1. Introduction
The motivation for studying the topological properties of normed Boolean algebras arises from probability theory, more precisely from its subfield of stochastic geometry. Nowadays, the mathematical theory of random sets is very popular. The books [1] and [2] provide basic definitions, notions and theoretical results on closed random sets (or compact random sets) as random elements with values in the family of closed subsets of locally compact Hausdorff second countable topological space.
An approach to defining a random set that takes values in a more general family of sets than closed or compact sets is presented in [3] . There, the random set is represented as a random element taking values in a normed Boolean algebra (N.B.A.), i.e. a complete Boolean algebra endowed with a strictly positive finite measure, see [4] .) These random elements are defined using Borel subsets of N.B.A. generated by a distance on N.B.A. equal to the square root of a measure of symmetric difference between two elements.
If we want to study different types of convergence of these random sets taking values in N.B.A. or generalize some other concepts related to random variables, it is beneficial to ensure that the space of its values is a Polish space or a locally compact, Hausdorff and second countable topological space (LCSH space). This motivated us to study the topological properties of the N.B.A.s with respect to topology generated by the distance equal to the square root of a measure of symmetric difference between two elements.
Let us mention some conveniences we get when working with random elements with a separable metric space of values. In this setting, for every two random elements X and X', a set
is an event. The distance between two random elements is a random variable, which allows us to introduce convergence in probability (see [5] ). In this case, the space of simple random elements is a dense subspace. If the space of values of the random elements is complete and separable (Polish), then the conditional law can be defined and the Doob-Dynkin representation holds (see [6] ).
Locally compactness is also a desirable property when considering weak convergence of distributions of random elements (see [7] ).
It is worth mentioning that there are some other topologies that can be defined on N.B.A.s that can generate Borel sets (for more details see [4, Chapter 4]). The best known is the o-topology, which in our case coincides with the topology generated by the distance on N.B.A. mentioned above. However, in this paper, we will only focus on the topological properties of N.B.A. when considering the o-topology, since it was the most suitable for defining the notion of random set in [3] . To our knowledge, these topological properties have not yet been studied.
For each complete N.B.A.
where m is finite, there exists a finite measure space
such that the N.B.A.
and N.B.A.
resulting from the factorisation of initial σ-algebra by the ideal of negligible sets are isomorphic (see [4] ). Following this result, we derive that the N.B.A. is homeomorphic to the space of indicator functions
.
The measure space analysis approach allows us to apply some well-established properties of topologies on space of measurable functions to the topology on N.B.A.
We generalise this setting allowing μ and corresponding m to obtain infinite values. In this case, the above-mentioned homeomorphism does not hold.
As we mentioned before, if μ is finite, then topological properties of N.B.A. are equivalent to topological properties of a subset of indicators in L2 space. Following results concerning the separability of Lp spaces are established. If μ is σ-finite and
is countably generated, then
is separable for
(see [8, Proposition 3.4.5.]). Since every metric subspace of separable metric space is separable [9, Theorem VIII, p.~160] if these conditions hold the space of indicators is separable as well.
If measure μ is not finite, then
is not homeomorphic to the space of indicator functions in
. In this case, we also consider
, which is homeomorphic to the space of indicator functions in
. In case
we prove the
and corresponding space of indicators is separable if measure μ is outer regular. Although
is countably generated, there are measures on
which are outer regular but not σ-finite.
The compactness of subsets of Lp-spaces has already been well studied, and some conditions for the compactness of generally bounded subsets of Lp-spaces can be found in [10] and ( [11] , Theorems 18, 20, 21 pp.297). Although these conditions can be verified for our case when μ is finite, we introduce conditions that are easier to verify, more intuitive in our setting and can be applied for verifying compactness of
in case when μ is infinite.
