New York Journal of Mathematics

New York J. Math. 14 (2008) 235–259.

On generators of shy sets on Polish topological vector spaces

G. R. Pantsulaia

Abstract. We introduce the notion of generators of shy sets in Polish topological vector spaces and give their various interesting applications. In particular, we demonstrate that this class contains specific measures which naturally generate implicitly introduced subclasses of shy sets. Moreover, such measures (unlike σ-finite Borel measures) possess many interesting, sometimes unexpected, geometric properties.

Contents

1. Preliminaries 235 2. Existence and uniqueness of generators 237 3. Quasigenerators in Polish topological vector spaces 243 4. On the quasifiniteness problem for Gaussian generators 246 5. On generators defined by Kharazishvili’s quasigenerators 247 6. Representation of cube null sets by P. Mankiewicz’s generator 253 7. On the geometric aspects of Preiss–Tišer generators 255 References 258

1. Preliminaries

The main goal of the present work is to construct specific Borel measures in Polish topological vector spaces which naturally generate classes of null sets playing an important role in studying the properties of a function space (see, for example, [1], [4], [6], [8], [12], [16], [20], [21], [23], [25]). In this direction, for a Polish topological vector space V, we introduce the notion of a generator of shy sets which is a Borel measure μ in V such that every

235^{ISSN 1076-9803/08}

Received December 15, 2007.

Mathematics Subject Classification. Primary 28Axx, 28Cxx, 28Dxx; Secondary 28C20, 28D10, 28D99.

Key words and phrases. Shy set, generator of shy set, Polish topological vector space.

set of μ-measure zero is a Haar null set (or a shy set). Note that since no nonzero σ-finite Borel measure possesses the above mentioned property, we have to choose measures in the class of nonzero non-σ-finite Borel measures defined on the entire spaces.

The paper is organized as follows.

In Section 2 we introduce the notion of a generator defined in a Polish topological vector space and prove that the class of generators is nonempty. We also show that in the class of generators there does not exist a generator with the property of uniqueness. We prove that every generator defined in the infinite-dimensional topological vector space is non-σ-finite. We demonstrate that a quasifinite translation-quasiinvariant Borel measure, as well as a quasifinite translation-invariant Borel measure, is a generator. Some interesting examples of generators [2], [3], and [19] are considered in this section.

In Section 3 we focus on the so-called quasigenerator μ which is a Borel measure whose every pair of shifts is equivalent or orthogonal. The main result of this section (Theorem 3.1) states that the structure of an arbitrary quasigenerator μ allows one to construct a new quasifinite generator G_{μ} such that the class of all shy sets S(μ) defined by any quasigenerator μ coincides with the σ-ideal of null sets N(G_{μ}) defined by the constructed generator G_{μ}.

Section 4 presents various examples of Gikhman–Skorokhod measures which were constructed in papers [5], [9], [10], [11], [13], [14], [24], [26]. A simple example of a Gaussian generator is given in an arbitrary Polish topological vector space. A general problem is stated whether a Gaussian generator is quasifinite.

In Section 5 we describe a construction of A.B. Kharazishvili’s quasigenerators and, using their structures, we construct an example of a semifinite quasifinite Baker’s generator in ℝ^{ℕ}.

In Section 6, A.B. Kharazishvili’s quasigenerator ν_{[0,1]^{ℕ}} is used to construct Mankiewicz’s generator G_{ν_{[0,1]^{N}}} which generates exactly the class of all cube null sets [16] in ℝ^{ℕ}. Also, we consider the generator defined by a finite-dimensional manifold in Polish topological vector space V. Here we establish that the class of all shy sets defined by any finite-dimensional manifold L ⊆ V coincides with the σ-ideal of G_{λ_{L}}-null sets, where λ_{L} denotes a Lebesgue measure concentrated on L.

In Section 7, we construct n-dimensional Preiss–Tišer generators in Banach spaces (in this context, see [20]) and establish their some interesting geometrical properties.

2. Existence and uniqueness of generators

Let V be a Polish topological vector space, by which we mean a vector space with a complete metric for which the addition and the scalar multiplication are continuous. Let B(V) be the σ-algebra of Borel subsets of V and μ be a nonzero nonnegative measure defined on B(V). We write X + a for the translation of a set X ⊆ V by a vector a ∈ V .

Definition 2.1 ([12, Definition 1, p. 221 ]). A measure μ is said to be transverse to a Borel set S ⊂ V if:

(1) There exists a compact set U ⊂ V for which 0 < μ(U) < ∞.

(2) μ(S + v) = 0 for every v ∈ V.

Definition 2.2 ([12, Definition 1, p. 222 ]). A Borel set S ⊂ V is called shy if there exists a Borel measure which is transverse to S. More generally, a subset of V is called shy if it is contained in a shy Borel set.

The class of all shy sets in V is denoted by S(V).

Definition 2.3 ([12, p. 226 ]). The complement of a shy set is called a prevalent set. We say that almost every element of V satisfies a given property if the subset of V on which the property holds is prevalent.

As Christensen [6, p. 119] notes, there is no σ-finite (equivalently, probability) measure μ such that S being shy is equivalent to μ(S) = 0. Slightly more can be said that any σ-finite measure μ must assign 0 to a prevalent set of points. On these grounds, we introduce the following:

Definition 2.4. A Borel measure μ in V is called a generator (of shy sets) in V, if

μ(X) = 0 ⇒ X ∈ S(V),

where μ denotes the usual completion of the Borel measure μ.

Definition 2.5. A Borel measure μ in V is called quasifinite if there exists a compact set U ⊂ V for which 0 < μ(U) < ∞.

Definition 2.6. A Borel measure μ in V is called semifinite if for X with μ(X) > 0 there exists a compact subset F ⊂ X for which 0 < μ(F) < ∞.

Definition 2.7. For a Borel measure μ in V and h ∈ V, the shift μ_{h} is defined by

(2.1)  μ_{h}(X) = μ(X + h)

for X ∈ B(V). We say that μ is translation-quasiinvariant if μ ∼ μ_{h} (i.e., μ is equivalent to μ_{h}) for all h ∈ V.

Definition 2.8. A Borel measure μ in V is called translation-invariant if μ_{h} = μ for all h ∈ V.

Definition 2.9. Let K be a class of measures on V. We say that a measure μ ∈ K has the property of uniqueness in the class K if μ and λ are equivalent for every λ ∈ K.

Now there naturally arises the following:

Question 2.1. Let V be a Polish topological vector space. Does there exist a generator in V?

Let V be a Polish vector space. Let L be a proper vector subspace of V. By the Axiom of Choice, we can construct a proper vector subspace F ⊂ V such that

(2.2)  L ∩ F = {0} and L + F = V,

where {0} denotes the zero of V.

In the sequel we denote by F such a class of vector subspaces F ⊂ V satisfying (2.2). We set L^{⊥} = τ(F), where τ denotes a global operator of choice. A vector subspace L^{⊥} is said to be a linear complement of the vector space L in V. In the sequel we will apply such a notation.

The next assertion gives a positive answer to Question 2.1.

Theorem 2.1. Let V be a Polish topological vector space. For every v_{0} ∈ V, there exists a semifinite inner regular generator λ such that

λ({αv_{0} : α ∈ [0, 1]}) = 1

and λ is non-σ-finite iff dim(V) ≥ 2.

Proof. Let L_{1} be the one-dimensional vector subspace defined by v_{0}. Let μ be the classical one-dimensional Borel measure in L_{1} = {αv_{0} : α ∈ ℝ} defined by μ(Y) = b_{1}(X) for Y = Xv_{0}, X ∈ B(ℝ).

We put

λ(X) = ∑_{v∈L_{1}^{⊥}} μ((X + v) ∩ L_{1})

for X ∈ B(V). First, we are to show the correctness of the definition of a functional λ. Indeed, let F be any element from the class F and assume that λ_{1} is defined by

λ_{1}(X) = ∑_{u∈F} μ((X + u) ∩ L_{1})

for X ∈ B(V)

We have L_{1} + F = L_{1} + L_{1}^{⊥} which means that

∪_{u∈F} (L_{1} + u) = ∪_{v∈L_{1}^{⊥}}(L_{1} + v) = V.

