On Bijective Continuous Images of Absolute Null Sets
The images of absolute null sets (spaces) under bijective continuous mappings are studied. It is shown that, in general, these images do not possess regularity properties from the viewpoint of topological measure theory.
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| author | Kharazishvili, A. B. Харазішвілі, А. Б. |
| author_facet | Kharazishvili, A. B. Харазішвілі, А. Б. |
| author_sort | Kharazishvili, A. B. |
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| description | The images of absolute null sets (spaces) under bijective continuous mappings are studied. It is shown that, in general, these images do not possess regularity properties from the viewpoint of topological measure theory. |
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UDC 517.9
A. Kharazishvili (A. Razmadze Math. Inst., Tbilisi, Georgia)
ON BIJECTIVE CONTINUOUS IMAGES OF ABSOLUTE NULL SETS
ПРО ВЗАЄМНО ОДНОЗНАЧНI НЕПЕРЕРВНI ВIДОБРАЖЕННЯ
МНОЖИН АБСОЛЮТНОЇ МIРИ НУЛЬ
The images of absolute null sets (spaces) under bijective continuous mappings are studied. It is shown that, in general,
such images do not possess regularity properties from the viewpoint of topological measure theory.
Вивчаються зображення множин (просторiв) абсолютної мiри нуль при взаємно однозначних вiдображеннях. До-
ведено, що (в загальному випадку) цi зображення нe мають властивостей регулярностi з точки зору топологiчної
теорiї мiри.
There are various concepts of so-called small sets (spaces). Classical examples of such sets are Luzin
subsets of the real line R and Sierpiński subsets of the same line (see, e. g., [1 – 5]). Here we would
like to discuss one concept of such spaces which is motivated by topological measure theory.
All topological spaces E considered below are assumed to have the following property: for any
point x ∈ E, the singleton {x} is a Borel set in E. In particular, any Hausdorff topological space has
this property.
A measure µ defined on some σ-algebra S of subsets of E is called diffused (or continuous) if
{x} ∈ S and µ({x}) = 0 for each point x ∈ E.
According to a well-known definition, E is said to be an absolute null space (or universal measure
zero space) if every Borel σ-finite diffused measure on E is identically equal to zero.
Example 1. Every Luzin subset of R (and, under Martin’s Axiom, every generalized Luzin
subset of R) is an absolute null space. Also, without assuming additional set-theoretical hypotheses,
there exist uncountable absolute null subspaces of R (see, for instance, [2, 3, 6, 8]). Moreover, any
nonempty perfect subset of R contains an uncountable absolute null space.
Example 2. Let E be an infinite set equipped with the discrete topology and let E′ denote
Alexandrov’s one-point compactification of E. Then these two assertions are equivalent:
(a) E′ is an absolute null space;
(b) the cardinal number card(E′) is nonmeasurable in the Ulam sense.
We thus see that there exist compact absolute null spaces whose cardinalities are sufficiently
large. Some other examples of nondiscrete Hausdorff absolute null spaces with sufficiently large
cardinalities can be found in [6] and [7].
It readily follows from the definition of absolute null spaces that:
(i) the topological product of finitely many absolute null spaces is also an absolute null space;
(ii) if E′ is an absolute null space and h : E → E′ is a Borel mapping such that h−1(y) is at
most countable for each point y ∈ E′, then E is an absolute null space (in particular, any subspace
of an absolute null space is also an absolute null space);
(iii) if {Ej : j ∈ J} is a family of absolute null spaces such that the cardinal number card(J) is
nonmeasurable in the Ulam sense, then the topological sum of this family is an absolute null space;
(iv) if {Ej : j ∈ J} is a countable family of absolute null subspaces of E, then ∪{Ej : j ∈ J}
is also an absolute null subspace of E;
c© A. KHARAZISHVILI, 2015
1134 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
ON BIJECTIVE CONTINUOUS IMAGES OF ABSOLUTE NULL SETS 1135
(v) if X is a Hausdorff absolute null space and Y is a Radon space, then the topological product
X × Y is a Radon space;
(vi) every absolute null subspace of R is zero-dimensional.