It is well known that a separable space is a space that is “well approximated by a countable subset” and a compact space is a space that is “well approximated by a finite subset”. We construct conditions for the corresponding measure space that follow this intuition. We call those conditions approximability and uniform approximability. We prove that if the measure can be well approximated by its values on a countable family or a finite family of measurable sets, then the corresponding N.B.A. is separable or a compact metric space, respectively.
Verifying the conditions of approximability and uniform approximability, we derive conditions and in some cases characterisation for separability and compactness of
and
based on properties of corresponding measure space
.
The outline of the paper is as follows.
In the Preliminaries section, we recall basic definitions and results concerning separability and compactness, we also mention some results from the measure theory we use for deriving results. The final subsection is dedicated to the terminology concerning Boolean algebras. The metric spaces
and
are introduced and their completeness is discussed.
In the Main result section, we introduce properties of approximability and uniform approximability of measure with respect to filtration. Separability and compactness are characterised using these terms. Further, we discuss separability and compactness of
and
based on the properties of the corresponding measure space
.
The paper is concluded by the Discussion section where the obtained results are summarised.
2. Preliminaries
2.1. Topological Properties
Let us first recall definitions and the basic relation of topological properties we study. The definitions and the results we present can be found in [12] and [13] .
For some
, let
be a class of subsets of X such that
.
is called a cover of A, and an open cover if each
is open. Furthermore, if a finite subclass of A is also a cover of A, i.e. if
such that
then
contains a finite subcover.
Definition 1 A subset A of a topological space X is compact if every open cover of A is reducible to a finite cover.
In other words, if A is compact and
, where the
are open sets, then one can select a finite number of the open sets
, so that
.
If X is a topological space, a neighbourhood of
is a subset V of X that includes an open set U such that
.
Definition 2 A topological space X is locally compact if every point in X has a compact neighbourhood.
Definition 3 A subset S of a metric space X is called a totally bounded subset of X if, and only if, for each
, there is a finite collection of balls of X of radius r that. covers S. A metric space X is said to be totally bounded if, and only if, it is a totally bounded subset of itself.
Theorem 2.1 A metric space is compact if and only if it is complete and totally bounded.
Theorem 2.2 A subspace Y of a complete metric space is complete if and only if Y is closed.
Theorem 2.3 Every closed subset of a compact space is compact.
Definition 4 A topological space X is said to be separable if it contains a countable dense subset.
Theorem 2.4 Every metric subspace of separable metric space is separable.
2.2. Measure theory
We will need the following definitions and results from measure theory.
Theorem 2.5 ( [14] ) Every open subset U of
,
, can be written as a
countable union of disjoint half-open cubes of form
,
,
.
Definition 5 Let
be a σ-algebra on
that includes the σ-algebra
of Borel sets. A measure μ on
is regular if
(a) (locally finite) each compact subset K of
satisfies
,
(b) (outer regular) Each set A in
satisfies
, and
(c) (inner regular) each open subset U of
satisfies
Theorem 2.6 ( [8] ) Any finite measure on
is regular.
The Lebesgue measure on
is a regular measure (see e.g. [8] ). However, not all σ-finite measures on
are regular ( [15] , Corollary 13.7]. Also, there are some outer regular measures that are not σ-finite. For example, defined
by
. It is easy to see that μ is a measure
on
that is not σ-finite but is outer regular.
Definition 6 If
is a measure space, a set
is called an atom of μ iff
and for every
with
, either
or
.
A measure without any atoms is called non-atomic.
A measure space
, or the measure μ, is called purely atomic if there is a collection
of atoms of μ such that for each
,
is the sum of the numbers
for all
such that
.
Lemma 2.1 ( [16] ) An atom of any finite measure μ on
is a singleton
such that
.
Lemma 2.2 ( [17] ) Any atom of a Borel measure on a second countable Hausdorff space includes a singleton of positive measure.
In particular, a Borel measure on a second countable Hausdorff space is nonatomic if and only if every singleton has measure zero.