The latter equalities imply that for every u ∈ F there exists f(u) ∈ L_{1}^{⊥} such that L_{1} + u = L_{1} + f(u). It is not hard to show that f is an injective function from F to L_{1}^{⊥}, and so is f^{−1} : L_{1}^{⊥} → F. Thus, f defines a bijection between F and L_{1}^{⊥}. Since a sum of an arbitrary family of nonnegative real numbers is invariant under their permutations, for X ∈ B(V), we have

λ_{1}(X) = ∑_{u∈F} μ((X + u) ∩ L_{1}) = ∑_{f(u)∈f(F)} μ((X + f(u)) ∩ L_{1})

= ∑_{v∈L_{1}^{⊥}} μ((X + v) ∩ L_{1}) = λ(X).

The latter equality shows the definition of the functional λ is correct.

Now let us show that λ is a measure. Let (X_{k})_{k∈ℕ} be a family of pairwise disjoint Borel sets in B. Then we get

λ(∪_{k∈ℕ}X_{k}) = ∑_{v∈L_{1}^{⊥}} μ(((∪_{k∈ℕ}X_{k}) + v) ∩ L_{1}) = ∑_{v∈L_{1}^{⊥}} μ((∪_{k∈ℕ}(X_{k} + v)) ∩ L_{1})

= ∑_{v∈L_{1}^{⊥}} ∑_{k∈ℕ}μ((X_{k} + v) ∩ L_{1}) = ∑_{k∈ℕ} ∑_{v∈L_{1}^{⊥}} μ((X_{k} + v) ∩ L_{1})

= ∑_{k∈ℕ}λ(X_{k}).

The measure λ is quasifinite. Indeed, for D = [0, 1]v_{0}, we have

λ(D) = ∑_{v∈L_{1}^{⊥}} μ((D + v) ∩ L_{1}) = μ((D + 0) ∩ L_{1}) = 1.

The measure λ is translation-invariant. Indeed, for h ∈ V we have a representation h = h_{1} + h_{2} where h_{1} ∈ L_{1} and h_{2} ∈ L_{1}^{⊥}. Therefore for X ∈ B(V), we have

λ(X + h) = ∑_{v∈L_{1}^{⊥}} μ((X + h + v) ∩ L_{1}) = ∑_{v∈L_{1}^{⊥}} μ((X + h_{1} + h_{2} + v) ∩ L_{1})

= ∑_{v∈L_{1}^{⊥}} μ((X + h_{2} + v) ∩ L_{1}) = ∑_{h_{2}+v∈h_{2}+L_{1}^{⊥}} μ((X + h_{2} + v) ∩ L_{1})

= ∑_{h_{2}+v∈L_{1}^{⊥}} μ((X + h_{2} + v) ∩ L_{1}) = ∑_{s∈L_{1}^{⊥}} μ((X + s) ∩ L_{1}) = λ(X).

Now let us show that λ is a generator. Let S be a subset of V with λ(S) = 0. Since λ is a completion of λ, there exists a Borel set S′ for which S ⊆ S′ and λ(S′) = 0. Taking into account that D is a compact set in V with λ(D) = 1 and applying a simple consequence of the translationinvariance of the measure λ, stating that λ(S′ +v) = 0 for v ∈ V, we deduce that λ is transverse to a Borel set S′. This means that S′ is a Borel shy set. Finally, since S is a subset of a Borel shy set S′, we conclude that S is a shy set. One can observe that the generator λ is non-σ-finite if and only if dim(V) ≥ 2.

Let us show that the generator λ is inner regular. For this, we are to show that for all X with 0 < λ(X) < ∞ and all ϵ > 0, there exists a compact F_{ϵ} ⊆ X such that λ(X \ F_{ϵ}) < ϵ.

Since 0 < λ(X) < ∞, there exists G_{0} ⊂ L_{1}^{⊥} such that card(G_{0}) ≤ ℵ_{0} and

0 < λ(X) = ∑_{g∈G_{0}} μ((X + g) ∩ L_{1}) < ∞.

We set G_{0} = (g_{m})_{m∈N}. Let n_{0} be a natural number such that

∑_{1≤m≤n_{0}} μ((X + g_{m}) ∩ L_{1}) > λ(X) − \frac{ϵ}{2}.

For ϵ > 0 there exists a compact set F_{m} ⊂ V such that F_{m} ⊆ (X+g_{m})∩L_{1} and

μ(((X + g_{m}) \ F_{m}) ∩ L_{1}) < \frac{ϵ}{2^{m+1}}

for 1 ≤ m ≤ n_{0}.

We set F_{ϵ} = ∪_{1≤k≤n_{0}}(F_{k} − g_{k}). It is obvious that F_{ϵ} is compact in V. Finally, we get

λ(X \ ∪_{1≤k≤n_{0}}(F_{k} − g_{k})) = ∑_{g∈G_{0}} μ(((X \ ∪_{1≤k≤n_{0}}(F_{k} − g_{k})) + g) ∩ L_{1})

= ∑_{m∈N} μ(((X \ ∪_{1≤k≤n_{0}}(F_{k} − g_{k})) + g_{m}) ∩ L_{1})

= ∑_{1≤m≤n_{0}} μ(((X \ ∪_{1≤k≤n_{0}}(F_{k} − g_{k})) + g_{m}) ∩ L_{1})

+ ∑_{m>n_{0}} μ(((X \ ∪_{1≤k≤n_{0}}(F_{k} − g_{k})) + g_{m}) ∩ L_{1})

≤ ∑_{1≤m≤n_{0}} μ(((X \ (F_{m} − g_{m})) + g_{m}) ∩ L_{1})

+ ∑_{m>n_{0}} μ((X + g_{m}) ∩ L_{1})

≤ ∑_{1≤m≤n_{0}} μ(((X + g_{m}) \ F_{m}) ∩ L_{1})

+ ∑_{m>n_{0}} μ((X + g_{m}) ∩ L_{1})

≤ ∑_{1≤m≤n_{0}} \frac{ϵ}{2^{m+1}} + \frac{ϵ}{2} ≤ ∑_{m∈N} \frac{ϵ}{2^{m+1}} + \frac{ϵ}{2} = \frac{ϵ}{2} + \frac{ϵ}{2}

= ϵ.

Let us show that λ is semifinite. Indeed, if λ(X) > 0, then there exists g_{0} ∈ L_{1}^{⊥} such that 0 < μ((X + g_{0}) ∩ L_{1}) < ∞. Using the property of inner regularity of μ we deduce that there exists a compact set U ⊆ (X + g_{0}) ∩ L_{1} with 0 < λ(U) < ∞. The latter relation means that λ is semifinite. □

Theorem 2.2. Let V be a Polish topological vector space whose dimension is not equal to one. Then there does not exist a generator with the property of uniqueness in the class of all generators in V.

Proof. Let us consider two linearly independent elements v_{0} and v_{1} in V and apply the construction used in the proof of Theorem 2.1 with respect to the above-mentioned elements. Hence we will construct two generators λ_{0} and λ_{1} such that

λ_{0}({αv_{0} : 0 ≤ α ≤ 1}) = 1,
λ_{1}({αv_{0} : 0 ≤ α ≤ 1}) = 0,

which means that λ_{0} and λ_{1} are not equivalent. From the latter relations it follows that there does not exist a generator in V with the property of uniqueness. □

Theorem 2.3. Every generator in an infinite-dimensional Polish topological vector space V is non-σ-finite.

Proof. Assume the contrary and let λ be a σ-finite generator in V. Then there exists a countable family of compact sets {K_{n} : n ∈ N} such that

λ(V \ ∪_{n∈N}K_{n}) = 0.

Note that the compact set K_{n} is shy for n ∈ N (cf. [12, Fact 8, p. 225]). Following Definition 2.4, the set V \ ∪_{n∈N}K_{n} is a shy set because it is of λ-measure zero. Thus, we get that V is shy, which is a contradiction since there does not exists a Borel measure in V which is transverse to the set V. □

Remark 2.1. An n-dimensional classical Borel measure in ℝ^{n}(n ≥ 2) is an example of a σ-finite generator, but on the same space an example of such a generator which is non-σ-finite can be constructed (cf. Theorem 2.1).

