail addresses: [email protected] (N. Kholshchevnikova), [email protected] (V. Skvortsov). http://dx.doi.org/10.1016/j.jmaa.2016.08.062 0022-247X/© 2016 Elsevier Inc. All rights reserved.

384

N. Kholshchevnikova, V. Skvortsov / J. Math. Anal. Appl. 446 (2017) 383–394

A set is called a set of restricted multiplicity or M0 -set for the considered system of functions if it is an M -set and the corresponding non-trivial series is a Fourier–Stieltjes series of some measure defined on the domain of the functions of the system. Otherwise the set is called a set of extended uniqueness or U0 -set. A good survey of the theory of sets of uniqueness and sets of multiplicity for the trigonometric system is given in [5] (see also [18]). The development of this theory was stimulated by some difficult long-standing problems. One of them was the so-called category problem. It is not difficult to verify that in the trigonometric case as well as in the case of many classical orthogonal systems every measurable U -set of uniqueness has measure zero. A topological analogue of σ-ideal of sets of measure zero is the class of sets of the first category (meager sets). So the corresponding question in the topological terms would be: should every U -set with Baire property (an analogue to the measurability of sets) be of the first category. In the trigonometric case it was an open problem for many decades, since 20th, which eventually was solved by Debs and Saint Raymond using methods of Kechris–Louveau–Wooding [5] as well as results of Piatetski-Shapiro [9]. Here we consider the system of characters of arbitrary zero-dimensional compact abelian group with the second axiom of countability (i.e., with topology given by a countable base). For various results in Fourier Analysis on zero-dimensional groups see, e.g., [1,4,8,14]. The most known examples of such a group are the Cantor dyadic group for which the dual group is the classical Walsh system (see [3,11,16]) and the group of p-adic integers (see [17]). We give a solution of the category problem for this system of characters by adjusting for our setting some methods used in the trigonometric case in [5] and in the Walsh case in [6], and also using some technical methods of [13]. Another problem we are concerned with here is how “thin” M0 -set can be. Menshov constructed a classical example of M0 -set of measure zero for trigonometric series. Ivashov-Musatov showed that this set can be of Hausdorff h-measure zero (see [5]). An analogue of this result for the Walsh system was obtained in [12] and generalized in [6]. It was extended to the case of the considered here system of characters in [13]. Generalizing the last result we show here that a closed M0 -set with zero Hausdorff h-measure can be constructed inside of any closed M0 -set. This theorem gives also a shorter and more elegant proof for the result of [13]. In Section 2 we recall some definitions and facts on the structure of a zero-dimensional compact abelian group and on the properties of series with respect to the group of characters of this group. In section 3 we prove some technical results related to Rajchman measures on a zero-dimensional group and obtain some properties of M0 -sets. The main results on category problem and on M0 -set of zero Hausdorff h-measure are obtained in Section 4. 2. Preliminaries Consider a zero-dimensional compact abelian group G with the second axiom of countability. It is known (see [1]) that a topology in such a group can be given by a chain of subgroups G = G0 ⊃ G1 ⊃ . . . ⊃ Gn ⊃ . . . , +∞ with {0} = n=0 Gn . The subgroups Gn are clopen sets with respect to this topology. As G is compact, the quotient groups G/Gn are finite for each n. Let pn be the order of the group Gn /Gn+1 . Then the order of G0 /G1 is p0 , and the order of G0 /Gn , n = 1, 2, ..., is mn = p0 · p1 · . . . · pn−1 with pi ≥ 2 for all i = 0, 1, 2, . . . (we put m0 = 1). We denote by Kn any coset of the subgroup Gn in G/Gn and by Kn (x) the coset which contains the element x, i.e.,

N. Kholshchevnikova, V. Skvortsov / J. Math. Anal. Appl. 446 (2017) 383–394

385

Kn (x) := x + Gn .  For any x ∈ G the sequence {Kn (x)} is decreasing and {x} = n Kn (x). It will be convenient to use the terminology of derivation basis theory (see for example [14]) and to refer to any coset Kn , n = 0, 1, 2, . . . , as a B-interval of rank n. We numerate all the B-intervals of rank n denoting them by Knj , j = 0, 1, . . . , mn − 1. Then

G=

m n −1

Knj .

