Good Measures on Locally Compact Cantor Sets
We study the set M(X) of full non-atomic Borel measures μ on a non-compact locally compact Cantor set X. The set Mμ = {x is in X : for any compact open set U (x is in U) we have μ(U) = ∞} is called defective. μ is non-defective if μ(Mμ) = 0. The set M⁰(X) is subset of M(X) consists of probability a...
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irk-123456789-1067232016-10-04T03:02:28Z Good Measures on Locally Compact Cantor Sets Karpel, O.M. We study the set M(X) of full non-atomic Borel measures μ on a non-compact locally compact Cantor set X. The set Mμ = {x is in X : for any compact open set U (x is in U) we have μ(U) = ∞} is called defective. μ is non-defective if μ(Mμ) = 0. The set M⁰(X) is subset of M(X) consists of probability and infinite non-defective measures. We classify the measures from M⁰(X) with respect to a homeomorphism. The notions of goodness and the compact open values set S(μ) are defined. A criterion when two good measures are homeomorphic is given.For a group-like set D and a locally compact zero-dimensional metric space A we find a good non-defective measure μ on X such that S(μ) = D and Mμ is homeomorphic to A. We give a criterion when a good measure on X can be extended to a good measure on the compactification of X. Изучается множество M(X) полных неатомарных борелевских мер μ на некомпактном локально-компактном канторовском множестве X. Множество Mμ = {x є X : для любого компактно-открытого множества U (x є U) имеем μ(U) = ∞} называется дефектным. m недефектна, если μ(Mμ) = 0. Класс M⁰(X), являющийся подмножеством M(X), состоит из вероятностных и бесконечных недефектных мер. Меры из M⁰(X) классифицируются с точностью до гомеоморфизма. Введены понятия хорошей меры и множества S(μ) значений меры на компактно-открытых подмножествах. Представлен критерий гомеоморфности для двух хороших мер. Для группоподобного множества D и локально-компактного нульмерного метрического пространства A найдена хорошая мера m на X, такая что S(μ) = D и Mμ гомеоморфно A. Дан критерий, когда хорошая мера на X может быть продолжена до хорошей меры на компактификации X. 2012 Article Good Measures on Locally Compact Cantor Sets/ O.M. Karpel // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 3. — С. 260-279. — Бібліогр.: 16 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106723 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We study the set M(X) of full non-atomic Borel measures μ on a non-compact locally compact Cantor set X. The set Mμ = {x is in X : for any compact open set U (x is in U) we have μ(U) = ∞} is called defective. μ is non-defective if μ(Mμ) = 0. The set M⁰(X) is subset of M(X) consists of probability and infinite non-defective measures. We classify the measures from M⁰(X) with respect to a homeomorphism. The notions of goodness and the compact open values set S(μ) are defined. A criterion when two good measures are homeomorphic is given.For a group-like set D and a locally compact zero-dimensional metric space A we find a good non-defective measure μ on X such that S(μ) = D and Mμ is homeomorphic to A. We give a criterion when a good measure on X can be extended to a good measure on the compactification of X. |
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Karpel, O.M. Good Measures on Locally Compact Cantor Sets Журнал математической физики, анализа, геометрии |
| author_facet |
Karpel, O.M. |
| author_sort |
Karpel, O.M. |
| title |
Good Measures on Locally Compact Cantor Sets |
| title_short |
Good Measures on Locally Compact Cantor Sets |
| title_full |
Good Measures on Locally Compact Cantor Sets |
| title_fullStr |
Good Measures on Locally Compact Cantor Sets |
| title_full_unstemmed |
Good Measures on Locally Compact Cantor Sets |
| title_sort |
good measures on locally compact cantor sets |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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http://dspace.nbuv.gov.ua/handle/123456789/106723 |
| citation_txt |
Good Measures on Locally Compact Cantor Sets/ O.M. Karpel // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 3. — С. 260-279. — Бібліогр.: 16 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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AT karpelom goodmeasuresonlocallycompactcantorsets |
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2025-07-07T18:54:15Z |
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1837015453196615680 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2012, vol. 8, No. 3, pp. 260–279
Good Measures on Locally Compact Cantor Sets
O.M. Karpel
Mathematics Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv 61103, Ukraine
E-mail: helen.karpel@gmail.com
Received March 28, 2012
We study the set M(X) of full non-atomic Borel measures µ on a non-
compact locally compact Cantor set X. The set Mµ = {x ∈ X :
for any compact open set U 3 x we have µ(U) = ∞} is called defective. µ is
non-defective if µ(Mµ) = 0. The set M0(X) ⊂ M(X) consists of probability
and infinite non-defective measures. We classify the measures from M0(X)
with respect to a homeomorphism. The notions of goodness and the compact
open values set S(µ) are defined. A criterion when two good measures are
homeomorphic is given. For a group-like set D and a locally compact zero-
dimensional metric space A we find a good non-defective measure µ on X
such that S(µ) = D and Mµ is homeomorphic to A. We give a criterion
when a good measure on X can be extended to a good measure on the
compactification of X.
Key words: Borel measures, locally compact Cantor set, compactifica-
tion, invariant measures.
Mathematics Subject Classification 2000: 37A05, 37B05 (primary); 28D05,
28C15 (secondary).
1. Introduction
The problem of classification of Borel finite or infinite measures on topological
spaces has a long history. Two measures µ and ν defined on the Borel subsets of a
topological space X are called homeomorphic if there exists a self-homeomorphism
h of X such that µ = ν ◦ h, i.e., µ(E) = ν(h(E)) for every Borel subset E of
X. The topological properties of the space X are important for the classification
of measures up to a homeomorphism. For instance, Oxtoby and Ulam [1] gave
a criterion for a Borel probability measure on the finite-dimensional cube to be
homeomorphic to the Lebesgue measure. Similar results were obtained for various
manifolds (see [2, 3]).
c© O.M. Karpel, 2012
Good Measures on Locally Compact Cantor Sets
A Cantor set (or a Cantor space) is a non-empty zero-dimensional compact
perfect metric space. For Cantor sets the situation is much more complicated
than for connected spaces. During the last decade, in the papers [4–8] Borel
probability measures on Cantor sets were studied. In [9] infinite Borel measures
on Cantor sets were considered. For many applications in dynamical systems the
state space is only locally compact. In this paper, we study Borel both finite and
infinite measures on non-compact locally compact Cantor sets.
It is possible to construct uncountably many full (the measure of every non-
empty open set is positive) non-atomic measures on a Cantor set X which are
pairwise non-homeomorphic (see [10]). This is due to the existence of a countable
base of clopen subsets of a Cantor set. The clopen values set S(µ) is the set of
finite values of a measure µ on all clopen subsets of X. The set provides an
invariant for homeomorphic measures, although it is not a complete invariant.
For the class of the so-called good probability measures, S(µ) is a complete
invariant. By definition, a full non-atomic probability or non-defective measure µ
is good if, whenever U , V are clopen sets with µ(U) < µ(V ), there exists a clopen
subset W of V such that µ(W ) = µ(U) (see [9, 11]). Good probability measures
are exactly invariant measures of uniquely ergodic minimal homeomorphisms of
Cantor sets (see [11, 12]). For an infinite Borel measure µ on a Cantor set X,
denote by Mµ the set of all points in X whose clopen neighbourhoods have only
infinite measures. The full non-atomic infinite measures µ such that µ(Mµ) = 0
are called non-defective. These measures arise as ergodic invariant measures for
homeomorphisms of a Cantor set and the theory of good probability measures
can be extended to the case of non-defective measures (see [9]).
