On countable almost invariant partitions of g-spaces
For any σ -finite G-quasiinvariant measure μ given in a G-space, which is G-ergodic and possesses the Steinhaus property, it is shown that every nontrivial countable μ-almost G-invariant partition of the G-space has a μ-nonmeasurable member.
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irk-123456789-1660052020-02-18T01:28:42Z On countable almost invariant partitions of g-spaces Kharazishvili, A.B. Статті For any σ -finite G-quasiinvariant measure μ given in a G-space, which is G-ergodic and possesses the Steinhaus property, it is shown that every nontrivial countable μ-almost G-invariant partition of the G-space has a μ-nonmeasurable member. Для будь-ям'ї σ -скінченної G-квазіінваріантної міри μ, що задана на G-прсторні, є G-ергодичною та має властивість Штейнхауса, показано, що кожне нетривіальне розбиття μ-майже G-інваріантного розбиття G-простору має μ-невимірний член. 2014 Article On countable almost invariant partitions of g-spaces / A.B. Kharazishvili // Український математичний журнал. — 2014. — Т. 66, № 4. — С. 510–517. — Бібліогр.: 14 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166005 512.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Kharazishvili, A.B. On countable almost invariant partitions of g-spaces Український математичний журнал |
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For any σ -finite G-quasiinvariant measure μ given in a G-space, which is G-ergodic and possesses the Steinhaus property, it is shown that every nontrivial countable μ-almost G-invariant partition of the G-space has a μ-nonmeasurable member. |
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On countable almost invariant partitions of g-spaces |
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On countable almost invariant partitions of g-spaces |
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On countable almost invariant partitions of g-spaces |
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On countable almost invariant partitions of g-spaces |
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On countable almost invariant partitions of g-spaces |
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on countable almost invariant partitions of g-spaces |
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On countable almost invariant partitions of g-spaces / A.B. Kharazishvili // Український математичний журнал. — 2014. — Т. 66, № 4. — С. 510–517. — Бібліогр.: 14 назв. — англ. |
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Український математичний журнал |
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UDC 512.5
A. B. Kharazishvili (A. Razmadze Math. Inst., Tbilisi, Georgia)
ON COUNTABLE ALMOST INVARIANT PARTITIONS OF G-SPACES
ПРО ЗЛIЧЕННI МАЙЖЕ IНВАРIАНТНI РОЗБИТТЯ G-ПРОСТОРIВ
For any σ-finite G-quasiinvariant measure µ given in a G-space which is G-ergodic and possesses the Steinhaus property,
it is shown that every nontrivial countable µ-almost G-invariant partition of the G-space has a µ-nonmeasurable member.
Для будь-якoї σ-скiнченноїG-квазiiнварiантної мiри µ,що задана наG-просторi, єG-ергодичною та має властивiсть
Штейнхауса, показано, що кожне нетривiальне розбиття µ-майже G-iнварiантного розбиття G-простору має µ-
невимiрний член.
Several interesting countable partitions of the real line R into pairwise congruent subsets are known
(see, e.g., [1, 2]).
Historically, the first example of such a partition was presented by Vitali [3] in 1905. With the aid
of an uncountable form of the Axiom of Choice, Vitali constructed a set V ⊂ R having the following
properties:
(a) (V + p) ∩ (V + q) = ∅ for any two distinct rational numbers p and q;
(b) the union of the sets V +q, where q ranges over the field Q of all rational numbers, coincides
with R.
Recall that the set V is, in fact, the first example of a Lebesgue nonmeasurable subset of R.
V is usually called a Vitali subset of R. Notice that Vitali type subsets of uncountable groups are
discussed in many articles, surveys and books (among relatively recent works, see, e.g., [4 – 9]).
After Vitali’s result [3], Sierpiński [10] gave another example of a countable partition of R into
pairwise congruent sets. Namely, he constructed a disjoint countable family {Ei : i ∈ I} of subsets
of R such that:
(1) all sets Ei, i ∈ I, are translates of each other and collectively cover R;
(2) all sets Ei, i ∈ I, are thick with respect to the Lebesgue measure on R; in particular, each Ei
is nonmeasurable in the Lebesgue sense.
The main purpose of this paper is to present some related results concerning countable almost
invariant partitions of the real line (of a finite-dimensional Euclidean space equipped with an appro-
priate transformation group).
Throughout the paper we use the following fairly standard notation:
ω = the set of all natural numbers, i.e.,
ω = N = {0, 1, 2, . . . , n, . . .};
simultaneously, ω stands for the least infinite ordinal (cardinal) number.
c = the cardinality of the continuum; naturally, we identify c with the least ordinal number which
is equinumerous with R.