(1)

j=0

We denote by I the algebra generated by family of all B-intervals. Note that the collection of all B-intervals forms a countable base for the considered here topology on G. It is obvious that G with this topology is a Hausdorff space. Being also compact, G is a normal space (see [2, Th. 2.48]). Let λ denote the normalized Haar measure on G with λ(G) = 1. As λ is translation invariant, then by (1) for any n ≥ 0 λ(Gn ) = λ(Kn ) =

1 . mn

(2)

Define B-derivative of a function F : I → C at a point x ∈ G by DB F (x) := lim

n→∞

F (Kn (x)) . λ(Kn (x))

We say that F is B-differentiable at x ∈ G if its B-derivative DB F (x) exists and is finite. Some theorems on recovering a function from its B-derivative (see for example [14] and [15]) imply Lemma 1. If a function F : I → C is additive on I and DB F (x) = 0 at any x ∈ K ∈ I then F (I) = 0 for any B-interval I ⊂ K. We say that F : I → C is B-continuous at x ∈ G if lim F (Kn (x)) = 0.

n→∞

Let Γ denote the dual group of G, i.e., the group of characters of the group G. It is known that under the assumptions imposed on G the group Γ is a discrete abelian group with respect to the point-wise multiplication of characters (see [1]). Moreover we can represent it as a union of increasing sequence of subgroups of finite order: Γ0 ⊂ Γ1 ⊂ Γ2 ⊂ . . . ⊂ Γn ⊂ . . . . +∞ Then Γ = i=0 Γi and Γ0 consists of one character γ0 with γ0 (x) = 1 for all x ∈ G. For each n ∈ N the group Γn is the annihilator of Gn , i.e., Γn = G⊥ n = {γ ∈ Γ : γ(x) = 1 for all x ∈ Gn }. The following lemma can be easily proved. Lemma 2. [14]. If γ ∈ Γn then γ is constant on each B-interval of rank n.

386

N. Kholshchevnikova, V. Skvortsov / J. Math. Anal. Appl. 446 (2017) 383–394

⊥ The quotient groups Γn+1 /Γn = G⊥ n+1 /Gn and Gn /Gn+1 are isomorphic (see [1]) and so they are of the same order pn for each n ∈ N. This implies that the group Γn has mn = p0 · p1 · . . . · pn−1 elements and the group Γ is countable and forms an orthonormal system on G with respect to λ. We can introduce a numeration of the characters in the following way. Having put γ0 = 1, we pick up a character γ ∈ Γn+1 \ Γn for each n = 0, 1, 2, ... and denote it by γmn . For

n=

kn 

0 ≤ tk ≤ pk − 1

t k mk ,

k=0

define γn :=

kn 

(γmk )tk .

(3)

k=0

This is so called Vilenkin–Paley numeration. The functions γ belonging to Γk \ Γk−1 are called the characters of rank k. In fact γn is of rank k if mk−1 ≤ n ≤ mk − 1. According to Lemma 2 all the characters of rank k are constant on B-intervals of rank k. If P is a polynomial with respect to Γ then its rank is defined as the maximal rank of its terms. The following statement can be easily checked. Lemma 3. Let G = ∪nj=1 Ij where Ij are disjoint B-intervals. Then any function which is constant on each Ij can be represented as a polynomial with respect to the system of characters and the rank of this polynomial is equal to the maximal rank of intervals Ij . Dirichlet kernels with respect to the system of characters satisfy the equality (see [1])

Dmn (x) =

m n −1

 γj (x) =

j=1

mn 0

if x ∈ Gn , if x ∈ G \ Gn .