In Sec. 2, we define a good probability measure and a good non-defective
measure on a non-compact locally compact Cantor set X and extend the results
concerning good measures on Cantor sets to non-compact locally compact Cantor
sets. For a Borel measure µ on X, the set S(µ) is defined as a set of all finite
values of µ on the compact open sets. The defective set Mµ is the set of all points
x in X such that every compact open neighbourhood of x has an infinite measure.
We prove the criterion when two good measures on non-compact locally compact
Cantor sets are homeomorphic. For every group-like subset D ⊂ [0, 1) we find a
good probability measure µ on a non-compact locally compact Cantor set such
that S(µ) = D. For every group-like subset D ⊂ [0,∞) and any locally compact
zero-dimensional metric space A (including A = ∅) we find a good non-defective
measure µ on a non-compact locally compact Cantor set such that S(µ) = D and
Mµ is homeomorphic to A.
In Sec. 3, the compactifications of non-compact locally compact Cantor sets
are studied. We investigate whether a compactification can be used to classify
the measures on non-compact locally compact Cantor sets. We consider only
the compactifications which are Cantor sets and extend the measure µ by giving
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 261
O.M. Karpel
the remainder of compactification a zero measure. It turns out that in some
cases a good measure can be extended to a good measure on a Cantor set, while
in other cases the extension always produces a measure which is not good. The
extensions of a non-good measure are always non-good. After compactification of
a non-compact locally compact Cantor set, new compact open sets are obtained.
We study how the compact open values set changes. Basing on this study, we
give a criterion when a good measure on a non-compact locally compact Cantor
set remains good after the compactification.
Section 4 contains the examples to the results of Secs. 2 and 3. For instance,
the Haar measure on the set of p-adic numbers and the invariant measure for
(C, F )-construction are good. We give the examples of good ergodic invariant
measures on the generating open dense subset of a path space of stationary Brat-
teli diagrams such that any compactification gives a non-good measure.
2. Measures on Locally Compact Cantor Sets
Let X be a non-compact locally compact metrizable space with no isolated
points and with a (countable) basis of compact and open sets. Hence X is totally
disconnected. The set X is called a non-compact locally compact Cantor set.
Every two non-compact locally compact Cantor sets are homeomorphic (see [13]).
Take a countable family of the compact open subsets On ⊂ X such that X =⋃∞
n=1 On. Denote X1 = O1, X2 = O2 \ O1, X3 = O3 \ (O1 ∪ O2),. . . . The
subsets Xn are compact open pairwise disjoint, and X =
⋃∞
n=1 Xn. Since X
is non-compact, we may assume without loss of generality that all Xn are non-
empty. Since X has no isolated points, every Xn has the same property. Thus,
we represent X as a disjoint union of a countable family of the compact Cantor
sets Xn.
Recall that a Borel measure on a locally compact Cantor space is called full
if every non-empty open set has a positive measure. It is easy to see that for
a non-atomic measure µ the support of µ in the induced topology is a locally
compact Cantor set. We can consider the measures on their supports to obtain
full measures. Denote by M(X) the set of full non-atomic Borel measures on X.
Then M(X) = Mf (X)tM∞(X), where Mf (X) = {µ ∈ M(X) : µ(X) < ∞} and
M∞(X) = {µ ∈ M(X) : µ(X) = ∞}. For a measure µ ∈ M∞(X), denote Mµ =
{x ∈ X : for any compact and open set U 3 x we have µ(U) = ∞}. It will be
shown that Mµ is a Borel set. Denote M0∞(X) = {µ ∈ M∞(X) : µ(Mµ) = 0}.
Let M0(X) = Mf (X) t M0∞(X). Throughout the paper we will consider only
the measures from M0(X). We normalize the measures from Mf (X) such that
µ(X) = 1 for any µ ∈ Mf (X).
Recall that µ ∈ M0(X) is locally finite if every point of X has a neighbourhood
of a finite measure. The properties of the measures from the class M0(X) are
collected in the following proposition.
262 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3
Good Measures on Locally Compact Cantor Sets
Proposition 2.1. Let µ ∈ M0(X). Then
(1) The measure µ is locally finite if and only if Mµ = ∅.
(2) The set X \Mµ is open. The set Mµ is Fσ.
(3) For any compact open set U with µ(U) = ∞ and any a > 0 there exists a
compact open subset V ⊂ U such that a ≤ µ(V ) < ∞.
(4) The set Mµ is nowhere dense.
(5) X =
⊔∞
i=1 Vi
⊔
Mµ, where each Vi is a compact open set of a finite measure
and Mµ is a nowhere dense Fσ and is of zero measure. The measure µ is σ-finite.
(6) µ is uniquely determined by its values on the algebra of the compact open sets.
P r o o f. (1) The condition Mµ = ∅ means that every point x ∈ X has a
compact open neighbourhood of a finite measure. Hence µ is locally finite and
vise versa.
(2) We have X \ Mµ = {x ∈ X : there exists a compact open set Ux 3
x such that µ(Ux) < ∞}. Then for every point x ∈ X\Mµ we have Ux ⊂ X\Mµ.
Hence X \Mµ is open. Therefore, for every n ∈ N the set Xn \Mµ is open and
Xn ∩Mµ is closed. Then Mµ =
⊔
n∈N (Xn ∩Mµ) is Fσ set.
(3) Let U be a non-empty compact open subset of X such that µ(U) = ∞.
Since µ ∈ M0(X), we have µ(U) = µ(U \Mµ). Since U is open, the set U \Mµ =
U∩(X\Mµ) is open. There are only countably many compact open subsets in X,
hence the open set U \Mµ can be represented as a disjoint union of the compact
open subsets {Ui}i∈N of a finite measure. We have µ(U) =
∑∞
i=0 µ(Ui) = ∞,
hence for every a ∈ R there is a compact open subset V =
⊔N
i=0 Ui such that
a ≤ µ(V ) < ∞.
(4) Let U be a compact open subset of X. It suffices to show that there exists
a non-empty compact open subset V ⊂ U such that V ∩Mµ = ∅. If µ(U) < ∞,
then U ∩Mµ = ∅. Otherwise, by (3), there exists a compact open subset V ⊂ U
such that 0 < µ(V ) < ∞. Obviously, V ∩Mµ = ∅.
(5) Follows from the proof of (3).
(6) Follows from (5).
For a measure µ ∈ M0(X) define the compact open values set as the set of all
finite values of the measure µ on the compact open sets:
S(µ) = {µ(U) : U is compact open in X and µ(U) < ∞}.
For each measure µ ∈ M0(X), the set S(µ) is a countable dense subset of the
interval [0, µ(X)). Indeed, the set S(µ) is dense in [0, µ(V )] for every compact
open set V of a finite measure (see [10]). By Proposition 2.1, S(µ) is dense in
[0, µ(X)).
Let X1, X2 be two non-compact locally compact Cantor sets. It is said that
the measures µ1 ∈ M(X1) and µ2 ∈ M(X2) are homeomorphic if there exists a
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 263
O.M. Karpel
homeomorphism h : X1 → X2 such that µ1(E) = µ2(h(E)) for every Borel subset
E ⊂ X1. Clearly, S(µ1) = S(µ2) for any homeomorphic measures µ1 and µ2.
We call the two Borel infinite measures µ1 ∈ M0∞(X1) and µ2 ∈ M0∞(X2) weakly
homeomorphic if there exists a homeomorphism h : X1 → X2 and a constant
C > 0 such that µ1(E) = Cµ2(h(E)) for every Borel subset E ⊂ X1. Then
S(µ1) = CS(µ2).