Notice, in connection with (i), that the topological product of countably many absolute null spaces is
not, in general, an absolute null space.
One can readily deduce from (ii) that if E is an absolute null space and f : E → E′ is a Borel
isomorphism, then E′ is an absolute null space. In particular, the class of absolute null spaces is
closed under the operation of taking homeomorphic images.
Example 3. If X is a Luzin subset of R and Y ⊂ R is a homeomorphic image of X, then
one cannot assert that Y is a Luzin set. Moreover, such a Y can be a nowhere dense subset of R.
Analogously, if X ′ is a Sierpiński subset of R and Y ′ ⊂ R is a homeomorphic image of X ′, then one
cannot assert that Y ′ is a Sierpiński set. Moreover, such a Y ′ can be a λ-measure zero subset of R,
where λ stands for the standard Lebesgue measure on R. On the other hand, it is easy to see that if f :
R → R is a bijection such that both f and f−1 preserve the σ-ideal of all first category sets in
R, then f transforms the class of all Luzin subsets of R onto it-self. Similarly to this fact, if g :
R→ R is a bijection such that both g and g−1 preserve the σ-ideal of all λ-measure zero sets, then
g transforms the class of all Sierpiński subsets of R onto it-self.
Example 4. If X is a Luzin set and f : X → R is a mapping having the Baire property, then
f(X) is an absolute null space (hence f(X) is totally imperfect as well). Analogously, if Y is a
Sierpiński set and a mapping g : Y → R is measurable with respect to the measure on Y induced by
λ, then g(Y ) does not contain any uncountable absolute null space (hence g(Y ) is totally imperfect
as well).
Theorem 1. Let E be a compact absolute null space, E′ be a Hausdorff space, and let φ :
E → E′ be a mapping such that the φ-pre-image of any open subset of E′ is of type Gδ in E (or,
equivalently, the φ-pre-image of any closed subset of E′ is of type Fσ in E).
Then either φ(E) is an absolute null space or φ(E) is not a Radon space.
Proof. We may assume, without loss of generality that φ is a surjection, i. e., E′ = φ(E). If E′
is an absolute null space, then there is nothing to prove. Suppose now that E′ is not an absolute null
space, so there exists a probability diffused Borel measure µ on E′. If this µ is not a Radon measure,
then E′ is not a Radon space, and we are done. So it remains to consider the case when µ is a Radon
measure. In this case, let us put
S =
{
φ−1(Y ) : Y ∈ B(E′)
}
.
The introduced class S of subsets of E is a σ-subalgebra of the Borel σ-algebra B(E), and we can
define a probability measure ν on S by putting:
ν
(
φ−1(Y )
)
= µ(Y )
(
Y ∈ B(E′)
)
.
Let us check that this ν is a Radon probability measure on S. For this purpose, take any set Y ∈ B(E′).
Since µ is a Radon measure, we may write
µ(Y ) = sup
{
µ(Kj) : j ∈ J
}
,
where J is a countable set of indices and, for each j ∈ J, the set Kj ⊂ Y is compact. It directly
follows from the above equality that
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1136 A. KHARAZISHVILI
ν
(
φ−1(Y )
)
= sup
{
ν(φ−1(Kj)) : j ∈ J
}
,
where all sets φ−1(Kj) (j ∈ J) are of type Fσ in E and all of them belong to S and are contained
in φ−1(Y ). Taking into account the compactness of E and the assumption on φ, all sets φ−1(Kj)
(j ∈ J) are σ-compact in E. This circumstance directly implies that ν is a Radon measure on S.