(4)

We can consider a series 

aγ γ

(5)

γ∈Γ

k−1 with respect to the system Γ in the above numeration with partial sums Sk := i=0 ai γi . We suppose that coefficients an are complex numbers. An important subsequence of partial sums of this series is Smn =

 i:γi ∈Γn

ai γ i =

m n −1

ai γ i

i=0

By Lemma 2 sums Sk with k ≤ mn are constant on each Kn . Let M denote the class of complex Borel measures on G. One of the main objects of our investigation here will be Fourier–Stieltjes series of a measure μ ∈ M, i.e., a series (5) with coefficients given by formula μ ˆ(k) =

γk dμ, k = 0, 1, 2, .... G

Using (4) it is easy to check that for partial sums of Fourier–Stieltjes series of a measure μ ∈ M formula

N. Kholshchevnikova, V. Skvortsov / J. Math. Anal. Appl. 446 (2017) 383–394

Smn (x, μ) =

Dmn (x + y)dμ(y) = mn

G

dμ =

μ(Kn (x)) , λ(Kn (x))

387

(6)

Kn (x)

holds at each point x ∈ G. The next lemma is a direct corollary of the equality (6). Lemma 4. The partial sums Smn (x, μ) converge to a function f at a point x if and only if the measure μ is B-differentiable at x and its B-derivative is equal to f (x). The following lemma gives a sufficient condition for B-continuity of measure μ. Lemma 5. Let the coefficients aγ of Fourier–Stieltjes series of a measure μ satisfy the condition lim {sup |aγ | : γ ∈ Γn+1 \ Γn }) = 0

n→∞

(in fact this means that the coefficients of the series converge to zero for any ordering of characters inside Γn+1 \ Γn ). Then the measure μ is B-continuous at each point x ∈ G. Proof. Recall that the order of the subgroup Γn is mn . For a fixed x ∈ G, having in mind (2) and the equality |γ(x)| = 1, we get |Smn (x, μ)|λ(Kn (x)) ≤

1  |aγ |. mn γ∈Γn

The right-hand side of the above inequality converges to zero as a sequence of arithmetic means of a sequence convergent to zero. This together with (6) implies B-continuity of the measure μ. Theorem 1. [13]. If for the series (5) with coefficients tending to zero some subsequence of partial sums of the form Smnk (x) converges to zero for all x on some open set O, then this series converges to zero on O. 3. Rajchman measures Denote by P the class of probability Borel measures on G and by M0 the class of measures μ ∈ M on G for which the sequence of Fourier–Stieltjes coefficients μ ˆ := {ˆ μ(0), μ ˆ(1), . . . , μ ˆ(k), . . .} is an element of c0 , i.e., the Banach space of sequences converging to zero, with sup norm. By Rajchman measure we call a measure from the class R := P ∩ M0 . The total variation |μ| of a complex Borel measure μ we define first on I. In this case the total variation |μ| of μ on a set E ∈ I can be defined as |μ|(E) = sup

n 

|μ(Ii )|,

i=1

where supremum is taken over all finite partitions of the set E into disjoint B-intervals Ii . It is easy to check that |μ| is an additive function on I. Then in a standard way it can be extended to be a positive Borel measure on G.

388

N. Kholshchevnikova, V. Skvortsov / J. Math. Anal. Appl. 446 (2017) 383–394

We recall that the support of a measure μ, denoted by supp(μ), is defined to be the complement of the largest open set O ⊂ G, such that |μ|(O) = 0. Let M(E), M0 (E), P(E), R(E), denote the classes of respective measures supported by sets contained in E ⊂ G. Theorem 2. A closed set E ⊂ G is an M0 -set if and only if there exists non-zero B-continuous measure μ ∈ M0 (E). Proof. Let E be an M0 -set. If E = G then we can put μ = λ (it is clear that λ ∈ M0 (G)). If E = G then there exists a Fourier–Stieltjes series of a measure μ ∈ M which converges to zero outside E. As |γ(x)| = 1 for all characters and for all x, then the convergence of the series implies that μ ˆ ∈ c0 , i.e., μ ∈ M0 . Applying Lemmas 1 and 4 we get μ(I) = 0 for any B-interval I ⊂ G \ E. So measure μ is supported by a set contained in E. B-continuity of μ follows from Lemma 5. To prove the converse, consider a nonzero measure μ ∈ M0 (E). There exists a B-interval I with μ(I) = 0.