Let D be a dense countable subset of the interval [0, a) where a ∈ (0,∞].
Then D is called group-like if there exists an additive subgroup G of R such that
D = G∩ [0, a). It is easy to see that D is group-like if and only if for any α, β ∈ D
such that α ≤ β we have β − α ∈ D (see [9, 11]).
Definition 2.2. Let X be a locally compact Cantor space (either compact or
non-compact) and µ ∈ M0(X). A compact open subset V of X is called good for
µ (or just good when the measure is understood) if for every compact open subset
U of X with µ(U) < µ(V ) there exists a compact open set W such that W ⊂ V
and µ(W ) = µ(U). A measure µ is called good if every compact open subset of
X is good for µ.
If µ ∈ M0(X) is a good measure and ν ∈ M0(X) is (weakly) homeomorphic to
µ, then, obviously, ν is good. It is easy to see that in the case of a compact Cantor
set the definition of a good measure coincides with the one given in [11]. For a
compact open subset U ⊂ X, let µ|U be the restriction of the measure µ to the
Cantor space U . Then the set U is good if and only if S(µ|U ) = S(µ|X)∩[0, µ(U)].
Denote by Hµ(X) the group of all homeomorphisms of a space X preserving the
measure µ. The action of Hµ(X) on X is called transitive if for every x1, x2 ∈ X
there exists h ∈ Hµ(X) such that h(x1) = x2. The action is called topologically
transitive if there exists a dense orbit, i.e., there is x ∈ X such that the set
O(x) = {h(x) : h ∈ Hµ(X)} is dense in X.
We extend naturally the notion of partition basis introduced in [4]. A partition
basis B for a non-compact locally compact Cantor set X is a collection of the
compact open subsets of X such that every non-empty compact open subset of
X can be partitioned by the elements of B.
The properties of good measures on non-compact locally compact Cantor sets
are gathered in the following proposition. The proofs for the measures on compact
Cantor spaces can be found in [4, 9, 11].
Proposition 2.3. Let X be a locally compact Cantor space (either compact
or non-compact). Let µ ∈ M0(X). Then
(a) If µ is good and C > 0, then Cµ is good and S(Cµ) = CS(µ).
(b) If µ is good and U is a non-empty compact open subset of X, then the
measure µ|U is good and S(µ|U ) = S(µ) ∩ [0, µ(U)].
(c) µ is good if and only if every compact open subset of a finite measure is
good.
264 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3
Good Measures on Locally Compact Cantor Sets
(d) µ is good if and only if for every non-empty compact open subset U of a
finite measure the measure µ|U is good.
(e) If µ is good, then S(µ) is group-like.
(f) If a compact open set U admits a partition by good compact open subsets,
then U is good.
(g) The measure µ is good if and only if there exists a partition basis B con-
sisting of the compact open sets which are good for µ.
(h) If µ is good, then the group Hµ(X) acts transitively on X \ Mµ. In
particular, the group Hµ(X) acts topologically transitively on X.
(i) If µ is a good measure on X and ν is the counting measure on {1, 2, . . . , n},
then µ× ν is a good measure on X × {1, 2, . . . , n}.
P r o o f. (a), (b) are clear.
(c) Suppose that every compact open subset of finite measure is good. Let
V be any compact open set with µ(V ) = ∞ and U be a compact open set with
µ(U) < ∞. By Proposition 2.1, there exists a compact open subset W ⊂ V
such that µ(U) ≤ µ(W ) < ∞. By assumption, W is good. Hence there exists a
compact open set W1 ⊂ W with µ(W1) = µ(U) and V is good.
(d) Suppose that for every non-empty compact open subset U of finite mea-
sure, the measure µ|U is good. We prove that every compact open subset of
finite measure is good, then use (c). Let U , V be compact open sets with
0 < µ(U) < µ(V ) < ∞. Set W = U ∪ V . Then W is a compact open set of finite
measure. Since µ|W is good, there exists W1 ⊂ V such that µ(W1) = µ(U).
(e) If µ is good, then for any α, β ∈ S(µ) such that β − α ≥ 0 we have
β − α ∈ S(µ). Hence S(µ) is group-like.
(f) See [4] for the case of the finite measure and [9] for the infinite measure.
(g) If there exists a partition basis B consisting of compact open sets which
are good for µ, then, by (f), every compact open set is good.
(h) For any x, y ∈ X \Mµ there exists a compact open set U of finite measure
such that x, y ∈ U . By (d), the measure µ|U is a good finite measure on a Cantor
space U . By Theorem 2.13 from [11], there exists a homeomorphism h : U → U
which preserves µ and h(x) = y. Define h1 ∈ Hµ(X) to be h on U and the identity
on X \ U . For every x ∈ X \Mµ we have O(x) = X \Mµ. By Proposition 2.1,
the set X \Mµ is dense in X. Hence Hµ(X) acts topologically transitively on X.
(i) The rectangular compact open sets U × {z}, where U is compact open
in X and z ∈ {1, 2, . . . , n}, form a partition basis for X × {1, 2, . . . , n}. Since
µ× ν(U × {z}) = µ(U), these sets are good. The measure µ is good by (g).
For G, an additive subgroup of R we call a positive real number δ a divisor
of G if δG = G. The set of all divisors of G is called Div(G). By a full measure
on a discrete countable topological space Y we mean a measure ν such that
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 265
O.M. Karpel
0 < ν({y}) < ∞ for every y ∈ Y . We will use the following theorem for Y = Z,
but the proof remains correct for any discrete countable topological space Y .
Theorem 2.4. Let µ be a good measure on a non-compact locally compact
Cantor space X. Let ν be a full measure on Z, where Z is endowed with discrete
topology. Let G be an additive subgroup of R generated by S(µ). Then µ × ν is
good on X ×Z if and only if there exists C > 0 such that ν({i}) ∈ C ·Div(G) for
every i ∈ Z.
P r o o f. Let us prove the “if” part. Suppose µ is good on X and
ν({i}) ∈ C · Div(G) for some C > 0 and every i ∈ Z. By Proposition 2.3
(g), it suffices to prove that a compact open set of the form U × {i} is good
for any compact open U ⊂ X and any i ∈ Z. Thus, it suffices to show that
S(µ×ν|U×{i}) = S(µ×ν|X×Z)∩[0, µ×ν(U×{i})]. The inclusion S(µ×ν|U×{i}) ⊂
S(µ × ν|X×Z) ∩ [0, µ × ν(U × {i})] is always true, hence we have to prove the
inverse inclusion. We have S(µ× ν|U×{i}) = ν({i})S(µ|U ) = CδS(µ|U ) for some
δ ∈ Div(G). Since µ is good on X, we obtain S(µ|U ) = G ∩ [0, µ(U)]. Hence
S(µ×ν|U×{i}) = CG∩[0, Cδµ(U)] = CG∩[0, µ×ν(U×{i})]. Note that Cδµ(U) ∈
CG because δ ∈ Div(G). Therefore, it suffices to prove that S(µ×ν|X×Z) ⊂ CG.
The set S(µ×ν|X×Z) consists of all finite sums
∑
i,j µ(Ui)ν({j}), where each Ui is
a compact open set in X and j ∈ Z. We have
∑
i,j µ(Ui)ν({j}) =
∑
i,j µ(Ui)Cδj ⊂
CG, here δi ∈ Div(G). Hence S(µ×ν|U×{i}) ⊃ S(µ×ν|X×Z)∩ [0, µ×ν(U ×{i})]
and U × {i} is good.