Now, according to the well-known Henry’s theorem, there exists a Radon probability measure ν ′
extending ν and defined on the whole Borel σ-algebra B(E). Clearly, the measure ν ′ is diffused. So
we come to a contradiction with the fact that E is an absolute null space.
The obtained contradiction finishes the proof.
Example 5. Let ω1 denote the least uncountable ordinal number, E = X ∪ {x} denote Alexan-
drov’s one-point compactification of a discrete topological space X of cardinality ω1, where x 6∈ X,
and let the closed interval E′ = [0, ω1] of ordinal numbers be equipped with its order topology. As
is well known, E′ carries a Borel probability diffused measure ν (the so-called Dieudonné measure)
and this ν is not Radon. Also, by virtue of Ulam’s classical theorem (see, e. g., [4, 5, 8]), E is an
absolute null space. Consider the mapping φ : E → E′ such that φ(x) = ω1 and the restriction φ|X
is a bijection between X and [0, ω1[. Then φ satisfies the assumption of Theorem 1 and the space
E′ = φ(E) is not Radon.
Remark 1. Theorem 1 and Example 5 show that the images of absolute null spaces may lack
natural measure-theoretical regularity properties even under quite good mappings.
In what follows, we are going to demonstrate that, similarly to Remark 1, among the bijective
continuous images of absolute null subspaces of R, one may meet those ones which possess very
bad properties from the point of view of topological measure theory. For this purpose, we need the
notion of absolutely nonmeasurable sets.
Let E be a base (ground) set and letM be a class of measures on E (in general, the domains of
measures fromM are diverse σ-algebras of subsets of E).
We say that a function f : E → R is absolutely nonmeasurable with respect toM if there exists
no measure µ ∈M such that f is µ-measurable. Accordingly, we say that a set Z ⊂ E is absolutely
nonmeasurable with respect toM if the characteristic function
χZ : E → {0, 1}
is absolutely nonmeasurable with respect toM.
Example 6. Any Bernstein subset of R is absolutely nonmeasurable with respect to the class
of completions of all nonzero σ-finite diffused Borel measures on R. Moreover, it can be proved
that there exists a Bernstein set in R which is absolutely nonmeasurable with respect to the class of
all nonzero σ-finite translation quasi-invariant measures on R. Many interesting and extraordinary
properties of Bernstein sets are considered in [1 – 5] and [8].
To proceed, we need four auxiliary propositions. The first of them is due to Erdös, Kunen, and
Mauldin [9].
Lemma 1. Under Martin’s Axiom, there exist two generalized Luzin sets L1 and L2 satisfying
the following two conditions:
(1) both L1 and L2 are vector spaces over the field Q of all rational numbers;
(2) the real line R considered as a vector space over Q is a direct sum of L1 and L2.
The proof of Lemma 1 may be found in [9] (see also [10] for a more general result).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
ON BIJECTIVE CONTINUOUS IMAGES OF ABSOLUTE NULL SETS 1137
Lemma 2. Under Martin’s Axiom, there exists a Hamel basis of R which simultaneously is a
generalized Luzin set.
Proof. We follow the idea of Sierpiński [11]. Let L1 and L2 be as in Lemma 1. Then the set
L = L1 ∪ L2 is a generalized Luzin set in R. Consider a maximal (with respect to the inclusion
relation) linearly independent (over Q) subset H of L. The existence of H immediately follows from
the widely known Kuratowski – Zorn lemma. We assert that H is a Hamel basis of R. Suppose
otherwise, i.e., there is an element x ∈ R such that x 6∈ spanQ(H). Since R is a direct sum of L1
and L2. we may write
x = l1 + l2, l1 ∈ L1, l2 ∈ L2.
Since x 6∈ spanQ(H), we have either l1 6∈ spanQ(H) or l2 6∈ spanQ(H). We may assume, without
loss of generality, that l1 6∈ spanQ(H). Then the set {l1} ∪ H is linearly independent over Q, is
contained in L and properly contains H. But this contradicts the maximality of H. The obtained
contradiction completes the proof.