Rm = the Euclidean space whose dimension is m, where m ∈ N.
λm = the standard m-dimensional Lebesgue measure on Rm. If m = 1, then, for the sake of
brevity, we write λ instead of λ1.
c© A. B. KHARAZISHVILI, 2014
510 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
ON COUNTABLE ALMOST INVARIANT PARTITIONS OF G-SPACES 511
If E is a set, then the symbol IdE denotes the identity transformation of E.
If X and Y are any two sets, then X4Y stands for the symmetric difference of X and Y.
Let E be a ground (base) set equipped with a transformation group G. In this case, for the sake
of brevity, we say that E is a G-space.
Let µ be a nonzero σ-finite measure on E. In the sequel, we denote by dom(µ) the σ-algebra of
all µ-measurable subsets of E. The symbol I(µ) stands for the σ-ideal generated by the family of
all µ-measure zero sets in E.
A subset X of E is called µ-thick (in E) if the equality µ∗(E \ X) = 0 holds true, where µ∗
denotes, as usual, the inner measure associated with µ.
We say that µ is a G-quasiinvariant measure on E if both dom(µ) and I(µ) are G-invariant
classes of sets.
We recall that µ is G-ergodic (or G-metrically transitive) if, for every µ-measurable set X ⊂ E
with µ(X) > 0, there exists a family {gj : j ∈ ω} ⊂ G such that
µ(E \ ∪{gj(X) : j ∈ ω}) = 0.
Let G be a group of transformations of E and suppose that G is endowed with some topology.
In general, we will not assume in our further consideration that this topology is compatible with the
algebraic structure of G.
We shall say that µ has (or possesses) the Steinhaus property if, for each µ-measurable set X ⊂ E
with µ(X) > 0, there exists a neighborhood U of the identity transformation IdE such that
(∀g ∈ U)(µ(g(X) ∩X) > 0).
Remark 1. Let (G, ·) be a σ-compact locally compact topological group, µ be the left Haar
measure on G and let µ′ denote the completion of µ. Further, let X be a µ′-measurable subset of G
satisfying the relations 0 < µ′(X) < +∞. Then, as is well known, the equality
lim
g→e
µ((g ·X)4X) = 0
holds true, where e stands for the neutral element of G. Obviously, the above equality implies the
Steinhaus property of µ′.
Let E be a G-space equipped with some σ-finite measure µ and let {Xi : i ∈ I} be a family of
subsets of E.
We say that {Xi : i ∈ I} is a µ-almost disjoint family if
µ(Xi ∩Xi′) = 0, i ∈ I, i′ ∈ I, i 6= i′.
We say that {Xi : i ∈ I} is µ-almost G-invariant if, for any g ∈ G, the family {g(Xi) : i ∈ I}
almost (more precisely, µ-almost) coincides with {Xi : i ∈ I}.
The latter phrase means that, for any index i ∈ I, there exists an index i′ = i′(i) such that
µ(Xi′4g(Xi)) = 0.
Recall that an abstract group (G, ·) (not necessarily commutative) is divisible if, for each element
g ∈ G and for each natural number n > 0, the equation xn = g is solvable in G.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
512 A. B. KHARAZISHVILI
Remark 2. The structure of all commutative divisible groups is well known (see, for instance,
§ 23 in monograph [11]). On the other hand, the structure of noncommutative divisible groups is still
unclear and may be very complicated. A simple example of a noncommutative divisible group is
provided by the group Is+2 of all orientation preserving isometric transformations of the Euclidean
plane R2. It is a widely known fact that any (commutative) group can be isomorphically embedded
in a (commutative) divisible group (see § 23 and § 67 in [11]).
Remark 3. It should be noticed that if (G,+) is an infinite commutative divisible group and X
is a nonempty proper subset of G, then the family {X + g : g ∈ G} is necessarily infinite. Actually,
this fact was first proved by Sierpiński (cf. [1]). Also, it worth noticing that there exists a Vitali set V
in the divisible commutative group (R,+) such that the family {V + t : t ∈ R} is countably infinite.
Let E be a G-space and suppose that the group G is endowed with some topology.
We shall say that G is admissible if, for any neighborhood U of IdE and for any element g ∈ G,
there exists a natural number n such that the equation xn = g has a solution belonging to U.
Example 1. Clearly, the topological group Tm of all translations of Rm is admissible. Also, the
topological group O+(m) of all rotations of Rm about its origin is admissible. These two simple
facts will be substantially exploited below.
Let E be again a G-space such that G is endowed with some topology.