Now we prove the “only if part”. Suppose that µ× ν is good on X×Z. Then
for any i ∈ Z we have S(µ × ν|X×{i}) = S(µ × ν|X×Z) ∩ [0, µ × ν(X × {i})].
Note that S(µ × ν|X×{i} = ν({i})S(µ|X). Denote by G̃ the additive subgroup
of R generated by S(µ × ν|X×Z). Let α = ν({i}). Then αG = G̃. Let j ∈ Z
and β = ν({j}). By the same arguments, we have βG = G̃. Then α
β ∈ Div(G).
Indeed, α
β G = 1
β G̃ = G. Hence α = βδ, where δ ∈ Div(G). Set C = ν({j}).
Then for every i ∈ Z we have ν({i}) = Cδi, where δi = ν({i})
ν({j}) ∈ Div(G).
Theorem 2.5. Let X, Y be non-compact locally compact Cantor sets. If
µ ∈ M0(X), ν ∈ M0(Y ) are good measures, then the product µ × ν is a good
measure on X × Y, and
S(µ× ν) =
{
N∑
i=0
αi · βi : αi ∈ S(µ), βi ∈ S(ν), N ∈ N
}
∩ [0, µ(X)× ν(Y )).
P r o o f. Let X =
⊔∞
m=1 Xn and Y =
⊔∞
n=1 Yn, where each Xn, Yn is a
Cantor set. Then X × Y =
⊔∞
m,n=1 Xm × Yn and µ × ν|Xm×Yn = µ|Xm × ν|Yn .
Since µ|Xn and ν|Yn are good finite or non-defective measures on a Cantor set,
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Good Measures on Locally Compact Cantor Sets
the measure µ × ν|Xm×Yn is good by Theorem 2.8 [4], Theorem 2.10 [9]. By
Proposition 2.3, µ× ν is good on X × Y .
Theorem 2.6. Let X, Y be non-compact locally compact Cantor spaces. Let
µ ∈ M0(X) and ν ∈ M0(Y ) be good measures. Let S(µ) = S(ν). Let M be
the defective set for µ and N be the defective set for ν. Assume that there is a
homeomorphism h : M → N where the sets M and N are endowed with the induced
topologies. Then there exists a homeomorphism h̃ : X → Y which extends h such
that µ = ν ◦ h̃.
Conversely, if µ ∈ M0(X) and ν ∈ M0(Y ) are good homeomorphic measures,
then S(µ) = S(ν), and there is a homeomorphism h : M → N.
P r o o f. The second part of the theorem is clear. We will prove the first
part. Let X =
⊔∞
i=1 Xi and Y =
⊔∞
j=1 Yj where Xi, Yj are compact Cantor
spaces.
First, consider the case when M = N = ∅, i.e., the measures µ, ν are either
finite or infinite locally finite measures. Since S(µ) = S(ν), we have µ(X1) ∈
S(ν). There exists n ∈ N such that ν(
⊔n−1
j=1 Yj) ≤ µ(X1) < ν(
⊔n
j=1 Yj). Since
S(ν) is group-like, we see that µ(X1)−ν(
⊔n−1
j=1 Yj) ∈ S(ν). Since ν is good, there
exists a compact open subset W ⊂ Yn such that ν(W ) = µ(X1) − ν(
⊔n−1
j=1 Yj).
Hence Z =
⊔n−1
j=1 YjtW is a compact Cantor set and µ(X1) = ν(Z). By Theorem
2.9 [11], there exists a homeomorphism h1 : X1 → Z such that µ|X1 = ν|Z ◦ h1.
Set h̃|X1 = h1. Consider (Yn \W )
⊔∞
j=n+1 Yj instead of Y, and
⊔∞
i=2 Xi instead of
X. Reverse the roles of X and Y . Proceed in the same way using Yn \W instead
of X1. Thus, we obtain countably many homeomorphisms {hi}∞i=1. Given x ∈ X,
set h̃(x) = hi(x) for the corresponding hi. Then h̃ : X → Y is a homeomorphism
which maps µ into ν.
Now, let M 6= ∅. If µ(X1) < ∞, we proceed as in the previous case. If
µ(X1) = ∞, then X1 ∩ M 6= ∅. Then h(X1 ∩ M) is a compact open subset of
N in the induced topology. Hence there exists a compact open set W ⊂ Y such
that W ∩N = h(X1 ∩M). Then, by Theorem 2.11 [9], the sets X1 and W are
homeomorphic via the measure preserving homeomorphism h1 and h1|X1∩M = h.
Since W is compact, there exists N such that W ⊂ ⊔N
n=1 Yn. Reverse the roles
of X and Y and consider
⊔N
n=1 Yn \W instead of X1.
The corollary for weakly homeomorphic measures follows:
Corollary 2.7. Let µ ∈ M0∞(X) and ν ∈ M0∞(Y ) be good infinite measures
on the non-compact locally compact Cantor sets X and Y . Let M be the defective
set for µ and N be the defective set for ν. Then µ is weakly homeomorphic to ν
if and only if the following conditions hold:
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O.M. Karpel
(1) There exists c > 0 such that S(µ) = cS(ν),
(2) There exists a homeomorphism h : M → N where the sets M and N are
endowed with the induced topologies.
R e m a r k 1. Let µ ∈ M0∞(X) be a good measure on a non-compact locally
compact Cantor set X and V be any compact open subset of X with µ(V ) < ∞.
Then µ on X is homeomorphic to µ on X \ V . Let S(µ) = G∩ [0,∞). Then µ is
homeomorphic to cµ if and only if c ∈ Div(G).
Corollary 2.8. Let µ be a good finite or non-defective measure on a non-
compact locally compact Cantor set X. Let U , V be two compact open subsets of
X such that µ(U) = ν(V ) < ∞. Then there is h ∈ Hµ(X) such that h(U) = V .
P r o o f. Set Y = U ∪ V . Then Y is a Cantor set with µ(Y ) < ∞. By
Proposition 2.11 from [11], there exists a self-homeomorphism h of Y such that
h(U) = V and h preserves µ. Set h to be the identity on X \ Y .
Corollary 2.9. Let µ and ν be good non-defective measures on non-compact
locally compact Cantor sets X and Y . Let M be the defective set for µ and N be
the defective set for ν. If there exist compact open sets U ⊂ X and V ⊂ Y such
that µ(U) = ν(V ) < ∞ and µ|U is homeomorphic to ν|V , then µ is homeomorphic
to ν if and only if M and N (with the induced topologies) are homeomorphic.
P r o o f. Let γ = µ(U) = ν(V ). Since µ|U is homeomorphic to ν|V , we have
S(µ|U) = S(ν|V ). Since µ and ν are good, we have S(µ) ∩ [0, γ] = S(ν) ∩ [0, γ]
by Proposition 2.3. Since S(µ) and S(ν) are group-like, we obtain S(µ) = S(ν).
Theorem 2.10. Let µ ∈ M0(X) be a good measure on a non-compact lo-
cally compact Cantor set X. Then the compact open values set S(µ) is group-
like and the defective set Mµ is a locally compact zero-dimensional metric space
(including ∅).
Conversely, for every countable dense group-like subset D of [0, 1), there is
a good probability measure µ on a non-compact locally compact Cantor set such
that S(µ) = D. For every countable dense group-like subset D of [0,∞) and any
locally compact zero-dimensional metric space A (including A = ∅) there is a good
non-defective measure µ on a non-compact locally compact Cantor set such that
S(µ) = D and Mµ is homeomorphic to A.
P r o o f. The first part of the theorem follows from Propositions 2.1, 2.3.
We will prove the second part. First, consider the case of the finite measure.