A measure space
is localizable if there is a collection
of disjoint measurable sets of finite measure, whose union is all of X, such that for every set
, B is measurable if and only if
for all
, and then
. Some examples of localisable measures are the σ-finite ones or counting measures on possibly uncountable sets.
Theorem 2.7 ( [18] ) Let
be a localisable measure space. Then there exist measures
and
such that
,
are purely atomic and
non-atomic.
2.3. Boolean Algebra
In this section, we present the basics concerning Boolean algebras of sets. For more details, see e.g. [4] or [19] .
Definition 7 A Boolean algebra (B.A.) is a structure
with two binary operations
and
, a unary operation
and two distinguished elements 0 and 1 such that for all A, B and C in
,
Definition 8 Let
be a B.A. The B.A.
is normed (N.B.A.) if there exists a σ-additive strictly positive finite measure μ (i.e.
implies
) defined on it. In this case, we use the notation
.
On
we can define a relation
by setting
if
. It is easy to verify that
is a partial order relation.
Definition 9 B.A.
is complete if every non-empty subset
has its infimum and supremum.
Let
be a finite measure space. We can define equivalence relation ~ on
by setting
if and only if
, where
is the symmetric difference between the sets A and B (
and
denote the complements of A and B, respectively).
Let
. Then
a quotient space of
by ~ is a complete N.B.A. endowed with the measure
defined by
.
The inverse result also holds. Namely, for each complete N.B.A.
, there exists a measure space
such that the N.B.A.
is isomorphic to
(see [4] ). Therefore, further on we focus on investigating properties of
. We generalise the above setting, by letting measure μ be arbitrary, possibly non-finite.
Define
by
It is easy to see that
is a metric on
possibly taking infinite values. We suppose the topology
is generated by
. We are interested in the topological properties of
.
Remark 1 Let us mention that there are many topologies introduced in B.A.s. The most popular among them is the order topology. It is known that the topology of the metric space
coincides with the ordered topology (see [4] ).
Denote
a Hilbert space of measurable functions that are square integrable with respect to the measure μ, where functions that agree μ almost everywhere are identified. Let
where
stands for the indicator of set A or a characteristic function of set A.
If
, we can define
by
. Since
and
are isometric.
Suppose that
converges to f in
. Since,
is complete,
. Let us show that f is an indicator function of some
measurable set. There exists a subsequence
such that
, μ-a.e. (see e.g. [20, Theorem 16.25]) Also, following
we conclude that
is closed. Following Theorem 2.2 we can conclude that
is complete metric subspace of
. Therefore, we have shown that in case μ is finite
is complete metric space. Let us show that this holds in a general case when μ is not finite.
Theorem 2.8
is complete metric space.
Proof. Suppose that
is a Cauchy sequence in
. Then for fixed,
there exists
such that for
. We define
,
. Since
, so
. Also,
is
a Cauchy sequence, and since
is complete, there exists
such
that
. Furthermore, there exists a subsequence
such that
, μ-a.e. It holds
which shows that
μ-a.e, or equivalently
. Following
, the sequence
is convergent, so
is complete.
Remark 2 In this case
, one can also consider
. It is easy to see that
is isometric to
, so it is a complete metric subset of
. However,
is not a B.A. since it is not e.g. closed under complements.
3. Main Result
Before we show the main result, in order to get intuition, we first start with a motivating example.
Suppose that
(an observation window) and consider
, where
is Borel σ-algebra on K and
Lebesgue measure.
If we consider a ball in
of radius
it holds:
For each
, we can partition K into
smaller squares
,
. Intuitively, we pixelise the unit square by a
net.
We show that for each
we can pixelise the unit square fine enough so that the error of approximation of the set B would be less than
. Denote by
.
Lemma 3.1 For an arbitrary
and an arbitrary
, there exists
and
such that
This result follows directly from Lemma 3.2 which we prove later in the paper.
Note that the family
is countable dense subset of
, so
is separable.