We shall say that G is weakly admissible if there are finitely many admissible subgroups
G1, G2, . . . , Gr of G such that
G = G1 ◦G2 ◦ . . . ◦Gr.
Example 2. Consider the topological group Is+m of all orientation preserving isometric transfor-
mations of Rm. If g is any element of Is+m, then g can be uniquely represented in the form g = h◦g0,
where h ∈ Tm and g0 ∈ O+
m. In view of Example 1, one may conclude that the group Is+m is weakly
admissible. On the other hand, if m ≥ 1, then the topological group Ism of all isometric transforma-
tions of Rm is not weakly admissible (cf. Example 5 at the end of this paper).
We will be dealing with countable µ-almost G-invariant partitions of a G-space E, where µ is a
nonzero σ-finite G-quasiinvariant measure on E and G is a weakly admissible group of transforma-
tions of E.
Let us formulate and prove the main statement of this paper.
Theorem 1. Let G be a weakly admissible group of transformations of E, let µ be a nonzero
σ-finite G-ergodic measure on E having the Steinhaus property, and let {Xi : i ∈ ω} be a µ-almost
disjoint µ-almost G-invariant covering of E. Then one of the following two assertions holds:
(1) there exists an index i ∈ ω such that the set Xi is not µ-measurable;
(2) there exists an index k ∈ ω such that µ(E \Xk) = 0, i.e., the given covering is trivial in the
sense of µ.
Proof. Suppose that (1) is not satisfied, i.e., all sets Xi, i ∈ ω, are µ-measurable. Then, since µ
is not identically equal to zero, there exists an index k ∈ ω such that µ(Xk) > 0. We assert that
µ(E \Xk) = 0.
Suppose the contrary that µ(E \Xk) > 0. Since µ is G-ergodic, we can find a family {gj : j ∈
∈ ω} ⊂ G for which the equality
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
ON COUNTABLE ALMOST INVARIANT PARTITIONS OF G-SPACES 513
µ(E \ ∪{gj(Xk) : j ∈ ω}) = 0
is valid. This equality implies the existence of an index k′ ∈ ω \ {k} such that
µ(gj(Xk) ∩Xk′) > 0
for some j ∈ ω. Therefore, taking into account the µ-almost disjointness and µ-almost G-invariance
of {Xi : i ∈ ω}, we must have
µ(Xk′4gj(Xk)) = 0.
Further, let us denote
Gk = {g ∈ G : µ(g(Xk)4Xk) = 0}.
Clearly, Gk is a subgroup of G and gj 6∈ Gk. Keeping in mind the inequality µ(Xk) > 0 and
remembering that µ has the Steinhaus property, we can find a neighborhood U of IdE such that
(∀g ∈ U)(µ(g(Xk) ∩Xk) > 0).
Further, since the group G is weakly admissible, we may write
G = H1 ◦H2 ◦ . . . ◦Hr
for some groups H1 ⊂ G, H2 ⊂ G, . . . , Hr ⊂ G, all of which are admissible. In particular, we have
the equality
gj = h1 ◦ h2 ◦ . . . ◦ hr,
where h1 ∈ H1, h2 ∈ H2, . . . , hr ∈ Hr. Since gj 6∈ Gk, at least one of the transformations
h1, h2, . . . , hr does not belong to Gk. Let p be a natural number from the set {1, 2, . . . , r} such that
hp 6∈ Gk.
Now, there exist a natural number n and an element h0 ∈ U for which we have hn0 = hp. Since
hp 6∈ Gk, we also have h0 6∈ Gk. On the other hand, the relation
µ(h0(Xk) ∩Xk) > 0
holds true by virtue of the Steinhaus property. Using once again the µ-almost disjointness and µ-
almost G-invariance of the family {Xi : i ∈ ω}, we obtain
µ(h0(Xk)4Xk) = 0, h0 ∈ Gk.
So we come to a contradiction with the relation h0 6∈ Gk. The obtained contradiction finishes the
proof.
The following statement is a direct consequence of the above theorem.
Theorem 2. Let G = Is+m be the topological group of all orientation preserving isometric trans-
formations of the space Rm, where m ≥ 1, and let µ be a nonzero σ-finite G-ergodic measure on
Rm possessing the Steinhaus property. Suppose that {Xi : i ∈ ω} is a µ-almost disjoint and µ-almost
G-invariant covering of Rm. Then either there exists at least one index i ∈ ω such that the set Xi is
not µ-measurable or there exists an index k ∈ ω such that µ(Rm \Xk) = 0.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
514 A. B. KHARAZISHVILI
Proof. It suffices to apply Theorem 1, keeping in mind the fact that G is a weakly admissible
group of transformations of Rm (see Example 2).