Let D ⊂ [0, 1) be a countable dense group-like subset. Then there exists a strictly
increasing sequence {γn}∞n=1 ⊂ D such that limn→∞ γn = 1. For n = 1, 2, . . . ,
268 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3
Good Measures on Locally Compact Cantor Sets
set δn = γn − γn−1. Denote Sn = D ∩ [0, δn]. Then Dn = 1
δn
(D ∩ [0, δn]) is a
group-like subset of [0, 1] with 1 ∈ Dn. In [11], it was proved that there exists a
good probability measure µn on a Cantor set Xn such that S(µn|Xn) = Dn. The
measure νn = δnµn is a good finite measure on Xn with S(νn|Xn) = D ∩ [0, δn].
Set X =
⊔∞
n=1 Xn and let µ|Xn = νn. Then µ is a good probability measure on
a non-compact locally compact Cantor space X and S(µ|X) = D.
Now consider the case of the infinite measure. Let γ ∈ D. Since D ⊂ [0,∞)
is group-like, we see that 1
γ D∩ [0, 1] is a group-like subset of [0, 1]. In [11], it was
proved that there exists a good probability measure µ1 on a Cantor space Y with
S(µ1) = 1
γ D ∩ [0, 1]. Set µ = γµ1. Then µ is a good finite measure on Y and
S(µ) = D ∩ [0, γ]. Endow the set Z with discrete topology. Let ν be a counting
measure on Z. Set X = Y × Z and µ̃ = µ× ν. Then, by Theorem 2.4, µ̃ is good
with S(µ̃) = D and Mµ̃ = ∅.
Suppose A is a non-empty compact zero-dimensional metric space. Then, by
Theorem 2.15 [9], there exists a good non-defective measure µ on a Cantor space
Y such that S(µ) = D and Mµ is homeomorphic to A. By the above, there
exists a good locally finite measure ν on a non-compact locally compact set X
with S(ν) = D and Mν = ∅. Set Z = Y tX and µ̃|Y = µ, µ̃|X = ν. Then µ̃ is
good on a non-compact locally compact Cantor set Z with S(µ̃) = D and Mµ̃ is
homeomorphic to A.
Suppose that A is a non-empty non-compact, locally compact zero-dimensional
metric space. Then A =
⊔∞
n=1 An where each An is a non-empty compact zero-
dimensional metric space. By Theorem 2.15 [9], for every n = 1, 2, . . . there exists
a good non-defective measure µn on a Cantor set Yn such that S(µn) = D and
Mµn is homeomorphic to An. Set X =
⊔∞
n=1 Yn and µ|Yn = µn. Then µ is
good on a non-compact locally compact Cantor set X with S(µ) = D, and Mµ̃
is homeomorphic to A.
Corollary 2.11. Let D be a countable dense group-like subset of [0,∞). Then
there exists an aperiodic homeomorphism of a non-compact locally compact Can-
tor set with good non-defective invariant measure µ̃ such that S(µ̃) = D.
P r o o f. We use the construction similar to the one in the proof of
Theorem 2.10. Let µ be a good measure on a Cantor set Y with S(µ) = D∩ [0, γ]
for some γ ∈ D. Let ν be a counting measure on Z. Set µ̃ = µ×ν on X = Y ×Z.
Then µ̃ is a good non-defective measure on a non-compact locally compact Cantor
set X with S(µ̃) = D. Since the measure µ is a good finite measure on Y , there
exists a minimal homeomorphism T : Y → Y such that µ is invariant for T
(see [11]). Let T1(x, n) = (Tx, n + 1). Then T1 is aperiodic homeomorphism of
X. The measure µ̃ is invariant for T1.
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O.M. Karpel
R e m a r k 2. The measure µ̃ built in Corollary 2.11 is invariant for any
skew product with the base (Y, T ) and the cocycle acting on Z.
Theorem 2.12. Let X be a non-compact locally compact Cantor set. Then
there exist continuum distinct classes of homeomorphic good measures in Mf (X).
There also exist continuum distinct classes of weakly homeomorphic good mea-
sures in M0∞(X).
P r o o f. There exist uncountably many distinct group-like subsets {Dα}α∈Λ
of [0, 1]. By Theorem 2.10, for each Dα there exists a good probability measure
µα on X such that S(µα) = Dα. By Theorem 2.6, the measures {µα}α∈Λ are
pairwise non-homeomorphic.
Let Y be a compact Cantor set. Let µ be a non-defective measure on Y .
Denote by [µ] the class of weak equivalence of µ in the set of all non-defective
measures on Y . There exist the continuum distinct classes [µα] of weakly home-
omorphic good non-defective measures on a Cantor set Y (see Theorem 2.18
in [9]). Moreover, if there exists C > 0 such that G(S(µα)) = CG(S(µβ)), then
µβ ∈ [µα]. Let ν be a counting measure on Z. Then, by Theorem 2.4, µα × ν
is a good measure on a non-compact locally compact Cantor set Y × Z, and
G(S(µα × ν)) = G(S(µα)). Hence, by Corollary 2.7, the measures µα × ν and
µβ × ν are weakly homeomorphic if and only if µβ ∈ [µα].
Proposition 2.13. If µ is a Haar measure for some topological group struc-
ture on a non-compact locally compact Cantor space X, then µ is a good measure
on X.
P r o o f. The ball B centered at the identity in the invariant ultrametric is a
compact open subgroup of X. Since µ is translation-invariant, by Proposition 2.3,
it suffices to show that µ|B is good for every such ball B. Since the restriction of
µ on B is a Haar measure on a compact Cantor space, µ|B is good by Proposition
2.4 in [4].
3. From Measures on Non-Compact Spaces to Measures
on Compact Spaces and Back
Let X be a non-compact locally compact Cantor space. A compactification of
X is a pair (Y, c) where Y is a compact space and c : X → Y is a homeomorphic
embedding of X into Y (i.e., c : X → c(X) is a homeomorphism) such that
c(X) = Y , where c(X) is the closure of c(X). In the paper, by compactification
we will mean not only a pair (Y, c) but also the compact space Y in which X can
be embedded as a dense subset. We will denote the compactifications of a space
X by the symbols cX, ωX, etc., where c, ω are the corresponding homeomorphic
embeddings.
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Good Measures on Locally Compact Cantor Sets
Let µ ∈ M0(X). We will consider only the compactifications cX where cX
is a Cantor set. Since c is a homeomorphism, the measure µ on X passes to a
homeomorphic measure on c(X). Since we are interested in the classification of
measures up to homeomorphisms, we can identify the set c(X) with X. Hence
X can be considered as an open dense subset of cX. The set cX \ X is called
the remainder of compactification. As far as X is locally compact, the remainder
cX \X is closed in cX for every compactification cX (see [14]). Since X = cX,
the set cX \X is a closed nowhere dense subset of cX.
The compactifications c1X and c2X of a space X are equivalent if there
exists a homeomorphism f : c1X → c2X such that fc1(x) = c2(x) for every
x ∈ X. We will identify the equivalent compactifications. For any space X one
can consider the family C(X) of all compactifications of X. The order relation
on C(X) is defined as follows: c2X ≤ c1X if there exists a continuous map
f : c1X → c2X such that fc1 = c2. Then we have f(c1(X)) = c2(X) and
f(c1X \ c1(X)) = c2X \ c2(X).
Theorem 3.1 (The Alexandroff compactification theorem). Every non-compact
locally compact space X has a compactification ωX with one-point remainder.
This compactification is the smallest element in the set of all compactifications
C(X) with respect to the order ≤.