Following Lemma 3.1, for arbitrary
the collection of balls
(3.1)
is an infinite (countable) open cover
. (Since for every
and arbitrary
there exists A in form
for some
such that
)
Suppose that
is compact, therefore the open cover (3.1) should
have a finite subcover. It means that there exists
such that the collection of open balls
covers
.
However, if we take
and define a set
(3.2)
where
it is easy to see (Left plot in Figure 1 provides a visualisation of the set T2) that
, for each A such that
for some
,
. Therefore
is not contained in any ball in
so
cannot be a
![]()
Figure 1. Left plot: Visualisation of set
(coloured in grey) defined by (3.2). Right plot: Visualisation of the set
for
(coloured in dark grey) defined by (3.3).
cover of
. So countable open cover
of
has no finite subcover. We can conclude that
is not compact.
In order to show that
is not locally compact, we will prove that closed ball with the centre in
and radius
denoted by
is not compact.
Since
covers
it also covers
We will show that open cover
cannot be reduced to a finite subcover. For that purpose, for
we define a set
(3.3)
where
of
disjoint triangles whose union has Lebesgue measure equal to
, so
for each
. The right plot in Figure 1 provides a visualisation of the set
for
. For each A such that
for some
, it holds
(3.4)
where the last inequality follows from the fact that from
follows
. So
but it is not in any ball in
. We conclude
cannot be a cover of
. Since the countable open cover
of
has no finite subcover,
is not compact. We conclude that
does not have a compact neighbourhood, so
is not locally compact.
In order to generalise these ideas on arbitrary measure space
we introduce the following definitions.
Definition 10 Let
be a measure space and
a filtration, i.e.
is σ-algebra such that
. Let
be an arbitrary subset of
.
Measure μ is approximable on
with respect to
if for each
and each
there exists
and
such that
Measure μ is uniformly approximable on
with respect to
if for each
there exists
such that for every
there exists
so that
For arbitrary
denote by
.
Theorem 3.1
is separable in
if and only if there exists
a filtration for which
is finite for each
such that μ is approximable on
with respect to
.
Proof. Suppose that μ is approximable on
wih respect to
. Denote by
Since
is countable,
is also countable.
Let us show that
is dense in
. For arbitrary
and arbitrary
there exists
such that
, so
, and therefore
is countable dense subset of
and
is separable.
If
is separable, then there exists a countable dense subset of
, denote it by
. W can represent
as
for some countable
. Take
. Then for each
and each,
there exists
such that
so that
.
Therefore, μ is aproximable on
with respect to
. □
Theorem 3.2
is totally bounded in
if and only if there exist
exists a filtration
such that μ is uniformly approximable on
with respect to
and
is finite for each
.
Proof. Suppose that
is totally bounded. Then for each
there exists
a finite family of sets
such that
. For arbitrary
,
, so there exists
such that
. If we consider
and take
we get that μ is uniformly approximable on
with respect to
. □
Conversely, if there exists
where
is finite for each
and μ is uniformly approximable on
with respect to
. For each
there exists n such that for each
there exists
,
. So
and therefore
, which shows that
is totally bounded.
Corollary 3.2.1.
(a)
is separable if and only if there exists
a filtration for which
is finite for each
such that μ is approximable on
with respect to
.
(b)
is compact if and only if there exists
a filtration for which
is finite for each
such that μ is approximable on
with respect to
.
Proof. The (a) part follows directly from Theorem 3.1. The (b) part follows from Theorem 2.1, Theorem 2.8 and Theorem 3.2. □
Intuitively speaking, we can imagine a finite filtration
as a way to pixelise E that in each step (as n grows) we get a finer “grid”. Following Theorem 3.1 and Theorem 3.2,
is separable if for each measurable set we can find a level of pixelization such that the error is smaller than arbitrary
and
is compact if for each
we can find a level of pixelisation such that all measurable sets are well approximated on this level, i.e. the error of pixelisation is smaller than
for each measurable set.
Let us now classify measures μ on
based on topological properties of corresponding
.