Theorem 2.4. Every quasifinite translation-quasiinvariant Borel measure μ defined in a Polish topological vector space V is a generator.

Proof. Let μ(S) = 0 for S ⊂ V. Since μ(S) = 0, there exists a Borel set S′ for which S ⊆ S′ and μ(S′) = 0. By using the property of translationquasiinvariance of the Borel measure μ, we have μ(S′ + h) = 0 for all h ∈ V. Thus, μ is transverse to the Borel set S′ and therefore S′ is a Borel shy set. S is a shy set because it is a subset of the Borel shy set S′. □

Corollary 2.1. Since every translation-invariant measure is at the same time translation-quasiinvariant, we deduce that every quasifinite translationinvariant Borel measure in V is a generator.

Let ℝ be the real line and ℝ^{ℕ} stand for the space of all real-valued sequences equipped with the Tychonoff topology (i.e., the product topology). Denote by B(ℝ^{ℕ}) the σ-algebra of all Borel subsets in ℝ^{ℕ}.

Example 2.1. Let R_{1} be the class of all infinite-dimensional rectangles R ∈ B(ℝ^{ℕ}) of the form

R = ∏_{i=1}^{∞}(a_{i}, b_{i}), −∞ < a_{i} ≤ b_{i} < +∞

such that

0 ≤ ∏_{i=1}^{∞}(b_{i} − a_{i}) := lim_{n→∞} ∏_{i=1}^{n}(b_{i} − a_{i}) < ∞.

Let τ_{1} be a set function on R_{1} defined by

τ_{1}(R) = ∏_{i=1}^{∞}(b_{i} − a_{i}).

Following R. Baker [2], the functional λ defined by

λ(X) = inf{∑_{j=1}^{∞} τ_{1}(R_{j}) : R_{j} ∈ R_{1} & X ⊆ ∪_{j=1}^{∞}R_{j}}

for X ∈ B(ℝ^{ℕ}) is a quasifinite translation-invariant Borel measure in ℝ^{ℕ}. By Corollary 2.1 we deduce that λ is a generator in ℝ^{ℕ}.

Example 2.2. Let R_{2} be the class of all infinite-dimensional rectangles R ∈ B(ℝ^{ℕ}) of the form

R = ∏_{i=1}^{∞} R_{i}, R_{i} ∈ B(ℝ)

such that

0 ≤ ∏_{i=1}^{∞} m(R_{i}) := lim_{n→∞} ∏_{i=1}^{n} m(R_{i}) < ∞,

where m denotes a one-dimensional classical Borel measure on ℝ. Let τ_{2} be a set function on R_{2}, defined by

τ_{2}(R) = ∏_{i=1}^{∞} m(R_{i}).

R. Baker [3] proved that the functional μ defined by

μ(X) = inf{∑_{j=1}^{∞} τ_{2}(R_{j}) : R_{j} ∈ R_{2} & X ⊆ ∪_{j=1}^{∞}R_{j}}

for X ∈ B(ℝ^{ℕ}) is a quasifinite translation-invariant Borel measure in ℝ^{ℕ}. By Corollary 2.1 we establish that μ is a generator in ℝ^{ℕ}.

Remark 2.2. Following [12, Theorem 15.2.1, p. 204] the generator μ is absolutely continuous with respect to the generator λ, and the generators λ and μ are not equivalent. This fact gives an answer to R. Baker’s one question posed in [3].

Example 2.3 ([19, Theorem 7.1, p. 119]). Let 𝕁 be any nonempty subset of the set of all natural numbers ℕ. Then, in the Solovay model [22] there exists a translation-invariant measure μ_{𝕁} on the power set of ℝ^{𝕁} such that μ_{𝕁}([0, 1]^{𝕁}) = 1. Following Corollary 2.1, the restriction of the measure μ_{𝕁} on B(ℝ^{𝕁}) is a generator in ℝ^{𝕁}.

3. Quasigenerators in Polish topological vector spaces

For a Borel measure μ in V we denote by N(μ) a class of all subsets which are of μ-measure zero, where μ denotes a completion of μ.

Definition 3.1. A set S ⊂ V is called μ-shy if it is a subset of a Borel set S′ with μ(S′ + v) = 0 for v ∈ V. The class of all μ-shy sets is denoted by S(μ).

It is clear that

S(V) = ∪_{μ}S(μ).

If μ has atoms, then S(μ) = {∅}. This means that the measures which matter in defining shy sets are the ones without atoms (equivalently, diffused measures) (cf. [23], p. 456).

In the context of the above representation of S(V) there naturally arises the following:

Problem 3.1. Let μ be a diffused Borel measure on V. Does there exist a quasifinite generator λ in V such that the following equality

S(μ) = N(λ)

holds?

Note that Problem 3.1 is not trivial and its solution depends on the structure of the measure μ. Below we identify a subclass of diffused Borel measures (cf. Definition 3.5) for which Problem 3.1 has a positive solution.

Definition 3.2. A quasifinite Borel measure μ in V is called a Gikhman– Skorokhod measure if for each h ∈ V, the shift μ_{h} (2.1) is either orthogonal (⊥) or equivalent to μ.

Definition 3.3. Let μ be a Borel measure in V. The set of admissible translations of μ in the sense of quasiinvariance is

Q_{μ} = {h ∈ V : μ_{h} ∼ μ}.

It is clear that Q_{μ} = V iff μ is translation-quasiinvariant.

Definition 3.4. Let μ be a Borel measure in V. The set of admissible translations of μ in the sense of invariance is

I_{μ} = {h ∈ V : μ_{h} = μ}.

It can be shown that I_{μ} = V iff μ is translation-invariant.

Remark 3.1. Obviously, I_{μ} ⊆ Q_{μ} for every measure μ. Note that every quasifinite Borel measure with Q_{μ} = V or I_{μ} = V is a Gikhman–Skorokhod measure in V.

Now there naturally arises a question whether an analogous result is valid when Q_{μ} or I_{μ} are everywhere dense linear manifolds in V.

In this context I.I. Gikhman and A.V. Skorokhod considered the following problem in [10, Chapter 7, Paragraph 2]:

Does there exist a probability Borel measure μ in the Hilbert space ℓ_{2} which satisfies the following conditions?

(i) The group Q_{μ} of all admissible translations (in the sense of quasiinvariance) is an everywhere dense linear manifold in ℓ_{2}.

(ii) There exists a ∈ ℓ_{2} \ Q_{μ} such that a measure μ is not orthogonal to the measure μ^{(a)}, where

μ^{(a)}(X) = μ(X − a)

for X ∈ B(ℓ_{2}). Gikhman–Skorokhod’s positive solution of this problem employs the tech-

nique of Gaussian measures in an infinite-dimensional separable Hilbert space. In [18], Gikhman–Skorokhod’s result was extended to invariant Borel measures in ℓ_{2}. In particular, a nonzero σ-finite Borel measure μ is constructed in ℓ_{2} which satisfies the following conditions:

(iii) The group I_{μ} of all admissible (in the sense of invariance) translations

for the measure μ is an everywhere dense linear manifold in ℓ_{2}. (iv) There exists a ∈ ℓ_{2} \ I_{μ} such that a measure μ^{(a)} is not orthogonal

to the measure μ.

Definition 3.5. A quasifinite Borel measure μ in V is called a quasigenerator if the following two conditions are satisfied:

(i) Q_{μ} is a linear manifold in V.

(ii) There exists a σ-compact F such that for all u ∈ Q_{μ} and v ∈ Q_{μ}^{⊥}\{0}, μ(V \ (F ∩ (F + u))) = 0 and μ(F ∩ (F + v)) = 0.