The topology on ωX is defined as follows. Denote by {∞} the point ωX \X.
The open sets in ωX are the sets of the form {∞}∪(X \F ), where F is a compact
subspace of X, together with all sets that are open in X.
For any Borel measure ν on the set cX\X with the induced topology, µ̃ = µ+ν
is a Borel measure on cX such that µ̃|X = µ. Since the aim of compactification
is the study of a measure µ on a locally compact set X, we will consider only the
extensions µ̃ on cX where µ̃(cX \X) = 0.
Lemma 3.2. Let X be a non-compact locally compact Cantor set and µ ∈
M0(X). Let c1X, c2X be the compactifications of X such that c1X ≤ c2X.
Denote by µ1 the extension of µ on c1X, and by µ2 the extension of µ on c2X.
Then S(µ) ⊆ S(µ1) ⊆ S(µ2).
P r o o f. Since c1X ≤ c2X, there exists a continuous map f : c2X → c1X
such that f(c2X \ X) = c1X \ X and fc2(x) = c1(x) for any x ∈ X. Since
f is continuous, it suffices to prove that f preserves measure, that is, µ1(V ) =
µ2(f−1(V )) for any compact open V ⊂ X. Recall that we can identify ci(X) with
X. Hence we can consider f as an identity on X ⊂ ciX and f preserves measure.
That is, for every compact open subset U of X we have µ(U) = µ1(U) = µ2(U).
Hence S(µ) ⊆ S(µ1). Since µ(ciX \ X) = 0, the measure of any clopen subset
of ciX is the sum of the measures of the compact open subsets of X. Hence the
measures of all clopen sets are preserved. Thus, S(µ1) ⊆ S(µ2).
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O.M. Karpel
R e m a r k 3. We can consider the homeomorphic embedding of a set X
into a non-compact locally compact Cantor set Y such that µ(Y \X) = 0. Then,
by the same arguments as above, the inclusion S(µ|X) ⊆ S(µ|Y ) holds.
Theorem 3.3. Let X be a non-compact locally compact Cantor set and µ ∈
M0(X) be a good measure. Let cX be any compactification of X. Then µ is good
on cX if and only if S(µ|cX) ∩ [0, µ(X)) = S(µ|X).
P r o o f. First, we prove the “if” part. Let V be a clopen set in cX. Consider
two cases. Let V ∩ (cX \X) = ∅. Then V is a compact open subset of X. Since
µ is good on X and S(µ|cX) ∩ [0, µ(X)) = S(µ|X), we see that V remains good
in cX. Now suppose that V ∩ (cX \ X) 6= ∅. Then V ∩ X is an open set and
µ(V ) = µ(V ∩X) = µ(
⊔∞
n=1 Vn) where each Vn is a compact open set in X. Let
U be a compact open subset of X with µ(U) < µ(V ). Then there exists N ∈ N
such that µ(U) < µ(
⊔N
n=1 Vn). The set Z =
⊔N
n=1 Vn is a compact open subset of
X. Since S(µ|cX)∩ [0, µ(X)) = S(µ|X), we have µ(U) ∈ S(µ|X). Since µ is good
on X, there exists a compact open subset W ⊂ Z such that µ(W ) = µ(U).
Now we prove the “only if” part. Assume the converse. Suppose that µ is a
good measure and the equality does not hold. Then there exists γ ∈ (0, µ(X))
such that γ ∈ S(µ|cX)\S(µ|X). Since S(µ|X) is dense in (0, µ(X)), there exists a
compact open subset U ⊂ X such that µ(U) > γ. Hence γ ∈ S(µ|cX) ∩ [0, µ(U)]
and γ 6∈ S(µ|U ). Thus U is not good and we get a contradiction.
R e m a r k 4. By Proposition 2.1, the set X \Mµ is a non-compact locally
compact Cantor set and X \Mµ = X. Thus, the set X \ Mµ can be homeo-
morphically embedded into X and then into some compactification cX. After
embedding X \Mµ into X, we add only compact open sets of infinite measure.
Hence if µ was good on X \Mµ, it remains good on X and S(µ|X\Mµ
) = S(µ|X).
We can consider X as a step towards compactification of X \ Mµ and include
Mµ into cX \X. The measure µ ∈ M0(X) is locally finite on X \Mµ, so we can
consider only locally finite measures among infinite ones.
If µ is not good on a locally compact Cantor set X, then clearly µ is not good
on any compactification cX.
Corollary 3.4. Let µ be a good infinite locally finite measure on a non-
compact locally compact Cantor set X. Then µ is good on ωX.
P r o o f. By definition of topology on ωX, the “new” open sets have a
compact complement. Since µ is locally finite on X, the measure of compact
subsets of X is finite. Hence the measure of each new clopen set is infinite. By
Theorem 3.3, µ is good on ωX.
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Good Measures on Locally Compact Cantor Sets
Theorem 3.5. Let µ be a good measure on a non-compact locally compact
Cantor set X. Then for any γ ∈ [0, µ(X)) there exists a compactification cX
such that γ ∈ S(µ|cX).
P r o o f. The set S(µ|X) is dense in [0, µ(X)). Hence for every γ ∈
[0, µ(X)] there exist {γn}∞n=1 ⊂ S(µ|X) such that limn→∞ γn = γ. Since µ is good,
there exist the disjoint compact open subsets {Un}∞n=1 such that µ(Un) = γn.
Then U =
⊔∞
n=1 Un is a non-compact locally compact Cantor set. Consider the
compactification cX = ωU t c(X \U), where c(X \U) is any compactification of
X \ U . Then ωU is a clopen set in cX, and µ(ωU) = γ ∈ S(µ|cX).
From Theorems 3.3, 3.5 the corollary follows:
Corollary 3.6. For any measure µ on a non-compact locally compact Cantor
space X there exists a compactification cX such that µ is not good on cX.
If a measure µ ∈ M0(X) is a good probability measure, then, by Theorem 3.3,
the measure µ is good on cX if and only if S(µ|cX) = S(µ|X) ∪ {1}.
Proposition 3.7. Let X be a non-compact locally compact Cantor set and µ ∈
Mf (X). If there exists a compactification cX such that S(µ|cX) = S(µ|X) ∪ {1},
then 1 ∈ G(S(µ|X)).
P r o o f. Let γ ∈ S(µ|cX) ∩ (0, 1). Since the complement of a clopen set is
a clopen set, we have 1 − γ ∈ S(µ|cX). Since S(µ|cX) = S(µ|X) ∪ {1}, we have
γ, 1− γ ∈ S(µ|X). Hence 1 ∈ G(S(µ|X)).
Thus, if 1 6∈ G(S(µ|X)), then for any compactification cX the set S(µ|X)
cannot be preserved after the extension. The examples are given in the last
section.
The corollary follows from Proposition 3.7 and Theorem 3.3.
Corollary 3.8. Let µ be a probability measure on a non-compact locally com-
pact Cantor set X, and 1 6∈ G(S(µ|X)). Then for any compactification cX of X,
µ is not good on cX.
Theorem 3.9. Let µ be a good probability measure on a non-compact locally
compact Cantor set X. Then µ is good on the Alexandroff compactification ωX
if and only if 1 ∈ G(S(µ|X)).
P r o o f. By Proposition 3.7 and Theorem 3.3, if µ is good on ωX, then
1 ∈ G(S(µ|X)).