Further on, denote by
,
, a
d-dimensional half-open interval in
. We prove that an arbitrary Borel set in
can be approximated by the finite union of disjoint half-open d-intervals in a sense that the measure of symmetric difference between the Borel set and the union is arbitrary small.
Lemma 3.2 If μ is a outer regular measure on
then for an arbitrary
such that
and an arbitrary
, there exists
and finite
such that
Proof. Let us take an arbitrary
,
and an arbitrary
.
Space
can be represented as a decreasing union of the half-open d-intervals
,
. It holds that
The last equality follows from continuity of the measure μ form below with
respect to the increasing sequence
.
Since
for given
there exists
such that for each
(3.5)
Since μ is outer regular, for
and arbitrary
there exist an open set O such that
and
Following Theorem 2.5, O can be represented as a countable union of almost disjoint half-open cubes
. Since
for chosen
there exists
such that
so since
Set
. Note that if
, each
can be represented as finite union of disjointed d-intervals
.
We take
and define
.
Note that
Therefore,
Theorem 3.3 Let
where μ is an outer regular measure. Then
is separable. □
Proof. For each,
the family
is finite (see Figure 2 for visualization). Denote by
Note that since
forms a finite partition of
,
. It holds that
. Also note that
. Let
, and note that
is countable. Note that from Lemma 3.2 it follows that the μ is approximable on
with respect to
), and from Theorem 3.1 it follows that
is separable. □
![]()
Figure 2. Visualisation of sets in family
for
(black),
(blue) and
(light blue).
Corollary 3.3.1 For any outer regular Borel measure μ on
,
is a Polish space (i.e. complete separable metric space).
Remark 3 Since
is isometric to set of indicators in
, we can also conclude that in case of outer regular measure μ set of indicators in
is a Polish space.
Corollary 3.3.2 For any finite Borel measure μ on
,
is a Polish space (i.e. complete separable metric space).
Note that Corollary 3.3.2 could be proven using separability of set indicators in
.
As it has been already mentioned in Introduction, if
is countably generated and μ is a σ-finite measure then
and set of indicators in
are separable. For a finite μ, since
and
is homeomorfic to set of indicators, we can conclude
is separable. We provide an alternative proof of this fact using the notion of approximability.
Theorem 3.4 Suppose that there exists
a countable family of subsets of E such that
and
is a finite measure space. Then
is separable.
Proof. Without loss of generality, we can suppose that the family
is a family of disjointed sets that cover E and
. We define
. For an arbitrary
there exists
such that
and I is at most countable and
. It holds
. If I is finite, we can take
and for
it holds
. If I is countable, we can suppose
. Since
, for
there exists
such that
. If we take
and
, it holds
. So, μ is approximable on
with respect
to finite
and therefore
is separable. □
Theorem 3.5 Suppose
and suppose that
is not (uniformly) approximable on
with respect to a finite filtration, then μ is not (uniformly) approximable on
with respect to any finite filtration.
Proof. We prove the result for uniformity approximability since the proof in a case of approximability is similar.
Since
is not uniformly approximable on
with respect to any finite filtration, for arbitrary
and
there exists
such that
for all
. But then
from which follows that μ is not uniformly approximable approximable on
with respect to finite filtration. □
Localizable measures can be decomposed into a non-atomic part and a purely atomic part (Theorem 2.7). Following Theorem 3.5, measure μ is (uniformly) approximable if its non-atomic and purely atomic parts are (uniformly) approximable. In other words,
is separable (compact) if non-atomic and purely atomic part of μ are (uniformly) approximable. Therefore, we focus on separability and compactness properties of
, first in a case when μ is non-atomic and then in a case of purely atomic μ.
Theorem 3.6 If μ is non-atomic measure and
, then
is not separable.