Remark 3.2. Note that every quasigenerator is a Gikhman–Skorokhod measure, but the converse statement is not always valid. Indeed, consider an arbitrary generator λ in an infinite-dimensional Polish topological vector space V. Obviously, Q_{λ} = V, which means that λ is a Gikhman–Skorokhod measure. If we assume that λ is a quasigenerator, then there will be a σcompact set F such that λ(V \(F ∩(F +0))) = 0 because 0 ∈ Q_{λ}. Therefore, λ(V \ F) = 0, from which by Definition 2.4 it follows that V \ F ∈ S(V). So F ∈ S(V) (cf. [12, Fact 8, p. 225] ), we get V = (V \ F) ∪ F ∈ S(V), which is a contradiction.

Let μ be an arbitrary Borel measure in V. Let us define a functional G_{μ} by

G_{μ}(X) = ∑_{v∈Q_{μ}^{⊥}} μ(X + v)

for X ∈ B(V). The main result of the present section is formulated as follows.

Theorem 3.1. Let μ be an arbitrary Borel measure in V. Then G_{μ} is a generator in V such that

S(μ) = N(G_{μ}),

where G_{μ} denotes a usual completion of G_{μ}. If μ is a quasigenerator, then the generator G_{μ} is quasifinite.

Proof. Step (i) We show that if G_{μ}(X) = 0 then G_{μ}(X + h) = 0 for every h ∈ V. For h ∈ V we have the representation h = h_{1} + h_{2}, where h_{1} ∈ Q_{μ} and h_{2} ∈ Q_{μ}^{⊥}. Thus

G_{μ}(X) = 0 ⇔ ∑_{v∈Q_{μ}^{⊥}} μ(X + v) = 0 ⇔ ∑_{v∈Q_{μ}^{⊥}} μ(X + h_{1} + v) = 0

⇔ ∑_{v∈Q_{μ}^{⊥}} μ(X + h_{1} + h_{2} + v) = 0 ⇔ ∑_{v∈Q_{μ}^{⊥}} μ((X + h) + v) = 0

⇔ G_{μ}(X + h) = 0.

Step (ii) We show the measure G_{μ} is quasifinite if μ is a quasigenerator. Indeed, in such a situation there exists a σ-compact set F such that for all u ∈ Q_{μ} and v ∈ Q_{μ}^{⊥} \ {0}, μ(V \ (F ∩ (F + u))) = 0 and μ(F ∩ (F + v)) = 0.

Since μ(V \F) = 0, there exists a compact set U ⊂ F with 0 < μ(U) < ∞. Thus, μ(U + v) = 0 for all v ∈ Q_{μ}^{⊥} \ {0, as

μ(U + v) = μ((U + v) ∩ F) ≤ μ((F + v) ∩ F) = 0

for v ∈ Q_{μ}^{⊥} \ {0}.

Hence

G_{μ}(U) = ∑_{v∈Q_{μ}^{⊥}} μ(U + v) = μ(U + 0) = μ(U).

Thus G_{μ} is a quasifinite generator.

Step (iii) We show that S(μ) = N(G_{μ}). Let X ∈ S(μ). This means that there exists a Borel set X′ such that X ⊆ X′ and μ(X′ + v) = 0 for v ∈ V. The latter relation means that

∑_{v∈Q_{μ}^{⊥}} μ(X′ + v) = 0

which implies that X′ ∈ N(G_{μ}) and X ∈ N(G_{μ}).

Now let X ∈ N(G_{μ}). This means that there exists a Borel set X′, where X ⊆ X′ and

∑_{v∈Q_{μ}^{⊥}} μ(X′ + v) = 0.

The latter relation implies that μ(X′ + v) = 0 for v ∈ Q_{μ}^{⊥}.

Now let h = h_{1} + h_{2} be an arbitrary element of V, where h_{1} ∈ Q_{μ} and h_{2} ∈ Q_{μ}^{⊥}. Since μ(X′+h_{2}) = 0 and h_{1} ∈ Q_{μ}, we conclude that μ(X′+h) = 0, which means that X′ ∈ S(μ). Therefore X ∈ S(μ) because X ⊆ X′. □

4. On the quasifiniteness problem for Gaussian generators

The problem of equivalence and orthogonality relations between two measures in infinite-dimensional topological vector spaces, which underlies the notion of a quasigenerator, has been investigated by many authors. In this direction, a special mention should be made of the result of S. Kakutani [13] stating that if one has equivalent probability measures μ_{i} and ν_{i} on the σ-algebra L_{i} of subsets of a set Ω_{i}, i = 1, 2, …, and if μ and ν denote respectively the infinite product measures ∏_{i∈ℕ} μ_{i} and ∏_{i∈ℕ} ν_{i} on the infinite product σ-algebra generated on the infinite product set Ω, then μ and ν are either equivalent or orthogonal. Similar dichotomies have revealed themselves in the study of Gaussian stochastic processes. C. Cameron and W.E. Martin showed in [5] that if one considers the measures induced on a path space by a Wiener process on the unit interval, then the measures are orthogonal provided that the variances of the processes are different. Results of this kind were generalized by many authors (cf. [9], [11], [26] and others).

Let μ be a Gaussian measure in V. The question whether G_{μ} is quasifinite depends on the structure of μ. Below we give a simple example of such a generator in an infinite-dimensional Polish topological vector space V.

Example 4.1. Let e_{1}, …, e_{k} be any family of linearly independent vectors in V. We set L(e_{1}, …, e_{k}) = span(e_{1}, …, e_{k}). We denote by λ_{L(e_{1},...,e_{k})} a Gaussian Borel measure on L(e_{1}, …, e_{k}) defined by

λ_{L(e_{1},...,e_{k})}({t_{1}e_{1} + ⋯ + t_{k}e_{k} : (t_{1}, …, t_{k}) ∈ X}) = γ_{k}(X),

where X ∈ B(ℝ^{k}) and γ_{k} is a standard k-dimensional Gaussian Borel measure in ℝ^{k}.

Clearly, λ_{L(e_{1},...,e_{k})} is a quasigenerator in V and G_{λ_{L(e_{1},...,e_{k})}} is a quasifinite Gaussian generator in V such that

S(λ_{L(e_{1},...,e_{k})}) = N(G_{λ_{L(e_{1},...,e_{k})}}).

Remark 4.1. Let μ_{k} be the standard Gaussian probability Borel measure in ℝ for k ∈ ℕ. Let X = ℝ^{ℕ} and μ = ∏_{k∈ℕ} μ_{k} be the canonical Gaussian Borel probability measure in ℝ^{ℕ}. Following Kakutani [14], Q_{μ} = ℓ_{2}, which implies that μ is a Gikhman–Skorokhod measure. It follows from Theorem 3.1 that G_{μ} is a generator and

N(G_{μ}) = S(μ),

but we do not know whether G_{μ} is a quasifinite.

Remark 4.2. We remind the reader that a Borel measure μ in an infinitedimensional separable Hilbert space is called Gaussian if an arbitrary continuous linear functional ℓ_{z}(x) = (z, x)(z, x ∈ H) is a normally distributed random variable (cf. [10, Chapter v, Paragraph 6]).

Following A.M. Vershik [26], a group Q_{μ} of arbitrary Gaussian measure μ in an infinite-dimensional separable Hilbert space is linear manifold. In particular, if μ is a Gaussian measure in H with zero mean and correlation operator B, then Q_{μ} = B^{\frac{1}{2}} (H) (cf. [10]).

Following Phelps [21], a Borel set is called Gaussian null if it is null for every Gaussian measure in H. A set is called Gaussian null if it is contained in a Borel Gaussian null set.

The class of all Gaussian null sets in H is denoted by GN(H). Let Γ be a class of all Gaussian measures in H. Then the representation

GN(H) = ∩_{γ∈Γ}N(G_{γ})

is valid. The proof of this fact employs the result of Theorem 3.1.

In the context of Example 4.1, there naturally arises a question whether any Gaussian measure γ in H is a quasigenerator or whether G_{γ} is quasifinite.