Suppose µ is good on X and 1 ∈ G(S(µ|X)). Since µ is good, any compact
open subset of µ is good, hence for every compact open U ⊂ X we have µ(U) =
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O.M. Karpel
G(S(µ|X))∩[0, µ(U)] = S(µ)∩[0, µ(U)]. Every clopen subset of ωX has a compact
open subset of X as a complement. Hence for every clopen V ⊂ ωX we have
µ(V ) = 1−µ(X \V ) ∈ G(S(µ|X))∩(0, 1) = S(µ|X). So, S(µ|ωX) = S(µ|X)∪{1}.
Hence µ is good on ωX by Theorem 3.3.
For a Cantor set Y, denote by M0(Y ) the set of all either finite or non-defective
measures on Y (see [9]). Since an open dense subset of a Cantor set is a locally
compact Cantor set, the corollary follows:
Corollary 3.10. Let Y be a (compact) Cantor set and the measure µ ∈
M0(Y ). Let X ⊂ Y be an open dense subset of Y of full measure. If µ is
good on Y, then µ is good on X.
P r o o f. The set X is a locally compact Cantor set, and Y is a compacti-
fication of X. Any compact open subset U of X is a clopen subset of Y and all
clopen subsets of U are compact open sets. Thus, a µ|Y -good compact open set
in X is, a fortiory, µ|X -good.
Thus, the extensions of a non-good measure are always non-good. The corol-
lary follows from Lemma 3.2, Theorem 3.3 and Corollary 3.10.
Corollary 3.11. Let X be a non-compact locally compact Cantor set and
µ ∈ M0(X). Let c1X, c2X be compactifications of X such that c1X ≥ c2X. Let
µ be good on c1X. Then µ is good on c2X. Moreover, if µ ∈ Mf (X), then µ|c1X
is homeomorphic to µ|c2X .
R e m a r k 5. Recall that the Alexandroff compactification ωX is the smallest
element in the set of all compactifications of X. Hence, if µ is not good on ωX,
then µ is not good on any compactification cX of X.
The following theorem can be proved using the results of Akin [11] for mea-
sures on compact sets.
Theorem 3.12. Let X, Y be non-compact locally compact Cantor spaces, and
µ ∈ M0
f (X), ν ∈ M0
f (Y ) be good measures such that their extensions to ωX, ωY
are good. Then µ|X and ν|Y are homeomorphic if and only if S(µ|X) = S(ν|Y ).
P r o o f. The “only if” part is trivial, we prove the “if” part. Since µ|ωX
and ν|ωY are good by Theorem 3.3, we have S(µ|ωX) = S(ν|ωY ). Denote by x0 =
ωX \X and y0 = ωY \ Y . By Theorem 2.9 [11], there exists a homeomorphism
f : ωX → ωY such that f∗µ = ν and f(x0) = y0. Hence f(X) = Y and the
theorem is proved.
In Example 1, we present a class of good measures on non-compact locally
compact Cantor sets such that these measures are not good on the Alexandroff
compactifications. Thus, these measures are not good on any compactification of
the corresponding non-compact locally compact Cantor sets.
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Good Measures on Locally Compact Cantor Sets
4. Examples
E x a m p l e 1. (Ergodic invariant measures on stationary Bratteli diagrams).
Let B be a non-simple stationary Bratteli diagram with the matrix A transpose
to the incidence matrix. Let µ be an ergodic R-invariant measure on B (see [6,
9, 15]). Let α be the class of vertices that defines µ. Then µ is good as a measure
on a non-compact locally compact set Xα. The measure µ on Xα can be either
finite or infinite, but it is always locally finite. The set XB is a compactification
of Xα. Since µ is ergodic, we have µ(XB \ Xα) = 0. In [6, 9], the criteria
of goodness for probability or non-defective measure µ on XB were proved in
terms of the Perron–Frobenius eigenvalue and eigenvector of A corresponding to
µ (see Theorem 3.5 [6] for probability measures and Corollary 3.4 [9] for infinite
measures). It is easy to see that these criteria are particular cases of Theorem 3.3.
We consider now a class of the stationary Bratteli diagrams and give a crite-
rion when a measure µ from this class is good on the Alexandroff compactification
ωXα. Fix an integer N ≥ 3 and let
FN =
2 0 0
1 N 1
1 1 N
be the incidence matrix of the Bratteli diagram BN . For AN = F T
N we can
easily find the Perron–Frobenius eigenvalue λ = N + 1 and the corresponding
probability eigenvector
x =
(
1
N
,
N − 1
2N
,
N − 1
2N
)T
.
Let µN be the measure on BN determined by λ and the eigenvector x. The
measure µN is good on ωXα if and only if for there exists R ∈ N such that
2(N+1)R
N−1 is an integer. This is possible if and only if N = 2k + 1, k ∈ N. For
instance, the measure µN is good on ωXα for N = 3, 5 but is not good for N = 4.
Note that the criterion for goodness on ωXα here is the same as for goodness
on XB. This example is a particular case of more general result (the notation
from [6] is used below):
Proposition 4.1. Let B be a stationary Bratteli diagram defined by a distin-
guished eigenvalue λ of the matrix A = F T . Denote by x = (x1, . . . , xn)T the
corresponding reduced vector. Let the vertices 2, . . . , n belong to the distinguished
class α corresponding to µ. Then µ is good on XB if and only if µ is good on
ωXα.
P r o o f. By Theorem 3.1 and Corollary 3.11, if µ is good on XB, then µ is
good on ωXα. We prove the converse. By Theorem 3.5 in [6] and Theorem 3.9,
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 275
O.M. Karpel
it suffices to prove that 1 ∈ G(S(µ|Xα)) only if there exists R ∈ N such that
λRx1 ∈ H(x2, . . . , xn). Note that G(S(µ|Xα)) =
(⋃∞
N=0
1
λN H(x2, . . . , xn)
)
, where
H(x2, . . . , xn) is an additive group generated by x2, . . . , xn. Suppose that 1 ∈
G(S(µ|Xα)). Since
∑n
i=1 xi = 1, we see that x1 ∈ G(S(µ|Xα)), hence there exists
R ∈ N such that λRx1 ∈ H(x2, . . . , xn).
Return to a general case of ergodic invariant measures on the stationary Brat-
teli diagrams. If µ is a probability measure on Xα and S(µ|Xα)∪{1} = S(µ|XB
),
then, by Lemma 3.2, we have S(µ|ωXα) = S(µ|XB
). By Theorem 3.3, the mea-
sure µ is good on ωXα. Hence µ|ωXα is homeomorphic to µ|XB
(see [11]). If µ is
infinite, then the measures µ|ωXα and µ|XB
are not homeomorphic since Mµ|ωXα
is one point and Mµ|XB
is a Cantor set (see [9]).
E x a m p l e 2. Let X be a Cantor space and µ be a good probability
measure on X with S(µ) = {m
2n : m ∈ N∩ [0, 2n], n ∈ N} (for example a Bernoulli
measure β(1
2 , 1
2)). Clearly, µn = 1
2n µ is a good measure for n ∈ N with S(µn) =
1
2n S(µ) ⊂ S(µ). Let {Xn, µn}∞n=1 be a sequence of Cantor spaces with measures
µn. Let A =
⊔∞
n=1 Xn be the disjoint union of Xn. Denote by ν a measure on
A such that ν|Xn = µn. Then ν is a good measure on a locally compact Cantor
space A with S(ν) = S(µ) ∩ [0, 1).
Consider the one-point compactification ωA and the extension ν1 of ν to ωA.
We add to S(ν) the measures of sets which contain {∞} and have a compact open
complement. Hence we add the set Γ = {1−γ : γ ∈ S(ν)}. Since Γ ⊂ S(ν)∪{1},
we have S(ν1) = S(ν) ∪ {1}. By Theorem 3.3, the measure ν1 is good on ωA.