Proof. Suppose that
is separable. Then there exists a filtration
,
on
such that μ is approximable on
with respect to
. We can suppose that
, where
are disjoint and
. Since
, for each n we can find
,
such that
and since
we can choose
in a way that
,
. Since μ is non-atomic, we can construct inductively a sequence of measurable sets
in a following way:
and
. Note that
are disjointed. Let
. For each
, and an arbitrary
,
So,
for each
which contradicts the assumption of approximability. □
Theorem 3.7 If measure μ on
is non-atomic than
is not compact or locally compact.
Proof. Suppose that μ is non-atomic, then for each
such that
, there exists
such that
and
. So for each
and for each
,
there exists a measurable set
such that
.
Let
be an arbitrary filtration on
, such that
is finite for each
. In this case, we can assume that
where
are disjoint,
and
is a finite set of indices.
Suppose first that μ is finite. Let
and
. Note that
. For each
we can find Borel set
such that
. If we define set
, calculation similar to (3.4)
yields
for each
. This shows that μ is not uniformly approximable
, so
is not compact.
Let
. If
for all
, we can take
. Then
and
.
Otherwise, let
. Let
and
. For each
we take measurable set
such that
. If we define set
, again we have
and
.
Let
Previous discussion show that μ is not uniformly approximable on
with respect to any filtration containing finite σ-algebras. We conclude that
is not compact, and each closed ball
is not totally bounded, but it is also not compact since
is closed. So,
has no compact neighbourhood. We
conclude that
is not locally compact. □
From Theorems 3.6 and 3.7 we see that in the case of non-atomic measure μ,
is not separable if μ is an infinite measure and also not compact when μ is an arbitrary measure.
Further on, let μ be a purely atomic measure on
. Suppose that atoms of the μ are singletons. If E is second countable Hausdorff and
is a Borel σ-algebra on E, following Lemma 2.2, every measure μ on
satisfies the condition.
We define set
of all atoms with a finite measure and set
of all atoms with an infinite measure. To prove that corresponding
is separable if and only if
is countable we need the following result.
Lemma 3.3 Let
be a measurable space. Let
be a filtration on
such that
can be represented as
where
and
is a finite partition of E. If
is a uncountable subset of E, there exists a decreasing sequence
,
such that
is infinite for each
and
.
Proof. It holds
. It follows that
. Note that partition
refines partition
, so if
then
.
Suppose conversely, that for each
,
such that
is infinite for each
,
. Then
, where
. If we denote by
, in this case it holds
Since
is at most countable (as a subset of all finite sequences
,
,
) and
, it follows that
is at most
countable, which is contradicts the assumption that
is uncountable. □
Theorem 3.8 Let μ be a purely atomic measure on
where all the atoms are singletons.
(a)
is separable if and only if
and
is finite.
(b)
is separable if and only if
is countable.
Proof. Suppose
is countable, so it can be written in a form
. We first prove that
is separable. Denote by
. We set
,
.
For
it holds
, where
. For arbitrary
and arbitrary
there exists
finite
such that
. If we choose
,
. We conclude that μ is approximable on
with respect to
, so
is separable.
Further on, suppose
and
is finite, i.e.
for fixed
and let
,
. For
such that
it holds
and
. For
we have already proven that for
there exists n and
such that
. If we take
it holds
, so μ is approximable on
with respect to
and we conclude
is separable.
To prove the separability of
implies
is countable, we suppose conversely that
is separable and
is uncountable. Since
is separable, there exists filtration
such that
is finite and μ is approximable on
with respect to
. Each
can be represented as
where
and
is a finite partition of E.
Since
is uncountable, following Lemma 3.3 there exists
such that
is uncountable. In other words, there exists a decreasing sequence
such that
is infinite for each
and the intersection
is non-empty. We take
, set
and take
such that
. For arbitrary
, if
such
then
, since the sum is the uncountable sum of non-negative numbers. If
then
. So, for each
and each
it holds
This is a contradiction to the fact that μ is approximable on
with respect to
and therefore
is not separable.