5. On generators defined by Kharazishvili’s quasigenerators

The problem of the existence of a partial analog of a Lebesgue measure in an infinite-dimensional topological vector space is interesting and important in itself and has been studied for over 50 years by many authors. Among their results the result of V. Sudakov [24] should be mentioned specially. This result asserts that an arbitrary σ-finite quasiinvariant Borel measure in an infinite-dimensional locally convex topological vector space is identically zero. According to this result, the properties of the σ-finiteness and the translation-invariance are not consistent for nonzero Borel measures in infinite-dimensional topological vector spaces. A.B. Kharazishvili [15] constructed an example of a nonzero non-translation-invariant σ-finite Borel measure in the Hilbert space ℓ_{2} which is invariant with respect to an everywhere dense (in ℓ_{2}) linear manifold. R. Baker gave constructions of quasifinite non-σ-finite translation-invariant Borel measures in the infinitedimensional topological vector space ℝ^{ℕ} (cf. Examples 2.1–2.2). A similar construction (cf. [17]) is given in the well-known Solovay’s model [22].

Below we present the construction of A.B. Kharazishvili’s quasigenerators in ℝ^{ℕ}.

Let (a_{i})_{i∈ℕ} and (b_{i})_{i∈ℕ} be sequences of real numbers such that a_{i} < b_{i} for all i ∈ ℕ.

We set

A_{n} = ℝ_{0} × ⋯ × ℝ_{n} × ∏_{i>n} Δ_{i},

for n ∈ ℕ, where ℝ_{i} = ℝ for i = 1, …, n and Δ_{i} = [a_{i}, b_{i}[ for i > n. We also set

Δ = ∏_{i∈ℕ}Δ_{i} and B_{Δ} = ∪_{n∈ℕ}A_{n}.

For an arbitrary natural number i ∈ ℕ, consider the Lebesgue measure μ_{i} in the space ℝ_{i} and satisfying the condition μ_{i}(Δ_{i}) = 1. Let us denote by λ_{i} the normalized Lebesgue measure defined on the interval Δ_{i}.

For an arbitrary n ∈ ℕ, let us denote by ν_{n} the measure defined by

ν_{n} = ∏_{1≤i≤n} μ_{i} × ∏_{i>n} λ_{i},

and by ν_{n} the Borel measure in the space ℝ^{ℕ} defined by

ν_{n}(X) = ν_{n}(X ∩ A_{n})

for X ∈ B(ℝ^{ℕ}.

Lemma 5.1 ([19, Lemmas 15.3.2–15.3.3, p. 207]). The functional ν_{Δ} defined by

ν_{Δ}(X) = lim_{n→∞} ν_{n}(X),

is a quasifinite Borel measure for which

I_{ν_{Δ}} = Q_{ν_{Δ}} = \{h = (h_{1}, h_{2}, …) ∈ ℝ^{ℕ} : ∑_{i∈ℕ} \frac{|h_{i}|}{b_{i} − a_{i}} is convergent\}

and for all u ∈ Q_{ν_{Δ}} and v ∈ Q_{ν_{Δ}}^{⊥} \ {0},

μ(ℝ^{ℕ} \ (∪_{n∈N}A_{n} ∩ (∪_{n∈N}A_{n} + u))) = 0,
μ((∪_{n∈N}A_{n}) ∩ (∪_{n∈N}A_{n} + v)) = 0.

As a consequence of Lemma 5.1 we get:

Corollary 5.1. The functional ν_{Δ} is a quasigenerator in ℝ^{ℕ}.

Remark 5.1. The construction of ν_{Δ} belongs to A.B. Kharazishvili [15]. For this reason, a quasigenerator ν_{Δ} is called A.B. Kharazishvili’s quasigenerator. Note that a generator G_{ν_{Δ}} is a semifinite inner regular translationinvariant Borel measure in ℝ^{ℕ}, which assigns value one to Δ (cf. [19]).

Now let K be the class of all positive sequences (a_{k})_{k∈ℕ} ∈ ℝ^{ℕ} such that

0 < ∏_{k∈ℕ}a_{k} < ∞.

Let (a_{k})_{k∈ℕ}, (b_{k})_{k∈ℕ} ∈ K. We say that (a_{k})_{k∈ℕ} ≃ (b_{k})_{k∈ℕ} if

ν_{∏_{k∈ℕ}[0,a_{k}]} = ν_{∏_{k∈ℕ}[0,b_{k}]}.

Lemma 5.2 ([19, Lemma 15.3.3, p. 207]). The relation ≃ is an equivalence relation on K.

Lemma 5.3 ([19, Lemma 15.3.4, p. 207]). Suppose (a_{k})_{k∈ℕ}, (b_{k})_{k∈ℕ} ∈ K are not equivalent. Then ν_{∏_{k∈ℕ}[0,a_{k}]} ⊥ ν_{∏_{k∈ℕ}[0,b_{k}]}.

Definition 5.1. Suppose R_{1} is the class of all infinite-dimensional rectangles R ∈ B(ℝ^{ℕ}) of a form

R = ∏_{k=1}^{∞}(a_{i}, b_{i}), −∞ < a_{i} ≤ b_{i} < ∞

such that 0 ≤ ∏_{k=1}^{∞}(b_{k} − a_{k}) < ∞. Following [2], a translation-invariant Borel measure λ in ℝ^{ℕ} is called Baker’s generator if

λ(R) = ∏_{k=1}^{∞}(b_{k} − a_{k})

for R ∈ R_{1}.

Theorem 5.1 ([19, Theorem 15.3.1, p. 208]). There exists a semifinite inner regular Baker’s generator λ in ℝ^{ℕ}.

Proof. Let us consider the equivalence classes (K_{i})_{i∈I} of K generated by the equivalence relation ≃ (cf. [19, Lemma 15.3.3]). Let (a_{k}^{(i)})_{k∈ℕ} ∈ K_{i} for

i ∈ I. We set Δ_{i} = ∏_{k∈ℕ}[0, a_{k}^{(i)}], B_{i} = B_{Δ_{i}} and μ_{i} = ν_{Δ_{i}}. We put

λ(X) = ∑_{i∈I} ∑_{g∈ℓ_{1}^{⊥}} μ_{i}((X − g) ∩ B_{i}),

for X ∈ B(ℝ^{ℕ}), where ℓ_{1}^{⊥} denotes a linear complement of the vector subspace ℓ_{1} in ℝ^{ℕ}.

We shall show that λ is a translation-invariant Borel measure in ℝ^{ℕ} such that

λ(∏_{k=1}^{∞}(a_{k}, b_{k})) = ∏_{k=1}^{∞}(b_{k} − a_{k}).

We do this in steps.

Step (i) We show λ is σ-additive. Let (X_{k})_{k∈ℕ} be a sequence of pairwise disjoint Borel subsets in ℝ^{ℕ}. Then

λ(∪_{k∈ℕ}X_{k}) = ∑_{i∈I} ∑_{g∈ℓ_{1}^{⊥}} μ_{i}((∪_{k∈ℕ}X_{k} − g) ∩ B_{i})

= ∑_{i∈I} ∑_{g∈ℓ_{1}^{⊥}} μ_{i}((∪_{k∈ℕ}(X_{k} − g)) ∩ B_{i})

= ∑_{i∈I} ∑_{g∈ℓ_{1}^{⊥}} ∑_{k∈ℕ}μ_{i}((X_{k} − g) ∩ B_{i})

= ∑_{k∈ℕ} ∑_{i∈I} ∑_{g∈ℓ_{1}^{⊥}} μ_{i}((X_{k} − g) ∩ B_{i}) = ∑_{k∈ℕ}λ(X_{k}).