As usual, we denote by c the cardinality of the continuum.
Lemma 3. Assume Martin’s Axiom and let E be a separable metric space equipped with the
completion of some nonzero σ-finite diffused Borel measure on E. Then there exists a set A ⊂ E
such that card(A) = c and no subset of A having the same cardinality c is an absolute null space.
Proof. The argument is very similar to the classical Sierpiński construction producing Sierpiński
subsets of R (cf. [2 – 5] or [8]), so we omit it here.
Lemma 4. Assume Martin’s Axiom and let E be an arbitrary metric space whose cardinality
does not exceed c. Then there exists a subset B of E which is absolutely nonmeasurable with respect
to the classM of completions of all nonzero σ-finite diffused Borel measures on E.
Proof. If E is an absolute null space, then there is nothing to prove. So suppose thatM 6= ∅.
In this case, the cardinality of E is equal to c and all subsets of E whose cardinalities are strictly
less than c are absolute null subspaces of E. Now, we may apply Bernstein’s classical transfinite
construction (cf. [1 – 5] or [8]) to the family of all closed separable subsets of E having cardinality
c. By virtue of this construction, there exists a set B ⊂ E such that, for any closed separable subset
F of E with card(F ) = c, the equalities
card(F ∩B) = card(F ∩ (E \B)) = c
are fulfilled. Then it is not difficult to show that the set B is absolutely nonmeasurable with respect
to the classM. Lemma 4 has thus been proved.
Remark 2. Let us underline that in the above proof the completeness of E is not needed. In
the literature the existence of Bernstein sets is usually established for uncountable Polish spaces
(cf. [1 – 5]).
Theorem 2. Under Martin’s Axiom, there exist an absolute null subspaceZ of R with card(Z) =
= c and a continuous injection φ : Z → R such that no subset of φ(Z) having cardinality c is absolute
null.
Proof. Let L1 and L2 be as in Lemma 1, i. e., both L1 and L2 are generalized Luzin sets, are
vector spaces over the field Q, and the real line R also considered as a vector space over Q is a
direct sum of L1 and L2. Let us introduce the product set
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
1138 A. KHARAZISHVILI
L = L1 × L2 ⊂ R2
and define a mapping φ : L→ R by the formula
φ(z) = pr1(z) + pr2(z) (z ∈ L1 × L2).
According to (i), the product set L is an absolute null subspace of R2. Moreover, L being the product
of two zero-dimensional spaces is a zero-dimensional subspace of R2 (see (vi)). Since R is a direct
sum of L1 and L2, the additive mapping φ is a continuous bijection of L onto R. Keeping in mind
the circumstance that R is equipped with the nonzero σ-finite diffused Lebesgue measure λ, we may
apply Lemma 3 to E = R. Consequently, there exists a set A ⊂ R with card(A) = c such that no
subset of A having the same cardinality c is an absolute null space (actually, A can be a generalized
Sierpiński subset of R). Now, put Z = φ−1(A) and observe that card(Z) = c and Z, being a subset
of L, is a zero-dimensional subspace of R2. Finally, identifying Z with its topological copy in R,
we obtain the required result.
In a similar manner, we are able to prove the following statement.
Theorem 3. Under Martin’s Axiom, there exist an absolute null subset Y of R with card(Y ) =
= c, a continuous injection φ : Y → R, such that the set φ(Y ) is absolutely nonmeasurable with
respect to the class of completions of all nonzero σ-finite diffused Borel measures on R.
Proof. The argument is analogous to the previous one. We preserve the notation of the proof of
Theorem 2. Let L and φ be as earlier. According to Lemma 4, there exists a set B ⊂ R absolutely
nonmeasurable with respect to the class of completions of all nonzero σ-finite diffused Borel measures
on R (actually, B is a Bernstein subset of R). Let us put Y = φ−1(B). Then card(Y ) = c and
Y ⊂ L is a zero-dimensional absolute null subspace of R2. Identifying Y with its topological copy
in R, we obtain the required result.