Consider the two-point compactification of A. Let A = A1 t A2 where
A1 =
⊔∞
k=1 X2k−1 and A2 =
⊔∞
j=1 X2j . Then cA = ωA1tωA2 is a two-point com-
pactification of A. Let ν2 be the extension of ν to cA. Then ν2(A1) = 2
3 6∈ S(ν).
Hence, by Theorem 3.3, the measure ν2 is not good on cA.
In the same example, we can make a two-point compactification which pre-
serves S(ν|A). Since µn is good for n ∈ N, there is a compact open partition
X
(1)
n tX
(2)
n = Xn such that µn(X(i)
n ) = 1
2n+1 for i = 1, 2. Let Bi =
⊔∞
n=1 X
(i)
n for
i = 1, 2. Consider c̃A = ωB1 t ωB2. Then it can be proved in the same way as
above that S(ν|c̃A) = S(ν|A) ∪ {1}.
E x a m p l e 3. Let µ = β(1
3 , 2
3) be a Bernoulli (product) measure on
the Cantor space Y = {0, 1}N generated by the initial distribution p(0) = 1
3 ,
p(1) = 2
3 . Then µ is not good but S(µ) = { a
3n : a ∈ N ∩ [0, 3n], n ∈ N} is group-
like (see [10]). Let X be an open dense subset of Y such that µ(Y \ X) = 0.
Thus, Y is a compactification of a non-compact locally compact Cantor space X
and µ extends from X to Y . Then µ is not good on X.
The compact open subsets of X are exactly the clopen subsets of Y that lie
in X. The compact open subset of X is a union of the finite number of the com-
pact open cylinders. Consider any compact open cylinder U = {a0, . . . , an, ∗}
276 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3
Good Measures on Locally Compact Cantor Sets
which consists of all points in z ∈ Y such that zi = ai for 0 ≤ i ≤ n. Then U is a
disjoint union of two subcylinders V1 = {a0, . . . , an, 0, ∗} and V2 = {a0, . . . , an, 1, ∗}
with µ(V2) = 2µ(V1). Let the numerator of the fraction µ(V1) be 2k. Then the
numerator of the fraction µ(V2) is 2k+1. Moreover, for any compact open W ⊂ V2
the numerator of the fraction µ(W ) will be divisible by 2k+1. Since S(µ) con-
tains only finite sums of the measures of the cylinder compact open sets and the
denominators of the elements of S(µ) are the powers of 3, there is no compact
open subset W ⊂ V2 such that µ(W ) = µ(V1). Hence µ is not good on X.
Moreover, let x = {00 . . .} be a point in Y which consists only of zeroes. Then
S(µ|Y \{x}) S(µ|Y ) while S(µ|Y \{y}) = S(µ|Y ) for any y 6= x.
Consider the case y 6= x. Let, for instance, y = {111 . . .}, all other cases are
proved in the same way. Let Un = {z ∈ Y : z0 = . . . = zn−1 = 1, zn = 0}.
Then Y \ {y} =
⊔∞
n=1 Un t {0∗}. Denote SN = µ(
⊔N
n=1 Un) and S0 = 0. Then
limN→∞ SN = 2
3 . Let G = { a
3n : a ∈ Z, n ∈ N}. Then G is an additive subgroup
of reals, and S(µ|Y ) = G ∩ [0, 1]. We prove that S(µ|Y \{y}) = G ∩ [0, 1), i.e.,
for every n ∈ N and a = 0, . . . , 3n − 1 there exists a compact open set W in
Y \ {y} such that µ(W ) = a
3n . Indeed, we have S(µ|{0∗}) = G ∩ [0, 1
3 ] and
[0, 1) = ∪∞n=0[SN , SN + 1
3 ]. Hence G∩ [0, 1) = ∪∞n=0(G∩ [SN , SN + 1
3 ]). For every
γ ∈ G there exists N ∈ N such that γ ∈ [SN , SN + 1
3 ]. There exists a compact
open subset W0 of {0∗} such that µ(W0) = γ−SN . Set W = UN tW0. Then W
is a compact open subset of Y \ {y} and µ(W ) = γ.
Now consider the set Y \ {x}. Every cylinder that lies in Y \ {x} has an even
numerator, hence S(µ|Y \{x}) S(µ|Y ). It can be proved in the same way as
above that S(µ|Y \{x}) = { 2k
3n : k ∈ N} ∩ [0, 1).
E x a m p l e 4. ((C, F )-construction). Denote by |A| the cardinality of a
set A. Given two subsets E, F ⊂ Z, by E +F we mean {e+ f |e ∈ E, f ∈ F} (for
more details see [13, 16]). Let {Fn}∞n=1, {Cn}∞n=1 ⊂ Z such that for each n:
(1) |Fn| < ∞, |Cn| < ∞,
(2) |Cn| > 1,
(3) Fn + Cn + {−1, 0, 1} ⊂ Fn+1,
(4) (Fn + c) ∩ (Fn + c′) = ∅ for all c 6= c′ ∈ Cn+1.
Set Xn = Fn×
∏
k>n Ck and endow Xn with a product topology. By (1), (2),
each Xn is a Cantor space.
For each n, define a map in,n+1 : Xn → Xn+1 such that
in,n+1(fn, cn+1, cn+2, . . .) = (fn + cn+1, cn+2, . . .).
By (1), (2) each in,n+1 is a well defined injective continuous map. Since Xn is
compact, we see that in,n+1 is a homeomorphism between Xn and in,n+1(Xn).
Thereby the embedding in,n+1 preserves topology. The set in,n+1(Xn) is a clopen
subset of Xn+1. Let im,n : Xm → Xn such that im,n = in,n−1◦in−1,n−2◦. . .◦im+1,m
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 277
O.M. Karpel
for m < n and in,n = id. Denote by X the topological inductive limit of the
sequence (Xn, in,n+1). Then X =
⋃∞
n=1 Xn. Since im,n = in,n−1 ◦ in−1,n−2 ◦
. . . ◦ im+1,m for m < n, we can write X1 ⊂ X2 ⊂ . . . . The set X is a non-
compact locally compact Cantor set. The Borel σ-algebra on X is generated by
the cylinder sets [A]n = {x ∈ X : x = (fn, cn+1, cn+2, . . .) ∈ Xn and fn ∈ A}.
There exists a canonical measure on X. Let κn stand for the equidistribution
on Cn and let νn = |Fn|
|C1|...|Cn| on Fn. The product measure on Xn is defined as
µn = νn × κn+1 × κn+2 × . . . , and a σ-finite measure µ on X is defined by the
restrictions µ|Xn = µn. The measure µ is a unique up to the scaling ergodic
locally finite invariant measure for a minimal self-homeomorphism of X (see [13,
16]). For every two compact open subsets U, V ⊂ X there exists n ∈ N such that
U, V ⊂ Xn. The measure µ is obviously good, since the restriction of µ onto Xn
is just the infinite product of the equidistributed measures on Fn and Cm, m > n.
We have S(µ) = { a
|C1|...|Cn| : a, n ∈ N} ∩ [0, µ(X)).
E x a m p l e 5. Let p be a prime number and Qp be the set of p-adic
numbers. Endowed with the p-adic norm, the set Qp is a non-compact locally
compact Cantor space. Thus the Haar measure µ on Qp is good and S(µ) =
{npγ |n ∈ N, γ ∈ Z}.
Acknowledgement. I am grateful to Prof. S. Bezuglyi for giving me the
idea of this work and for many helpful discussions of this paper.
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