Its characteristic function χI is a polynomial, according to Lemma 3. Then G χI dμ is a nonzero linear combination of coefficients of the Fourier–Stieltjes series of μ. So there exist nonzero coefficients of this series. The convergence to zero on G \ E of the Fourier–Stieltjes series of μ follows from Lemma 4 and Theorem 1. So E is an M0 -set. By adapting to our case a proof used in the trigonometric setting (see [18, Ch. XII, Th. (10.2)]), we prove the following Theorem 3. If μ ∈ M0 (E), where E ⊂ G is closed, then also |μ| ∈ M0 (E). Proof. It is clear that |μ| has the same support as μ. Fix ε > 0 and find a partition {Ii }pi=1 of G such that |μ|(G) −

p 

|μ(Ii )| < ε.

(7)

i=1

 We estimate the coefficients |μ|(k) = G γk d|μ| under condition that the rank of a character γk is big enough. More precisely, let the rank n of γk be greater than the maximal rank of intervals in the partition (7). Then γk is constant on each of the B-intervals Knj . Denote these values by γk (Knj ). Each Knj is a subinterval of i )| some Ii . Define a function P (x) := |μ(I μ(Ii ) = sgn(μ(Ii )), x ∈ Ii , i = 1, 2, . . . , p (so we agree that P (x) = 0 if μ(Ii ) = 0). According to Lemma 3 this function is a polynomial with respect to Γ. It is constant on each Ii , q i = 1, 2, . . . , p, and by this also on each Kni . Let P = j=1 cj γj and denote by P (Kni ) the constant value of P on Kni . Consider sk :=

γk P dμ =

q 

cj

j=1

G

γk γj dμ. G

Let γk γj = γl(k,j) . It is easy to see from (3) that l(k, j) → ∞ together with k. So sk is a linear combination of q Fourier–Stieltjes coefficients μ ˆ(l) of μ ∈ M0 , and l → ∞. Hence lim sk = 0.

k→∞

(8)

 Now we estimate the difference |μ|(k) − sk . Having in mind (7), additivity of |μ| and the fact that the j j j difference |μ|(Kn ) − P (Kn )μ(Kn ) is always non-negative, we get

N. Kholshchevnikova, V. Skvortsov / J. Math. Anal. Appl. 446 (2017) 383–394

 ||μ|(k) − sk | ≤



389

|γk (Knj )(|μ|(Knj ) − P (Knj )μ(Knj ))| =

j

=

  i

(|μ|(Knj ) − μ(Knj )

j j:Kn ⊂Ii

 |μ(Ii )| )= (|μ|(Ii ) − |μ(Ii )|) = μ(Ii ) i = |μ|(G) −

p 

|μ(Ii )| < ε.

i=1

 This estimation, together with (8), proves that |μ|(k) → 0 with k → ∞. Corollary 1. A closed set E ⊂ G is an M0 -set if and only if there exists a measure μ ∈ R(E).  denote the image of the injection μ → μ Let M ˆ from M into l∞ (the space of all complex bounded and R. Note that we can consider R as sequences with the sup norm). In the same way we use notation P a subset of c0 . μn } converges in Lemma 6. Let {μn } be a sequence of elements of R such that the correspondent sequence {ˆ the norm c0 to a ∈ c0 . Then there exists μ ∈ R such that μ ˆ = a and for some increasing subsequence {nj } and for any B-interval I lim μnj (I) = μ(I).

(9)

j→∞

Proof. As the family of all B-intervals is countable and all values of the measures μn are bounded by 1, the diagonal procedure yields a subsequence {nj } such that limj→∞ μnj (I) exists for each B-interval I, and we denote it by μ(I). In particular μ(G) = limj→∞ μnj (G) = 1. It is clear that B-interval function μ is an additive measure on the semiring of B-intervals and can be extended to a measure μ ∈ P. To show that it is a Rajchman measure with μ ˆ = a, fix a character γk . Let n be the rank of γk . Recall that γk is constant i on each of the B-intervals Kn with values γk (Kni ). Then by (9) we have for each fixed k lim μ ˆnj (k) = lim

j→∞

γk dμnj = lim

j→∞

j→∞

G

=

m n −1 i=0

m n −1

γk (Kni )μnj (Kni ) =

i=0

γk (Kni )μ(Kni ) =

γk dμ = μ ˆ(k). G

Therefore μ ˆ = a and so μ ˆ ∈ R. is closed in c0 . Note that this lemma in particular implies that R We consider now some subclass of closed M0 -sets. We say that a nonempty closed set E is an M0p -set if for any open set V ⊂ G, with V ∩ E = ∅, the set V ∩ E is also an M0 -sets. Lemma 7. A nonempty closed set E is an M0p -set if and only if there exists a measure μ ∈ R(E) such that supp(μ) = E. Proof. Let μ ∈ R(E) with supp(μ) = E. If V ⊂ G is an open set and V ∩ E = ∅, then there exists a B-interval I ⊂ V with I ∩ E = ∅. Define a measure ν by ν(A) := μ(A ∩ I) for any Borel set A. So q dν = χI (x)dμ. According to Lemma 3 χI is a polynomial P = j=1 cj γj . Then

