Georgian Mathematical Journal Volume 8 (2001), Number 3, 493–498

ON VECTOR SUMS OF MEASURE ZERO SETS

A. KHARAZISHVILI

Abstract. We consider the behaviour of measure zero subsets of a vector space under the operation of vector sum. The question whether the vector sum of such sets can be nonmeasurable is discussed in connection with the measure extension problem, and a certain generalization of the classical Sierpiński result [3] is presented.

2000 Mathematics Subject Classification: 28A05, 28D05.

Key words and phrases: Vector sum, measure zero set, transformation group, quasi-invariant measure, nonmeasurable set, absolutely negligible set.

It is a well-known fact that some nice descriptive properties of subsets of a topological vector space are not preserved under the operation of vector sum. The following two examples are typical in this respect.

Example 1. There exist two Borel subsets X and Y of the real line R, such that X + Y is not Borel (obviously, X + Y is an analytic subset of R).

Example 2. There exist two sets X ⊂ R and Y ⊂ R, both of Lebesgue measure zero, such that X + Y is not Lebesgue measurable.

In connection with Example 1, see, e.g., [1] or [2], Example 2 is due to Sierpiński (see his early paper [3]). He gave this example starting with a simple observation that there are two Lebesgue measure zero sets A ⊂ R and B ⊂ R for which A + B = R, and utilizing some properties of Hamel bases in R.

This paper is devoted primarily to some generalizations of Example 2 for nonzero σ-finite quasi-invariant measures in vector spaces. In particular, we shall demonstrate that, for certain extensions of quasi-invariant measures, the phenomenon described in Example 2 can always be realized.

As a rule, the measures considered below are assumed to be defined on some σ-algebras of subsets of a given uncountable vector space E (over the field Q of all rationals) and are supposed to be quasi-invariant under the group of all nondegenerate rational homotheties of this space. More precisely, we shall say that a mapping h : E → E is a nondegenerate rational homothety of E if h can be represented as

h(x) = qx + x_{0} (x ∈ E),

where q is a fixed nonzero rational number and x_{0} is a fixed element of E. The family of all above-mentioned homotheties forms a group with respect to the

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

usual composition operation. We denote this group by the symbol H_{E} (notice that H_{E} is not commutative).

If μ is a measure on E, then we put:

dom(μ) = the domain of μ;

I(μ) = the σ-ideal of all μ-measure zero subsets of E.

We shall say that μ is H_{E}-quasi-invariant if both classes of sets dom(μ) and I(μ) are invariant under all transformations from the group H_{E}.

Let μ be a nonzero σ-finite H_{E}-quasi-invariant measure on E. Motivated by Example 2, we may pose the following question: do there exist two sets X ∈ I(μ) and Y ∈ I(μ) for which X + Y ∉ dom(μ)? It can easily be shown that, in general, the answer to this question is negative. Moreover, various examples of a situation where

(∀X ∈ I(μ))(∀Y ∈ I(μ))(X + Y ∈ I(μ))

can be constructed without any difficulties (see, e.g., [4]). Also, as demonstrated in the same monograph [4], the question posed above can be reduced to another much easier problem. More precisely, we have the following statement.

Theorem 1. Let E be an uncountable vector space (over Q). Then, for any nonzero σ-finite H_{E}-quasi-invariant measure μ on E, the following eight assertions are equivalent:

1) there exist two sets X ∈ I(μ) and Y ∈ I(μ) such that X + Y ∉ I(μ);

2) there exists a set X ∈ I(μ) such that X + X ∉ I(μ);

3) there exists a set X ∈ I(μ) such that lin_{Q}(X) ∉ I(μ) where lin_{Q}(X) stands for the linear hull (over Q) of X;

4) there exists a linearly independent (over Q) set X ∈ I(μ) such that lin_{Q}(X) ∉ I(μ);

5) there exist two sets X ∈ I(μ) and Y ∈ I(μ) such that X + Y ∉ dom(μ);

6) there exists a set X ∈ I(μ) such that X + X ∉ dom(μ);

7) there exists a set X ∈ I(μ) such that lin_{Q}(X) ∉ dom(μ);

8) there exists a linearly independent (over Q) set X ∈ I(μ) such that lin_{Q}(X) ∉ dom(μ).

The proof is presented in [4]. Notice that the argument is essentially based on some properties of the so-called Ulam transfinite matrix (see, e.g., [6]).

The equivalence of assertions 1)–8) shows us that, in order to obtain a positive answer to the question formulated above, we need only the existence of two sets X ∈ I(μ) and Y ∈ I(μ) for which X + Y ∉ I(μ). Clearly, the question will be solved positively if the existence of two sets X ∈ I(μ) and Y ∈ I(μ) is established, for which X + Y = E.

Our goal is to demonstrate that there always exists an H_{E}-quasi-invariant extension μ′ of μ such that the vector sum of some two μ′-measure zero subsets of E is identical with the whole space E. For this purpose, several auxiliary notions and propositions are necessary.