Let us now prove that if
is separable then
is countable and
is finite. We suppose, conversely,
is separable and
is uncountable or
infinite. If
is uncountable then
is not separable. Since
is a subspace of
, using Theorem [9, Theorem VIII, p.~160] we can conclude
is not separable.
If
is infinite, its partitive set is also infinite. So, for an arbitrary filtration
with
we can find
such that
,
. It holds
for all
,
. So, μ is not approximable on
with respect to any
where
and therefore
is not separable. □
Theorem 3.9 Let μ be purely atomic measure on
where all the atoms are singletons.
(a)
is compact if and only if
is finite and
is finite.
(b)
is compact if and only if
is finite.
Proof. First, let us prove that
implies that
is compact. Since
implies that
is countable (see e.g. ( [21]
Theorem 3.12.6., p. 131)). We can assume that
. We denote
. It holds
So, for every
there exists
such that
. Define
. Let us show that μ is uniformly approximable on
with respect to
. If we take arbitrary
and set
it holds that
so following Theorem 3.2,
is compact.
Suppose
, for fixed
. To prove
is compact, we set
,
. For
such that
it holds
and
. We have shown that
there exists n such that for every
there exists
such that
. If we take
it holds
, so μ is uniformly approximable on
with respect to
and we conclude
is compact.
Let us prove that if
is compact then
. Suppose conversely, that
is compact and
. If
is uncountable, then following Theorem 3.8
is not separable, so it cannot be compact. Suppose now that
is compact and
is countable and
. Since
is compact there
exists a filtration
such that
is finite for each
and μ is uniformly approximable on
with respect to
.
For each,
it holds that
where
and
is a finite partition of E. We take an arbitrary
. For each n
there exists
such that
is infinite,
and
. We an find
such that
. If we take
, then it is easy to see that
for each
since if
and if
,
. So,
cannot be uniformly approximable on
with respect to
and
is not compact.
Let us now prove that if
is compact then
and
is finite. We suppose, conversely,
is compact and
or
is infinite. If
then
is not compact. Since
is a closed subspace of
, using Theorem [9, Theorem VIII, p.~160] we can conclude
is not compact.
If
is infinite,
is not separable, so it cannot be compact. □
We conclude the main part of the paper with a few examples.
Example 1 Let μ be a counting measure on
, i.e.
where
stands for the cardinal number of A if A is finite and
otherwise. Since
is countable,
is separable, but it is not compact since
. Also
is not separable. For arbitrary finite B and
arbitrary
,
. So, the closed ball with a radius of less than 1 around each element
and
is compact since it is finite. Therefore, they are both locally compact.
Example 2 Let μ be a counting measure on
,
where
stands for the cardinal number of A if A is finite and
otherwise. Since
is uncountable, the corresponding
is not separable and not compact. However,
and
are locally compact. Similarly to the previous example, for arbitrary finite B and arbitrary
,
. So, the closed ball with a radius less than 1 around each element of
is compact since it is finite.
Example 3 Let μ be a counting measure and
a Lebesgue measure on
. Let
,
, where
stands for a unit ball with centre at the origin. It is easy to see
and
are not separable not compact (since for
,
is uncountable) and not locally compact.
4. Discussion
Using newly defined terms of approximability and uniformly approximability, the conditions for separability and compactness of
and
can be summarised in Table 1. Table 2 provides the topological properties of
and
based on finiteness and atomicity properties of the corresponding measure space
.
![]()
Table 1. Condition on
for separability and compactness of
and
.
![]()
Table 2. Separability and compactness of
and
depending on whether μ is finite or infinite, purely atomic or non-atomic.
Future research will involve using the results from this paper to generalise the notions of different types of convergence of random sets as random elements taking values in N.B.A. and exploring their properties.
Funding
Supported in a part by the Czech Science Foundation, project No. 19-04412S, and by the Grant Agency of the Czech Technical University in Prague, project No. SGS21/056/OHK3/1T/13.