N. Kholshchevnikova, V. Skvortsov / J. Math. Anal. Appl. 446 (2017) 383–394

390

νˆ(k) =

γk dν =

G

γk P dμ =

q  j=1

G

cj

γk γj dμ. G

Now, as it was done in the proof of Theorem 3, we get that the right-hand side above tends to zero with k → ∞. Besides supp(ν) = ∅ and so νˆ(0) = 0. Hence the closed set I ∩ E is an M0 -set and this implies that E is an M0p -set. For the necessity consider an M0p -set E and a B-interval I. If I ∩ E = ∅ then I ∩ E is an M0 -set and by Corollary 1 there exists μ ∈ R(I ∩ E) supported by some set E  ⊂ I ∩ E. Let {In } be a sequence of all ∞ B-intervals with In ∩E = ∅ and with corresponding sets En supporting measures μn . Put μ := n=1 2−n μn . It is clear that for each k μ ˆ(k) =

∞ 

2−n μ ˆn (k)

n=1

and limk→∞ μ ˆ(k) = 0. Hence μ ∈ R(E). Besides, μ is supported by closure of ∪n En which coincides with E because the family {In ∩ E} forms a basis of topology in E. Lemma 8. Let a nonempty closed set E ⊂ G be an M0p -set and let a set V be dense and open in E in the induced topology of E. Then R(E) ⊂ R(V ),

(10)

where sets R(E), R(V ) and the closure of R(V ) are considered with topology of c0 , i.e., (10) means that for any measure μ ∈ R(E) and any ε > 0 there exists a measure ν ∈ R(V ) such that ||ˆ μ − νˆ||c0 < ε. Proof. We show first that R(V ) is dense in R(E) with respect to the weak topology of c0 . Put  A=

ϕ:ϕ=

∞  k=0

If ϕ ∈ A, ϕ =

∞ k=0

ak γ k ,

∞ 

|ak | < ∞ .

k=0

ak (ϕ)γk then a(ϕ) = (a0 (ϕ), a1 (ϕ), . . . , ak (ϕ), . . .) ∈ l1 .

∞ If we define ||ϕ||A := k=0 |ak |, then the space A is isometrically isomorphic to l1 = c∗0 . For any μ ∈ M0 and ϕ ∈ A we can identify a(ϕ) with a functional on c0 and compute its value at μ ˆ: a(ϕ), μ ˆ =

∞  k=0

μ ˆ(k)ak (ϕ) = lim

N →∞

N  k=0

μ ˆ(k)ak (ϕ) = lim

N →∞

 N G k=0

ak (ϕ)γk dμ =

ϕdμ.

(11)

G

Now fix any μ ∈ R(E) and the respective μ ˆ ∈ c0 . Take ε > 0 and fix ϕ1 , ..., ϕp ∈ A, which define a neighborhood of μ ˆ in the weak topology of c0 : Ua(ϕ1 ),...,a(ϕp ),ε = {b ∈ c0 : |a(ϕ1 ), b − μ ˆ| < ε, . . . , |a(ϕp ), b − μ ˆ| < ε}. Choose N such that

(12)

N. Kholshchevnikova, V. Skvortsov / J. Math. Anal. Appl. 446 (2017) 383–394 ∞ 

|ak (ϕi )| ≤

k=N +1

ε , i = 1, 2, . . . , p. 2

391

(13)

Put m αm = μ(KN ), m = 0, 1, . . . , mN − 1.