Let E be an arbitrary set, G be a group of transformations of E and let Y be a subset of E. We say (cf. [5]) that Y is G-absolutely negligible (in E) if, for any σ-finite G-quasi-invariant measure μ on E, there exists a G-quasi-invariant measure μ′ on E extending μ and such that μ′(Y) = 0. The concept of an absolutely negligible set is thoroughly discussed in monograph [5] where a significant role of this concept is emphasized for various questions concerning extensions of quasi-invariant (invariant) measures.

Below, the symbol ω denotes the first infinite ordinal (cardinal) and ω_{1} stands for the first uncountable ordinal (cardinal).

Let E be a set and let {X_{i} : i ∈ I} be a partition of E.

A set X ⊂ E is called a partial selector of {X_{i} : i ∈ I} if card(X ∩ X_{i}) ≤ 1 for all indices i ∈ I. Accordingly, a set X ⊂ E is called a selector of the same partition if card(X ∩ X_{i}) = 1 for all i ∈ I.

Our starting point is the following lemma (cf. [5]).

Lemma 1. Let E be a set of cardinality ω_{1}, let G be a group of transformations of E, such that card(G) = ω_{1} and

card({x ∈ E : g(x) = h(x)}) ≤ ω

for any two distinct transformations g ∈ G and h ∈ G. Further, let {G_{ξ} : ξ < ω_{1}} be an increasing (with respect to inclusion) ω_{1}-sequence of subgroups of G and let {X_{ξ} : ξ < ω_{1}} be a partition of E, such that:

1) card(G_{ξ}) ≤ ω for all ordinals ξ < ω_{1};

2) card(X_{ξ}) ≤ ω for all ordinals ξ < ω_{1};

3) ∪{G_{ξ} : ξ < ω_{1}} = G;

4) for each ξ < ω_{1}, the set X_{ξ} is G_{ξ}-invariant.

Then every partial selector of {X_{ξ} : ξ < ω_{1}} is a G-absolutely negligible subset of E.

A detailed proof of this proposition can be found in [5].

Let (G, +) be a commutative group. The group of all translations of G is obviously isomorphic to G, and we can identify these two groups in our further considerations.

Lemma 2. Let (G, +) be a commutative group of cardinality ω_{1} and let X be an arbitrary uncountable subset of G. Then there exists a G-absolutely negligible set Y ⊂ G such that X + Y = G.

Proof. Let {x_{ξ} : ξ < ω_{1}} be an injective family of all elements of G. Put E = G and equip E with the group of all translations of G. Let {G_{ξ} : ξ < ω_{1}} and {X_{ξ} : ξ < ω_{1}} be two families satisfying the conditions of Lemma 1. We now define an injective ω_{1}-sequence {y_{ξ} : ξ < ω_{1}} of elements of E. Suppose that, for an ordinal ξ < ω_{1}, the partial ξ-sequence {y_{ζ} : ζ < ξ} has already been defined. For each ordinal ζ < ξ, let X_{η(ζ)} be such that y_{ζ} ∈ X_{η(ζ)}. We denote

Z_{ξ} = ∪{X_{η(ζ)} : ζ < ξ}

and observe that card(Z_{ξ}) ≤ ω. Since the given set X is uncountable, we must have

(E \ Z_{ξ}) ∩ (x_{ξ} − X) ≠ ∅.

Choose any element of (E \ Z_{ξ}) ∩ (x_{ξ} − X) and denote it by y_{ξ}. Continuing in this manner, we will be able to construct the desired ω_{1}-sequence {y_{ξ} : ξ < ω_{1}} of elements of E. Now, putting

Y = {y_{ξ} : ξ < ω_{1}}

and taking into account the relation

(∀ξ < ω_{1})(x_{ξ} ∈ y_{ξ} + X),

we see that

X + Y = E = G.

Also, in accordance with Lemma 1, the set Y is G-absolutely negligible in E. This completes the proof of Lemma 2.

Remark 1. It is not hard to verify that Lemma 2 remains true for an arbitrary group G of cardinality ω_{1}. In addition to this, suppose that E is a vector space over Q with card(E) = ω_{1} and let G = H_{E}. Then card(G) = ω_{1}, too, and

card({x ∈ E : g(x) = h(x)}) ≤ 1

for any two distinct transformations g ∈ G and h ∈ G. The argument used in the proof of Lemma 2 shows us that, for every uncountable set X ⊂ E, there exists an H_{E}-absolutely negligible set Y ⊂ E for which we have X + Y = E.