Step (ii)We show that λ is translation-invariant. Taking into account the equality G_{Δ} = ℓ_{1} (cf. [19, Lemma 15.3.2, p. 214]) and the fact that an arbitrary element h ∈ ℝ^{ℕ} can be written in the form h = h_{1} + h_{2}, where h_{1} ∈ ℓ_{1} and h_{2} ∈ ℓ_{1}^{⊥}, we get

λ(X + h) = ∑_{i∈I} ∑_{g∈ℓ_{1}^{⊥}} μ_{i}(((X + h_{1} + h_{2}) − g) ∩ B_{i})

= ∑_{i∈I} ∑_{g∈ℓ_{1}^{⊥}} μ_{i}(((X + h_{2}) − g + h_{1}) ∩ B_{i})

= ∑_{i∈I} ∑_{g∈ℓ_{1}^{⊥}} μ_{i}(((X + h_{2}) − g) ∩ B_{i})

= ∑_{i∈I} ∑_{g∈ℓ_{1}^{⊥}} μ_{i}((X − (g − h_{2})) ∩ B_{i})

= ∑_{i∈I} ∑_{g−h_{2}∈ℓ_{1}^{⊥}−h_{2}} μ_{i}((X − (g − h_{2})) ∩ B_{i})

= ∑_{i∈I} ∑_{g−h_{2}∈ℓ_{1}^{⊥}} μ_{i}((X − (g − h_{2})) ∩ B_{i})

= ∑_{i∈I} ∑_{g̃∈ℓ_{1}^{⊥}} μ_{i}((X − g̃) ∩ B_{i}) = λ(X).

Step (iii) We now show

λ(∏_{k=1}^{∞}(a_{k}, b_{k})) = ∏_{k=1}^{∞}(b_{k} − a_{k}).

Assume that K_{i_{0}} is a class of equivalence of K such that (b_{k} − a_{k})_{k∈ℕ} ∈ K_{i_{0}}. By using the translation-invariance of λ we have

λ(∏_{k=1}^{∞}(a_{k}, b_{k})) = λ(∏_{k=1}^{∞}(a_{k}, b_{k}) − (a_{k})_{k∈ℕ}) = λ(∏_{k=1}^{∞}(0, b_{k} − a_{k}))

= ∑_{i∈I} ∑_{g∈ℓ_{1}^{⊥}} μ_{i}((∏_{k=1}^{∞}(0, b_{k} − a_{k}) − g) ∩ B_{i})

= μ_{i_{0}}((∏_{i=1}^{∞}(0, b_{k} − a_{k}) − 0) ∩ B_{i_{0}}) = ∏_{k=1}^{∞}(b_{k} − a_{k}).

Finally, one can prove that if ∏_{k=1}^{∞}(b_{k} − a_{k}) = 0, then λ(∏_{k=1}^{∞}(a_{k}, b_{k})) = 0.

Step (iv) We now show the generator λ is inner regular. To this end, we must to show that for all X with 0 < λ(X) < ∞ and all ϵ > 0, there exists a compact F_{ϵ} ⊆ X with λ(X \ F_{ϵ}) < ϵ.

Since 0 < λ(X) < ∞, there exist I_{0} ⊆ I and G_{0} ⊆ ℓ_{1}^{⊥} with card(I_{0}×G_{0}) ≤ ℵ_{0} and

0 < λ(X) = ∑_{i∈I_{0}} ∑_{g∈G_{0}} μ_{i}((X − g) ∩ B_{i}) < ∞.

We set I_{0} × G_{0} = (i_{k}, g_{m})_{k,m∈ℕ}. Let n_{0} be a natural number such that

∑_{1≤k≤n_{0}} ∑_{1≤m≤n_{0}} μ_{i_{k}}((X − g_{m}) ∩ B_{i_{k}}) > λ(X) − \frac{ϵ}{2}.

For ϵ > 0 there exists a compact set F_{k,m} such that F_{k,m} ⊆ (X −g_{m})∩B_{i_{k}} and

μ_{i_{k}}(((X − g_{m}) ∩ B_{i_{k}}) \ F_{k,m}) < \frac{ϵ}{2^{k+m+1}}

for 1 ≤ k, m ≤ n_{0}.

We set F_{ϵ} = ∪_{1≤p,q≤n_{0}}(F_{p,q} + g_{q}). It is clear that F_{ϵ} ⊆ X. We get

λ(X \ F_{ϵ}) = λ(X \ ∪_{1≤p,q≤n_{0}}(F_{p,q} + g_{q}))

= ∑_{i∈I} ∑_{g∈ℓ_{1}^{⊥}} μ_{i}(((X \ ∪_{1≤p,q≤n_{0}}(F_{p,q} + g_{q})) − g) ∩ B_{i})

= ∑_{i∈I_{0}} ∑_{g∈G_{0}} μ_{i}(((X \ ∪_{1≤p,q≤n_{0}}(F_{p,q} + g_{q})) − g) ∩ B_{i})

+ ∑_{(i,g)∈I×ℓ_{1}^{⊥}\I_{0}×G_{0}} μ_{i}(((X \ ∪_{1≤p,q≤n_{0}}(F_{p,q} + g_{q})) − g) ∩ B_{i})

≤ ∑_{i∈I_{0}} ∑_{g∈G_{0}} μ_{i}(((X \ ∪_{1≤p,q≤n_{0}}(F_{p,q} + g_{q})) − g) ∩ B_{i})

+ ∑_{(i,g)∈I×ℓ_{1}^{⊥}\I_{0}×G_{0}} μ_{i}((X − g) ∩ B_{i})

= ∑_{i∈I_{0}} ∑_{g∈G_{0}} μ_{i}(((X \ ∪_{1≤p,q≤n_{0}}(F_{p,q} + g_{q})) − g) ∩ B_{i})

= ∑_{k∈ℕ} ∑_{m∈ℕ}μ_{i_{k}}(((X \ ∪_{1≤p,q≤n_{0}}(F_{p,q} + g_{q})) − g_{m}) ∩ B_{i_{k}})

= ∑_{1≤k≤n_{0}} ∑_{1≤m≤n_{0}} μ_{i_{k}}(((X \ ∪_{1≤p,q≤n_{0}}(F_{p,q} + g_{q})) − g_{m}) ∩ B_{i_{k}})

+ ∑_{(k,m)∈ℕ×ℕ\{1,...,n_{0}}×{1,...,n_{0}}}μ_{i_{k}}(((X \ ∪_{1≤p,q≤n_{0}}(F_{p,q} + g_{q})) − g_{m}) ∩ B_{i_{k}})

≤ ∑_{1≤k≤n_{0}} ∑_{1≤m≤n_{0}} μ_{i_{k}}((X \ (F_{k,m} + g_{m}) − g_{m}) ∩ B_{i_{k}})

+ ∑_{(k,m)∈ℕ×ℕ\{1,...,n_{0}}×{1,...,n_{0}}} μ_{i_{k}}((X − g_{m}) ∩ B_{i_{k}})

= ∑_{1≤k≤n_{0}} ∑_{1≤m≤n_{0}} μ_{i_{k}}(((X − g_{m}) \ F_{k,m}) ∩ B_{i_{k}})

+ ∑_{(k,m)∈ℕ×ℕ\{1,...,n_{0}}×{1,...,n_{0}}} μ_{i_{k}}((X − g_{m}) ∩ B_{i_{k}})

= ∑_{1≤k≤n_{0}} ∑_{1≤m≤n_{0}} μ_{i_{k}}(((X − g_{m}) ∩ B_{i_{k}}) \ (F_{k,m} ∩ B_{i_{k}}))

+ ∑_{(k,m)∈ℕ×ℕ\{1,...,n_{0}}×{1,...,n_{0}}} μ_{i_{k}}((X − g_{m}) ∩ B_{i_{k}})

since F_{k,m} ⊆ (X − g_{m}) ∩ B_{i_{k}} we have F_{k,m} ∩ B_{i_{k}} = F_{k,m}

= ∑_{1≤k≤n_{0}} ∑_{1≤m≤n_{0}} μ_{i_{k}}(((X − g_{m}) ∩ B_{i_{k}}) \ F_{k,m})

+ ∑_{(k,m)∈(ℕ×ℕ)\({1,...,n_{0}}×{1,...,n_{0}})} μ_{i_{k}}((X − g_{m}) ∩ B_{i_{k}})

≤ ∑_{1≤k,m≤n_{0}} \frac{ϵ}{2^{k+m+1}} + \frac{ϵ}{2}

≤ ∑_{k,m∈ℕ} \frac{ϵ}{2^{k+m+1}} + \frac{ϵ}{2} = \frac{ϵ}{2} + \frac{ϵ}{2} = ϵ.