(14)

mN −1 m Then m=0 αm = 1. By Lemma 2 γk with 0 ≤ k ≤ N < mN are constant on each KN . So for any m ξm ∈ KN we can write γk dμ =

m N −1

m γk (ξm )μ(KN ).

m=0

G

If αm = 0 then we can choose ξm so that m ξm ∈ V ∩ KN .

Then there exists a B-interval Jm of rank greater than N such that m ξm ∈ Jm ∩ E ⊂ V ∩ KN

and γk (x) = γk (ξm ) for x ∈ Jm (0 ≤ m ≤ mN − 1, αm = 0, 0 ≤ k ≤ N ).

(15)

The definition of M0p -set implies that the closed set Jm ∩ E is an M0 -set. Hence, by Corollary 1, there exists a Rajchman measure νm ∈ R(Jm ∩ E), Jm ∩ E ⊂ V ∩ E. Note that by (15) γk (x)dνm = γk (ξm ). G

Let 

σ :=

αm νm .

0≤m≤mN −1, αm =0

Then σ(G) =

mN −1 m=0

αm = 1 and so σ ∈ R(V ). We have γk dσ =





αm γk (ξm ) =

0≤m≤mN −1, αm =0

G

γk dμ G

for 0 ≤ k ≤ N . By this, by formula (11) and by condition (13) we get           N N      ϕi dμ − ϕi dσ  ≤  ak (ϕi )γk dμ − ak (ϕi )γk dσ  +       k=0  G G G G k=0         ∞ ∞       +  ak (ϕi )γk dμ +  ak (ϕi )γk dσ  <  k=N +1   k=N +1  G

G

ε ε < + = ε, i = 1, 2, . . . , p. 2 2

392

N. Kholshchevnikova, V. Skvortsov / J. Math. Anal. Appl. 446 (2017) 383–394

Hence        |a(ϕi ), μ ˆ − a(ϕi ), σ ˆ | =  ϕi dμ − ϕi dσ  < ε, i = 1, 2, . . . , p.   G

G

). So we have proved that in an arbitrary weak neighborhood (12) of μ ˆ there exists an element σ ˆ ∈ R(V ) in c0 . The set R(V ) is obviously convex in c0 . Then This means that μ ˆ belongs to the weak closure of R(V  by known Mazur theorem (see [10, Th. 3.12]), the weak closure of R(V ) coincides with its strong closure in c0 . Therefore μ ∈ R(V ) and so R(E) ⊂ R(V ) (where the inclusion is understood as indicated in the formulation of Lemma). Lemma 9. If a nonempty closed set E ⊂ G is an M0p -set and O ⊂ E is a dense Gδ -set in E (in the topology of E induced from G), then O contains a closed M0 -set and so O is an M0 -set. Proof. We have O = ∩∞ n=1 On , where On are open and dense in E. By Lemma 7 there exists a measure μ0 ∈ R(E). According to Lemma 8 we have an inclusion R(E) ⊂ R(O1 ) and so there exists μ1 ∈ R(O1 ) such that ||ˆ μ0 − μ ˆ1 ||c0 <

1 . 2

We proceed by induction. Suppose that measures μ0 , μ1 , μ2 , . . . , μn and closed sets E = E1 , E2 , . . . , En are defined so that for k = 1, 2, . . . , n the following conditions are satisfied: Ek is an M0p -set,

(16)

Om ∩ Ek (m = 1, 2, 3, . . .) are open and dense in Ek ,

(17)

μk ∈ R(Ok ∩ Ek ),

(18)

1 , 2k (k = 2, 3, ..., n).

ˆk ||c0 < ||ˆ μk−1 − μ Ek ⊂ Ek−1 ∩ Ok−1

(19) (20)