Now, we are going to give an application of Theorem 2 to that case where the role of µ is played
by the standard Lebesgue measure λm on Rm.
For this purpose, we need one auxiliary statement (probably, it is well known but, for the sake of
completeness, we present its proof here).
Theorem 3. Let G be a subgroup of the group Ism. The following three assertions are equiva-
lent:
(1) the measure λm is G-metrically transitive;
(2) there exists a point y ∈ Rm whose G-orbit G(y) is dense in Rm;
(3) for any point z ∈ Rm, its G-orbit G(z) is dense in Rm.
Proof. The equivalence (2) ⇔ (3) is easy to show and, actually, this equivalence remains true in
a much more general situation (e.g., in the case of a metric space E equipped with some group G
of isometric transformations of E). So, in our further consideration, we will be focused only on the
proof of the equivalence (1)⇔ (2).
Suppose that (1) holds and consider an arbitrary point y ∈ Rm. Let ε be a strictly positive real
number and let U(y) be the open ε-neighborhood of y. Let us denote
V (0) = U(y)− y.
Then V (0) is the open ε-neighborhood of 0. Since λm(V (0)) > 0 and λm is G-metrically transitive,
there exists a family {gi : i ∈ ω} ⊂ G such that
λm(R
m \ ∪{gi(V (0)) : i ∈ ω}) = 0.
Further, since λm(U(y)) > 0, there is an index i ∈ ω such that
λm(gi(V (0)) ∩ U(y)) > 0
and, consequently,
gi(V (0)) ∩ U(y) 6= ∅.
So we infer that the point gi(0) belongs to the (2ε)-neighborhood of y. Since ε was taken arbitrarily
small, we conclude that the orbit G(0) is dense in the space Rm, i.e., (2) holds true.
Suppose now that (2) is satisfied for a group G ⊂ Ism. Without loss of generality, we may assume
that G is at most countable and the orbit G(0) is dense in Rm.
Let X be a λm-measurable subset of Rm with λm(X) > 0. We assert that
λm(R
m \G(X)) = 0.
Suppose otherwise, i.e., λm(Rm \ G(X)) > 0, and denote Z = Rm \ G(X). Since λm(X) > 0,
there exists a density point x of X. Analogously, since λm(Z) > 0, there exists a density point z of
Z. Let B be a ball in Rm whose center is 0 and whose radius is so small that
λm(X ∩ (B + x)) ≥ (2/3)λm(B), λm(Z ∩ (B + z)) ≥ (2/3)λm(B).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
ON COUNTABLE ALMOST INVARIANT PARTITIONS OF G-SPACES 515
Further, let g ∈ G be such that
λm(g(B + x) ∩ (B + z)) > (2/3)λm(B).
Then a straightforward calculation enables one to conclude that
λm(g(X ∩ (B + x)) ∩ (Z ∩ (B + z))) > 0
and, therefore,
λm(g(X) ∩ Z) > 0,
which contradicts the obvious equality G(X) ∩ Z = ∅. The obtained contradiction establishes the
validity of the implication (2)⇒ (1).
This finishes the proof of Theorem 3.
Combining Theorems 2 and 3, we obtain the next result.
Theorem 4. Let G be a subgroup of Is+m with the property that at least one point of the space
Rm has dense G-orbit, and let {Xi : i ∈ ω} be a λm-almost disjoint and λm-almost G-invariant
covering of Rm. Then either there exists an index i ∈ ω such that the set Xi is not λm-measurable
or there exists an index k ∈ ω such that λm(E \Xk) = 0.
We would like to underline that the proof of Theorem 1 uses the following two conditions:
(i) the G-ergodicity of a given measure µ;
(ii) the Steinhaus property of µ.
Now, we are going to show by relevant examples that none of the conditions (i) and (ii) can be
omitted.
To present these examples, we must consider some quasiinvariant and invariant extensions of the
Lebesgue measure. Notice that various constructions of extensions of such a type are given, e.g., in
[6, 8, 9, 12, 13].
Example 3. Let m ≥ 1 be a natural number. There exists a partition {An : n < ω} of Rm such
that:
(a) for any n < ω and g ∈ Is+m, we have card(g(An)4An) < c;
(b) if Z is a Borel subset of Rm with λm(Z) > 0 and n is any natural index, then card (Z∩An) =
= c; in particular, all sets An (n < ω) are λm-thick in Rm.
The transfinite construction of {An : n < ω} is fairly standard and may be found in several works
(see, e.g., [12, 13]).
Consider the family F of all those sets which admit a representation in the form
∪{An ∩ Zn : (∀n < ω)(Zn ∈ dom(λm))}.