Step (v) We show that λ is semifinite. Indeed, if λ(X) > 0, then there exist i_{0} ∈ I and g_{0} ∈ ℓ_{1}^{⊥} such that 0 < μ_{i_{0}}((X − g_{0}) ∩ B_{i_{0}}). Using the property of inner regularity of μ_{i_{0}} we deduce that there exists a compact set F ⊆ X with 0 < λ(F) < ∞. This means that λ is a semifinite measure. □

Remark 5.2. If we consider R. Baker’s measure constructed in [2], we observe that it is a nonsemifinite quasifinite Baker’s generator. Indeed, it takes the value +∞ on the set (3ℤ)^{ℕ}, while every compact subset of (3ℤ)^{ℕ} is null with respect to the same measure. Note that the set (3ℤ)^{ℕ} is null with respect to A.B. Kharazishvili’s generator λ, because μ_{i}(((3ℤ)^{ℕ}−g)∩B_{i})) = 0 for all i ∈ I and g ∈ ℓ_{1}^{⊥}.

6. Representation of cube null sets by P.

Mankiewicz’s generator

The following assertion plays key role in our further discussion.

Theorem 6.1 ([19, Theorem 15.3.2, p. 209]). There exists a semifinite inner regular translation-invariant Borel measure λ_{1} in ℝ^{ℕ} which satisfies the following conditions:

(1) λ_{1}([0, 1]^{ℕ}) = 1.

(2) There exists a rectangle ∏_{k=1}^{∞}(a_{k}, b_{k}) with 0 < ∏_{k=1}^{∞}(b_{k} − a_{k}) < ∞ such that

λ_{1}(∏_{k=1}^{∞}(a_{k}, b_{k})) = 0.

Remark 6.1. The inner regularity and semifiniteness can be proved by the scheme of the proof of Theorem 5.1.

Remark 6.2. Note that the generator λ_{1} is absolutely continuous with respect to R. Baker’s generator λ [2] but the converse relation is not valid (cf. Theorem 6.1, condition (2)). This means that the generators λ and λ_{1} are not equivalent.

Let (μ_{k})_{k∈ℕ} be a family of linear Lebesgue measures on [0, 1] such that μ_{k}([0, 1]) = 1 for all k. We set μ = ∏_{k∈ℕ} μ_{k}.

Definition 6.1. A Borel subset X ⊂ ℝ^{ℕ} is said to be a standard cube null set if μ((X + a) ∩ [0, 1]^{ℕ}) = 0. for all a ∈ ℝ^{ℕ}. More generally, a set is called a standard cube null set if it is contained in a Borel standard cube null set.

The class of all standard cube null sets in ℝ^{ℕ} is denoted by SCN (ℝ^{ℕ}). It is clear that SCN (ℝ^{ℕ}) ⊂ S(ℝ^{ℕ}).

Now naturally there arises the following:

Problem 6.1. Does there exist a semifinite inner-regular generator ν in ℝ^{ℕ} such that N (ν) = SCN (ℝ^{ℕ})?

The answer to this problem is contained in the following assertion.

Theorem 6.2 ([19, Theorem 15.3.3, p. 211]). The generator λ_{1} is a solution of Problem 6.1, i.e.,

N (λ_{1}) = SCN (ℝ^{ℕ}).

Remark 6.3. Following Theorem 6.2, the measure λ_{1} is a standard cube generator which can be called P. Mankiewicz’s generator in ℝ^{ℕ}. It is reasonable to note that P. Mankiewicz’s generator λ_{1} is defined by A.B. Kharazishvili’s standard quasigenerator ν_{[0,1]^{ℕ}} , i.e.,

λ_{1} = G_{ν_{[0,1]^{ℕ}}},

and the following equality holds:

SCN (ℝ^{ℕ}) = N (G_{ν_{[0,1]^{ℕ}}}).

Since the quasigenerator ν_{[0,1]^{ℕ}} can be defined uniquely by the Haar measure μ in [0, 1]^{ℕ}, we can use the notation G(μ) for the generator G_{ν_{[0,1]^{ℕ}}}. Such notation will be applied below.

Remark 6.4. Let λ and μ be R. Baker’s generators in ℝ^{ℕ} considered in Examples 2.1 and 2.2, respectively. Since^{1}

G_{ν_{[0,1]^{ℕ}}} ≪ μ ≪ λ,

we deduce that the following strict inclusions

N (λ) ⊂ N (μ) ⊂ N (G_{ν_{[0,1]^{ℕ}}})(= B(ℝ^{ℕ}) ∩ SCN (ℝ^{ℕ}))

are valid.

Now we consider one interesting application of Theorem 6.2 in a separable Banach space with basis.

Let B be a separable Banach space with basis e_{1}, e_{2}, … such that

∑_{k∈ℕ}||e_{k}|| < ∞.

Definition 6.2. Following P. Mankiewicz [16], a Borel set X ⊂ B is called a cube null set, if it is null for every nondegenerate cube measure. (Nondegenerate cube measures in B can be defined as distributions of the random variables of the form a + ∑_{k∈ℕ} X_{k}e_{k}, where a ∈ B and (X_{k})_{k∈N} are uniformly distributed independent random variables with values in [0, 1].)

The class of all cube null sets in B is denoted by CN S(B).

Let us denote by C the class of all cube measures defined in B. For μ ∈ C we denote by G(μ) Mankiewicz’ generator defined by μ. Then it is clear that

CN S(B) = ∩_{μ∈C}N (G(μ)).

Again let V be a Polish topological vector space.

Definition 6.3 ([25, Definition 6, p. 225]). We call a finite-dimensional subspace L ⊂ V a probe for a set X ⊂ V if an n-dimensional classical Borel measure λ_{L} supported on L is transverse to a Borel set containing the complement of X.

1Let μ and λ be two measures such that dom(μ) = dom(λ). We say that μ is absolutely continuous with respect to λ and write μ ≪ λ, if λ(X) = 0 implies μ(X) = 0.

Let λ_{L} be an n-dimensional classical Borel measure in an n-dimensional vector subspace L ⊂ V, where n ∈ ℕ. It is obvious that G_{λ_{L}} is a quasifinite translation-invariant generator in V.

It can be easily established that

N (G_{λ_{L}}) ⊂ S(V)

for all finite-dimensional vector subspace L ⊂ V.

One can show that every subset X ⊂ V is of G_{λ_{L}}-measure zero if and only if the linear manifold L is a probe for the sets V \ X. The class of all subsets of V whose complement have n-dimensional probes is denoted by n(V). It is clear that ∪_{n∈ℕ}n(V) is a subclass of the σ-ideal S(V) and we have the equality

∪_{n∈ℕ}n(V) = ∪_{L⊂V} N (G_{λ_{L}}),

where the union on the right side is considered over all finite-dimensional manifolds.

7. On the geometric aspects of Preiss–Tišer generators

Definition 7.1. A Borel set S in a separable Banach space B is said to be an n-dimensional Preiss–Tišer null set if every Lebesgue measure μ concentrated on any n-dimensional vector space Γ is transverse to S. Any subset A of such a Borel set B is said to be an n-dimensional Preiss–Tišer null set.

We denote the class of all n-dimensional Preiss–Tišer null sets in B by PT N S(B, n).

Let (Γ_{i})_{i∈I} be a family of all n-dimensional vector spaces and let μ_{i} be an n-dimensional Lebesgue measure concentrated on Γ_{i} for i ∈ I.

Let Γ_{i}^{⊥} be a linear complement of the vector space Γ_{i} in B for i ∈ I. We put

λ(X) = ∑_{i∈I} ∑_{g∈Γ_{i}^{⊥}} μ_{i}((X − g) ∩ Γ_{i})

for X ∈ B(B).

Theorem 7.1. A functional λ is a quasifinite generator in B such that

PT N S(B, n) = N (λ).

Proof. By using the construction of the proof of Theorem 5.1 one can easily show that λ is a Borel measure in B.

Now we are to show that λ is quasifinite and translation-invariant.