Now we define μn+1 and En+1 in such a way that (16)–(20) are satisfied for k = n +1. By (18) supp(μn ) ⊂ On ∩ En . The sets supp(μn ) and En \On are disjoint and closed in En . As the group G is a normal space and the separation properties are obviously inherited by the induced topology on any subset of G, there exist open in En disjoint sets Un and Vn such that supp(μn ) ⊂ Un , En \On ⊂ Vn . Then U n ∩Vn = ∅, U n ∩(En \On ) = ∅. Hence U n ⊂ On ∩ En . We put En+1 := U n . Then (20) is satisfied for k = n + 1. Moreover supp(μn ) ⊂ En+1 ⊂ On ∩ En . We prove now that En+1 satisfies the conditions (16) and (17). Let I be a B-interval with I ∩ En+1 = ∅. As Un is open in En , there exists a B-interval J such that I ∩ En+1 ⊃ I ∩ Un ⊃ J ∩ En = ∅. By our assumption En is an M0p -set and so J ∩ En is an M0 -set. Hence I ∩ En+1 is an M0 -set and En+1 is an M0p -set. As Om ∩ En are sets which are dense in En , we have for the same B-intervals I and J: ∅ = Om ∩ En ∩ J ⊂ Om ∩ En+1 ∩ I.

N. Kholshchevnikova, V. Skvortsov / J. Math. Anal. Appl. 446 (2017) 383–394

393

So the sets Om ∩ En+1 are dense and open in En+1 . Then by Lemma 8 there exists a measure 0 = μn+1 ∈ R(On+1 ∩ En+1 ) such that ||ˆ μn − μ ˆn+1 ||c0 <

1 . 2n+1

In such a way we have defined measures μn and sets En with properties (16)–(20) for all natural n. μ−μ ˆn ||c0 = 0. By By (18), (19) and Lemma 6 there exists a measure μ ∈ R(E) such that limn→∞ ||ˆ properties (18) and (20) supp(μn+p ) ⊂ On+p ∩ En+p ⊂ On ∩ En ⊂ On . Then by condition (9) of Lemma 6 supp(μ) ⊂ On for n = 1, 2, .... Hence supp(μ) ⊂

∞ 

On = O.

n=1

Therefore according to Corollary 1 supp(μ) ⊂ O is an M0 -set and by this O is an M0 -set. 4. Category problem and M0 -sets of zero Hausdorff h-measure We recall that a subset B of a topological space has Baire property if there exists an open set O such that BO is of the first category. It is known [7] that B fas Baire property if and only if it can be represented as a union of Gδ -set and a set of the first category. Theorem 4. If a closed set E is an M0p -set and B ⊂ E is a U0 -set having Baire property in E, then B is of the first category in E. In particular each U -set having Baire property is of the first category. Proof. As B has Baire property in E it can be represented as B = O ∪ A where O is Gδ -set in E and A is of the first category in E. Moreover O is a U0 -set being a subset of B. We prove first that O can be covered by countably many closed U0 -sets. Let W be the union of all B-intervals I such that I ∩ O can be covered by countably many closed U0 -sets. We show that O ⊂ W . If not, O1 := O \ W is nonempty Gδ -set such that for any B-intervals I with I ∩ O1 = ∅ we have that I ∩ O1 is an M0 -set. This means that O1 is a nonempty M0p -set and O1 is a Gδ -set which is dense in O1 . Then by Lemma 9 the set O1 is an M0 -set ∞ contradicting the fact that O1 is a U0 -set. So O1 = O \ W = ∅ and O can be covered by a union n=1 Fn of closed U0 -sets Fn . If for some n the set Fn is dense in I ∩ E = ∅ with some B-intervals I then I ∩ Fn = I ∩ E because Fn is closed. As E is an M0p -set then I ∩ Fn is an M0 -set. But this is impossible because Fn is an U0 -set, and so all the sets Fn , n = 1, 2, . . . , are nowhere dense in E. Therefore O and also B = O ∪ A is of the first category in E. Some important subclasses of the space having Baire property deserve to be mentioned (see [7, 38.1, 39.1]): Corollary 2. Any Borel, analytic, or coanalytic U -set is of the first category. Definition 1. Let h : (0, ∞) → (0, ∞) be a nondecreasing function which satisfies h(0+) = 0 and let λ be the Haar measure on G. The Hausdorff h-measure Hh is defined by Hh (A) = sup inf



h(λ(In )) : In are B-intervals,



 In ⊃ A, λ(In ) ≤ δ .

δ>0

Now generalizing the results of [13] on existence of an M0 -set with zero Hausdorff h-measure, we show that such a set can be constructed inside of any closed M0 -set.