Theorem 2. Let (G, ·) be a group of cardinality ω_{1} (identified with the group of all its left translations). Then there exist two G-absolutely negligible sets X ⊂ G and Y ⊂ G such that X · Y = G. In particular, for any nonzero σ-finite left G-quasi-invariant (respectively, left G-invariant) measure μ on G, there exists a left G-quasi-invariant (respectively, left G-invariant) measure μ′ on G extending μ and satisfying the relations

X ∈ I(μ′), Y ∈ I(μ′), X · Y = G ∉ I(μ′).

Proof. Take any uncountable G-absolutely negligible set X ⊂ G (the existence of such a set easily follows from Lemma 1). In virtue of Lemma 2 (cf. Remark 1 above), there exists a G-absolutely negligible set Y ⊂ G satisfying the equality X · Y = G.

Remark 2. It would be interesting to generalize Theorem 2 to those groups whose cardinalities are greater than ω_{1}. In this connection, it can be shown that if G is an uncountable group and card(G) is a regular cardinal, then, for each set X ⊂ G with card(X) = card(G), there exists a G-absolutely negligible set Y ⊂ G such that X · Y = G (the argument is very similar to the proof of Lemma 2).

Lemma 3. Let E be a vector space (over Q) and let

E = E_{1} + E_{2} (E_{1} ∩ E_{2} = {0})

be a representation of E in the form of the direct sum of two vector subspaces E_{1} and E_{2} (over Q again). Suppose also that a set Y ⊂ E_{1} is H_{E_{1}}-absolutely negligible in E_{1}. Then the set Y + E_{2} turns out to be H_{E}-absolutely negligible in E.

The proof of this statement is presented in monograph [5].

Lemma 4. Let E be a vector space (over Q). Then, for each uncountable set X ⊂ E, there exists an H_{E}-absolutely negligible set Y ⊂ E such that X+Y = E.

Proof. We may assume, without loss of generality, that card(X) = ω_{1}. Denote by E_{1} the vector subspace of E (over Q again) generated by X. Evidently, we have card(E_{1}) = ω_{1}. Let us represent our E in the form of the direct sum of two vector subspaces:

E = E_{1} + E_{2} (E_{1} ∩ E_{2} = {0}).

Applying Lemma 2 (see also Remark 1), we can find an H_{E_{1}}-absolutely negligible set Y_{1} ⊂ E_{1} such that X + Y_{1} = E_{1}. Let us put

Y = Y_{1} + E_{2}.

Then, in view of Lemma 3, the set Y is H_{E}-absolutely negligible in E. Furthermore, we may write

X + Y = X + Y_{1} + E_{2} = E_{1} + E_{2} = E,

and the lemma is proved.

From Lemma 4 we easily obtain the following statement.

Theorem 3. Let E be a vector space (over Q) and let μ be a nonzero σ-finite H_{E}-quasi-invariant measure on E. Then, for each uncountable set X ∈ I(μ), there exist an H_{E}-quasi-invariant measure μ′ on E extending μ and a set Y ∈ I(μ′), for which we have X + Y = E ∉ I(μ′).

Proof. Let Y be an H_{E}-absolutely negligible set in E such that X + Y = E (the existence of Y was established in Lemma 4). The absolute negligibility of Y implies that there exists an H_{E}-quasi-invariant extension μ′ of μ for which μ′(Y) = 0. Thus, we see that the measure μ′ and the set Y are the required ones.

Finally, taking into account Theorem 1, we conclude that the following result is valid.

Theorem 4. Let E be an uncountable vector space (over Q) and let μ be a nonzero σ-finite H_{E}-quasi-invariant measure on E. Then there exists an H_{E}- quasi-invariant measure μ′ on E extending μ such that, for some sets X ∈ I(μ′) and Y ∈ I(μ′), we have X + Y ∉ dom(μ′).

The latter theorem can be regarded as a generalized version of Example 2 for nonzero σ-finite quasi-invariant measures in vector spaces. It would be interesting to extend this theorem to a wider class of uncountable groups equipped with nonzero σ-finite left quasi-invariant (in particular, left invariant) measures.

References

1. B. S. Sodnomov, An example of two G_{δ}-sets whose arithmetical sum is not Borel measurable. (Russian) Dokl. Akad. Nauk SSSR 99(1954), 507–510.

2. C. A. Rogers, A linear Borel set whose difference set is not a Borel set. Bull. London Math. Soc. 2(1970), 41–42.

3. W. Sierpiński, Sur la question de la mesurabilité de la base de M. Hamel. Fund. Math. 1(1920), 105–111.

4. A. B. Kharazishvili, Applications of point set theory in real analysis. Kluwer Academic Publishers, Dordrecht, 1998.

5. A. B. Kharazishvili, Invariant extensions of Lebesgue measure. (Russian) Tbilisi University Press, Tbilisi, 1983.

6. J. C. Oxtoby, Measure and category. Springer-Verlag, Berlin, 1971.

(Received 12.03.2001)

Author’s address:

I. Vekua Institute of Applied Mathematics

I. Javakhishvili Tbilisi State University

2, University St., Tbilisi 380043

Georgia