Let i_{0} ∈ I and let (e_{k})_{1≤k≤n} be any basis of Γ_{i_{0}}. The quasifiniteness is obvious because

0 < μ_{i_{0}}({ ∑_{i=1}^{n} α_{k}e_{k} : (α_{k})_{1≤k≤n} ∈ [0, 1]^{n}}) < ∞

and

λ({ ∑_{i=1}^{n} α_{k}e_{k} : (α_{k})_{1≤k≤n} ∈ [0, 1]^{n}})

= μ_{i_{0}}({ ∑_{i=1}^{n} α_{k}e_{k} : (α_{k})_{1≤k≤n} ∈ [0, 1]^{n}}).

Let us show that λ is translation-invariant. Indeed, let X ∈ B(B) and h ∈ B. For every i ∈ I we have h = h_{i}^{(0)} + h_{i}^{(1)}, where h_{i}^{(0)} ∈ Γ_{i} and h_{i}^{(1)} ∈ Γ_{i}^{⊥}. Then we obtain

λ(X + h) = ∑_{i∈I} ∑_{g∈Γ_{i}^{⊥}} μ_{i}(((X + h) − g) ∩ Γ_{i})

= ∑_{i∈I} ∑_{g∈Γ_{i}^{⊥}} μ_{i}((X + h_{i}^{(0)} + h_{i}^{(1)} − g) ∩ Γ_{i})

= ∑_{i∈I} ∑_{g∈Γ_{i}^{⊥}} μ_{i}((X + h_{i}^{(2)} − g) ∩ Γ_{i})

= ∑_{i∈I} ∑_{g∈Γ_{i}^{⊥}} μ_{i}((X − (g − h_{i}^{(2)})) ∩ Γ_{i})

= ∑_{i∈I} ∑_{g−h_{i}^{(2)}∈Γ_{i}^{⊥}−h_{i}^{(2)}} μ_{i}((X − (g − h_{i}^{(2)})) ∩ Γ_{i})

= ∑_{i∈I} ∑_{g−h_{i}^{(2)}∈Γ_{i}^{⊥}} μ_{i}((X − (g − h_{i}^{(2)})) ∩ Γ_{i})

= ∑_{i∈I} ∑_{s∈Γ_{i}^{⊥}} μ_{i}((X − s) ∩ Γ_{i}) = λ(X).

Now let us assume that Y ∈ PT N S(B, n). Then there exists a Borel set X such that Y ⊆ X and μ_{i}((X − g) ∩ Γ_{i})) = 0 for g ∈ Γ_{i}^{⊥}. Therefore,

∑_{i∈I} ∑_{g∈Γ_{i}^{⊥}} μ_{i}((X − g) ∩ Γ_{i})) = 0.

This means that λ(X) = 0 and we obtain that Y ∈ N (λ).

Now let λ(Y) = 0. This means that there exists a Borel set X such that Y ⊆ X and μ_{i}(X − g ∩ Γ_{i}) = 0 for g ∈ Γ_{i}^{⊥} and i ∈ I. Let Γ be an arbitrary n-dimensional vector subspace of B, and λ_{Γ} be a Lebesgue measure concentrated on Γ. We are to show that λ is transverse to X. Indeed, assume the contrary. Then for h ∈ B we have λ_{Γ}(X − h ∩ Γ) > 0. Let i_{0} be an element of I such that Γ_{i_{0}} = Γ. Then we have

λ_{Γ}((X − h) ∩ Γ) = μ_{i_{0}}((X − h) ∩ Γ_{i_{0}}) > 0.

Since h = h_{i_{0}}^{(0)} + h_{i_{0}}^{(1)}, where h_{i_{0}}^{(0)} ∈ Γ_{i_{0}} and h_{i_{0}}^{(1)} ∈ Γ_{i_{0}}^{⊥}, we get

0 < μ_{i_{0}}((X − h) ∩ Γ_{i_{0}}) = μ_{i_{0}}((X − h_{i_{0}}^{(1)}) ∩ Γ_{i_{0}})

≤ ∑_{i∈I} ∑_{g∈Γ_{i}^{⊥}} μ_{i}((X − g) ∩ Γ_{i}) = λ(X) = 0.

giving a contradiction. □

Remark 7.1. Following Theorem 7.1, the generator λ can be called an n-dimensional Preiss–Tišer generator.

Below we present one interesting property of the one-dimensional Preiss– Tišer generator in a Banach space whose dimension is larger than 1.

Theorem 7.2. Let (B, || · ||) be a Banach space whose dimension is larger than 1. Then there exists a one-dimensional Preiss–Tišer generator λ (not σ-finite) such that for an arbitrary broken line A_{0}A_{1} ⋯ defined by

A_{0}A_{1} ⋯ ≡ [A_{0}, A_{1}[ ∪ ⋯

we have

λ(A_{0}A_{1} ⋯) = ∑_{k=0}^{∞} ||A_{k+1} − A_{k}||.

Proof. Let B_{0} be the unit sphere in B, i.e.,

B_{0} = {h ∈ B : ||h|| = 1}.

For h_{1}, h_{2} ∈ B_{0} we say that h_{1} ≈ h_{2} if h_{1} and h_{2} are antipodal points of the unit sphere. It is clear that ≈ is an equivalence relation in B_{0}. Let (K_{i})_{i∈I} be the set of equivalent classes. Let (a_{i})_{i∈I} be a family of elements of B_{0} such that a_{i} ∈ K_{i} for i ∈ I. Let Γ_{i} be the one-dimensional vector subspace in B generated by the vector a_{i} for i ∈ I. Let Γ_{i}^{⊥} be a co-space of the vector space Γ_{i} for i ∈ I. Let μ_{i} be the one-dimensional classical Borel measure on Γ_{i} with

μ_{i}({ta_{i} : 0 ≤ t ≤ 1}) = 1.

We put

λ(X) = ∑_{i∈I} ∑_{g∈Γ_{i}^{⊥}} μ_{i}((X − g) ∩ Γ_{i}).

for X ∈ B(B). Now one can easily check that all conditions formulated for the measure λ in Theorem 7.2, are satisfied. □

In context of Theorem 7.2 the following assertion is of a certain interest.

Theorem 7.3. Let ℝ^{3} be a three-dimensional Euclidean vector space. Then there exists a two-dimensional Preiss–Tišer generator μ (not σ-finite) in ℝ^{3} such that for an arbitrary polyhedron P ⊂ ℝ^{3} we have

μ(B(P)) = S(B(P)),

258 G. R. Pantsulaia

where B(P) denotes the surface of the polyhedron P and S(B(P)) the surface area of P.

Proof. Let B_{0} be the unit sphere in ℝ^{3}, i.e.,

B_{0} = {h ∈ ℝ^{3} : ||h||_{3} = 1},

where ||·||_{3} denotes a usual norm in ℝ^{3}. For h_{1}, h_{2} ∈ B_{0} we say that h_{1} ≈ h_{2} if h_{1} and h_{2} are antipodal points. It is clear that ≈ is an equivalence relation in B_{0}. Let (K_{i})_{i∈I} be the set of equivalence classes. Let (a_{i})_{i∈I} be a family of elements of B_{0} such that a_{i} ∈ K_{i} for i ∈ I. Let Γ_{i} be a two-dimensional plane which has a normal a_{i} for i ∈ I and which contains the zero of ℝ^{3}. Let Γ_{i}^{⊥} be a linear complement of the vector subspace Γ_{i} for i ∈ I. Let μ_{i} be a standard two-dimensional classical Borel measure in Γ_{i} which takes the value π on the set Γ_{i} ∩ B_{0}.

We put

μ(X) = ∑_{i∈I} ∑_{g∈Γ_{i}^{⊥}} μ_{i}((X − g) ∩ Γ_{i}).

for X ∈ B(ℝ^{3}). Now one can easily check that all conditions formulated for the measure μ in Theorem 7.3 are satisfied. □

Remark 7.2. Other interesting geometric applications of generators of shy sets in infinite-dimensional separable Banach spaces with basis can be found in [19, Chapters 13 and 15].

Acknowledgments. The author expresses thanks to Professor Joseph Rosenblatt for his careful reading of the paper manuscript and some helpful remarks.

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Department of Mathematics, Georgian Technical University, 77 Kostava St. 0175 Tbilisi 75, Georgia gogi_{−}pantsulaia@hotmail.com

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