394

N. Kholshchevnikova, V. Skvortsov / J. Math. Anal. Appl. 446 (2017) 383–394

Theorem 5. Let E be a closed M0 -set. Then for any h as above there exists an M0 -set B ⊂ E whose Hausdorff h-measure equals zero. Proof. As E is a closed M0 -set, Corollary 1 implies that there exists μ ∈ R(E) supported by a closed set E1 ⊂ E. Then by Lemma 7 the set E1 is a M0p -set. We will construct Gδ -set B ⊂ E1 , which is dense in E1 and Hh (B) = 0. To this end take a countable set xm which is dense in E1 . Having fixed n choose for each xm a B-interval Im , of rank big enough so that ∞  1 1 and λ(Im ) ≤ h(λ(Im )) < . n n m=1

Note that Vn := E1 ∩ (∪m Im ) is an open and dense in E1 set. Then B := ∩n Vn is the required Gδ -set in the induced topology on E1 with Hh (B) = 0. According to Lemma 9 the set B contains a closed M0 -set and so it itself is an M0 -set. Corollary 3. Any set E ⊂ G of a positive Haar measure contains a closed M0 -set of Hausdorff h-measure zero (and also of zero Haar measure). Acknowledgments This research was supported by RFFI, grant no. 14-01-00417. References [1] G.N. Agaev, N.Ya. Vilenkin, G.M. Dzhafarli, A.I. Rubinstein, Multiplicative System of Functions and Harmonic Analysis on Zero-Dimensional Groups, ELM, Baku, 1981 (in Russian). [2] Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis, Springer-Verlag, Berlin, Heidelberg, 1999, 2006. [3] B.I. Golubov, A.V. Efimov, V.A. Skvortsov, Walsh Series and Transforms – Theory and Applications, Kluwer Academic Publishers, 1991. [4] D.J. Grubb, U-sets in compact, 0-dimensional, metric groups, Canad. Math. Bull. 32 (1989) 149–155. [5] A.S. Kechris, A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness, London Math. Soc. Lecture Series, vol. 128, Cambridge University Press, 1987. [6] N.N. Kholshchevnikova, On the category of U-sets for series in the Walsh system (in Russian), Mat. Zametki 53 (1993) 129–151, Engl. transl. in Math. Notes 53 (5–6) (1993) 539–554. [7] K. Kuratowski, Topology, vol. I, Academic Press, New York, 1966. [8] S.F. Lukomskii, Riesz multiresolution analysis on zero-dimensional groups, Izv. Math. 79 (2015) 153–184. [9] I.I. Piatetski-Shapiro, Supplement to the work “On the problem of uniqueness of the expansion of a function in a trigonometric series”, Mosk. Gos. Univ. Uc. Zap. Mat. 155 (1952) 54–72 (in Russian), English transl. in: Selected Works of Ilya Piatetski-Shapiro, AMS Collected Works, vol. 15, 2000. [10] W. Rudin, Functional Analysis, McGraw-Hill Inc., New York, 1991. [11] F. Schipp, W.R. Wade, P. Simon, J. Pal, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger Publishing, Ltd, Bristol and New York, 1990. [12] V.A. Skvortsov, The h-measure of M-sets for the Walsh system, Mat. Zametki 21 (1977) 335–340, Engl. transl. in Math. Notes 21 (1977) 186–189. [13] V.A. Skvortsov, On M0 -sets for series with respect to characters of compact zero-dimensional group, Tatra Mt. Math. Publ. 62 (2015) 165–174. [14] V.A. Skvortsov, F. Tulone, A Kurzweil–Henstock type integral on a zero-dimensional group and some of its applications, Czechoslovak Math. J. 58 (2008) 1167–1183. [15] V.A. Skvortsov, F. Tulone, Kurzweil–Henstock type integral in Fourier analysis on compact zero-dimensional group, Tatra Mt. Math. Publ. 44 (2009) 41–51. [16] Valentin Skvortsov, Francesco Tulone, Multidimensional dyadic Kurzweil–Henstock- and Perron-type integrals in the theory of Haar and Walsh series, J. Math. Anal. Appl. 421 (2015) 1502–1518. [17] V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, P-adic Analysis and Mathematical Physics, World Scientific Publ., 1994. [18] A. Zygmund, Trigonometric Series, vol. I, II, Cambridge Univ. Press, Cambridge, 2002.