Remark 3. As is well known, the existence of an absolute null subspace of R of cardinality c
cannot be proved within the framework of ZFC set theory. Moreover, there is a model of ZFC
in which ω1 < c and the cardinalities of all absolute null subspaces of R do not exceed ω1 (see,
e.ġ., [3] and references therein). Therefore, additional set-theoretical assumptions are necessary in
the formulations of Theorems 2 and 3.
1. Kuczma M. An introduction to the theory of functional equations and inequalities: cauchy’s equation and jensen’s
inequality. – Katowice: PWN, 1985.
2. Kuratowski K. Topology. – London; New York: Acad. Press, 1966. – Vol. 1.
3. Miller A. W. Special subsets of the real line // Handbook of Set-Theoretic Topology / Eds K. Kunen and J. E. Vaughan. –
Amsterdam: North-Holland Publ. Co., 1984.
4. Morgan II J. C. Point set theory. – New York: Marcel Dekker, Inc., 1990.
5. Oxtoby J. C. Measure and category. – New York: Springer-Verlag, 1971.
6. Pfeffer W. F., Prikry K. Small spaces // Proc. London Math. Soc. – 1989. – 58(3), № 3. – P. 417 – 438.
7. Kharazishvili A. B. Some properties of isodyne topological spaces // Bull. Acad. Sci. GSSR. – 1987. – 127, № 2. –
P. 261 – 264 (in Russian).
8. Kharazishvili A. B. Nonmeasurable sets and functions. – Amsterdam: Elsevier, 2004.
9. Erdös P., Kunen K., Mauldin R. D. Some additive properties of sets of real numbers // Fund. Math. – 1981. – 113,
№ 3. – P. 187 – 199.
10. Kharazishvili A. B. Sums of absolutely nonmeasurable functions // Georg. Math. J. – 2013. – 20, № 2. – P. 271 – 282.
11. Sierpiński W. Sur la question de la mesurabilité de la base de M.Hamel // Fund. Math. – 1920. – 1. – P. 105 – 111.
Received 12.06.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 8
|
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| last_indexed | 2026-03-24T02:17:45Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-20512019-12-05T09:49:28Z On Bijective Continuous Images of Absolute Null Sets Про взаємно однозначні неперервні відображення множин абсолютної міри нуль Kharazishvili, A. B. Харазішвілі, А. Б. The images of absolute null sets (spaces) under bijective continuous mappings are studied. It is shown that, in general, these images do not possess regularity properties from the viewpoint of topological measure theory. Вивчаються зображення множин (npocTopiB) абсолютної міри нуль при взаємно однозначних відображеннях. Доведено, що (в загальному випадку) ці зображення нє мають властивостей регулярності з точки зору топологічної теорії міри. Institute of Mathematics, NAS of Ukraine 2015-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2051 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 8 (2015); 1134-1138 Український математичний журнал; Том 67 № 8 (2015); 1134-1138 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2051/1119 https://umj.imath.kiev.ua/index.php/umj/article/view/2051/1120 Copyright (c) 2015 Kharazishvili A. B. |
| spellingShingle | Kharazishvili, A. B. Харазішвілі, А. Б. On Bijective Continuous Images of Absolute Null Sets |
| title | On Bijective Continuous Images of Absolute Null Sets |
| title_alt | Про взаємно однозначні неперервні відображення множин абсолютної міри нуль |
| title_full | On Bijective Continuous Images of Absolute Null Sets |
| title_fullStr | On Bijective Continuous Images of Absolute Null Sets |
| title_full_unstemmed | On Bijective Continuous Images of Absolute Null Sets |
| title_short | On Bijective Continuous Images of Absolute Null Sets |
| title_sort | on bijective continuous images of absolute null sets |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2051 |
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