Notice that F is a σ-algebra of subsets of Rm containing dom(λm). Define a functional µ on F by
the formula
µ(∪{An ∩ Zn : n < ω}) =
∑
{(1/2n+1)λm(Zn) : n < ω}.
This µ is well defined and is a measure extending λm. Moreover, µ can be uniquely extended to an
Is+m-quasiinvariant measure µ′ by adding to the domain of µ the family of all subsets of Rm whose
cardinalities are strictly less than c. For the extended measure µ′, one can readily conclude that:
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
516 A. B. KHARAZISHVILI
(i) µ′ is not Is+m-ergodic;
(ii) µ′ has the Steinhaus property;
(iii) {An : n < ω} is a nontrivial Is+m-invariant partition of Rm into countably many µ′-
measurable subsets of Rm, each of which is of strictly positive µ′-measure.
The next example essentially relies on the existence of a Hamel basis in R.
Example 4. Consider the real line R as a vector space over the field Q of all rational numbers.
Let {eξ : ξ < c} denote a Hamel basis of R. Now, take the vector space over Q generated by the
partial family {eξ : 0 < ξ < c}. Denote this vector space by H and observe that H is a hyperplane
in R complementary to the “line” Qe0. So we come to the countable partition
{Hn : n < ω} = {H + qe0 : q ∈ Q}
of R. Obviously, this partition is T1-invariant. Moreover, the following relations are satisfied:
(a) for each λ-measurable set Z with λ(Z) > 0 and for each n < ω, we have Z ∩Hn 6= ∅;
(b) for any two natural indices n and m, there exists h ∈ R such that h+Hn = Hm.
Now, we introduce the σ-algebra of sets
F = {∪{Hn ∩ Zn : n < ω} : (∀n < ω)(Zn ∈ dom(λ))}
and define a functional µ on F by the formula
µ(∪{Hn ∩ Zn : n < ω}) =
∑
{(1/2n+1)λ(Zn) : n < ω}.
It is not difficult to check that µ is a T1-quasiinvariant T1-ergodic extension of λ for which {Hn : n <
< ω} is a nontrivial µ-almost T1-invariant countable partition of R, and all sets Hn (n < ω) are
µ-measurable and have strictly positive µ-measure.
Consequently, in view of Theorem 1, µ does not possess the Steinhaus property.
Remark 4. Actually, for the measure µ of Example 4, the Steinhaus property fails in a very
strong form. Namely, one can see that
Hn ∩ (Hn + qe0) = ∅
for any n < ω and q ∈ Q \ {0}.
Let us present one more example which shows that the assumption on a transformation group G
be weakly admissible is very essential for the validity of Theorem 1.
Example 5. Let m ≥ 1 be a natural number. There exists a partition {A,B,C} of the space Rm
such that:
(a) for any g ∈ Is+m, we have
card(g(A)4A) < c, card(g(B)4B) < c;
(b) for any g ∈ Ism \ Is+m, we have
card(g(A)4B) < c, card(g(B)4A) < c;
(c) for any g ∈ Ism, we have
card(g(C)4C) < c;
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 4
ON COUNTABLE ALMOST INVARIANT PARTITIONS OF G-SPACES 517
(d) if Z is any Borel subset of Rm with λm(Z) > 0, then
card(Z ∩A) = card(Z ∩B) = c,
in particular, both sets A and B are λm-thick in Rm.
A detailed transfinite construction of the partition {A,B,C} is given in [14]. Notice, by the way,
that condition (c) directly follows from the conjunction of the conditions (a) and (b).
Now, consider the σ-algebra of sets
F = {(A ∩X) ∪ (B ∩ Y ) ∪ (C ∩ Z) : {X,Y, Z} ⊂ dom(λm)}
and define on F a functional µ by the formula
µ((A ∩X) ∪ (B ∩ Y ) ∪ (C ∩ Z)) = 1/2(λm(X) + λm(Y )).
It can be checked that µ is well defined and is a measure extending λm. Furthermore, by adding to
dom(µ) the class of all those subsets of Rm whose cardinalities are strictly less than c, we obtain
the measure µ′ which is Ism-invariant, Ism-ergodic and has the Steinhaus property.
However, we see that {A,B,C} is a µ′-almost Ism-invariant partition of Rm such that all three
sets A, B, C are µ′-measurable and
µ′(A) = µ′(B) = +∞.
This circumstance can be explained, by taking into account the fact that the group Ism is not weakly
admissible.
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1954. – 20.
2. Wagon S. The Banach – Tarski paradox. – Cambridge: Cambridge Univ. Press, 1985.
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Received 06.01.13,
after revision — 13.07.13
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