On some topological properties of normed Boolean algebras

This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The motivation arises from studying random elements talking values in N.B.A. Those topological properties are important assumptions that enable us to avoid possible difficulties when generalising concepts of random variable convergence, the definition of conditional law and others. For each N.B.A., there exists a finite measure space $(E, {\mathcal E}, \mu)$ such that the N.B.A. is isomorphic to $(\widetilde{\mathcal E}, \widetilde{\mu})$ resulting from the factorisation of initial $\sigma$-algebra by the ideal of negligible sets. We focus on topological properties of $(\widetilde{\mathcal E}, \widetilde{\mu})$ in general setting when $\mu$ can be an infinite measure. In case when $\mu$ is infinite, we also consider properties of $\widetilde{\mathcal E}_{fin} \subseteq \widetilde{\mathcal E}$ consisting of classes of measurable sets having finite measure. The compactness and separability of the N.B.A. are characterised using the newly defined terms of approximability and uniform approximability of the corresponding measure space. Finally, conditions on $(E,\mathcal E,\mu)$ are derived for separability and compactness of $\widetilde{\mathcal E}$ and $\widetilde{\mathcal E}_{fin}.$


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 [14] and [15] provide basic definitions, notions and theoretical results on closed random sets (or compact random sets).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 [10].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 [17].)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 generalise 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 {X = X ′ } is an event.The distance between two random elements is a random variable, which allows us to introduce convergence in probability (see [18]).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 [7]).
Locally compactness is also a desirable property when considering weak convergence of distributions of random elements (see [2]).
For each complete N.B.A. (X , m) where m is finite, there exists a finite measure space (E, E, µ) such that the N.B.A. (X , m) and N.B.A. ( E, µ) resulting from factorisation of initial σ-algebra by the ideal of negligible sets are isomorphic (see [17]).Following this result we derive that the N.B.A. is homeomorphic to the space of indicator functions in L p (E, E, µ).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 L 2 space.Following results concerning the separability of L p spaces are established.If µ is σ-finite and E is countably generated, then L p (E, E, µ) is separable for 1 ≤ p < +∞ (see [6,Proposition 3.4.5.]).Since every metric subspace of separable metric space is separable [22,Theorem VIII,p. 160 ] if these conditions hold the space of indicators is separable as well.
If measure µ is not finite, then ( E, µ) is not homeomorphic to the space of indicator functions in L p (E, E, µ).In this case we also consider we prove the E f in and corresponding space of indicators is a separable if measure µ is outer regular.Although B(R d ) is countably generated, there are measures on B(R d ) which are outer regular but not σ-finite.
The compactness of subsets of L p -spaces has already been well studied, and some conditions for the compactness of general bounded subsets of L p -spaces can be found in [4] and [9,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 E 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 a conditions and in some cases characterisation for separability and compactness of E and E f in based on properties of corresponding measure space (E, E, µ).
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 ( E, d µ ) and ( E f in , d µ ) 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 a filtration.Separability and compactness are characterised using these terms.Further, we discuss separability and compactness of ( E), d µ ) and ( E f in , d µ ) based on the properties of the corresponding measure space (E, E, µ).
The paper is concluded by the Discussion section where the obtained results are summarised.

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 [13] and [20] .
For some A ⊂ X, let A = {G i } be a class of subsets of X such that In other words, if A is compact and A ⊂ ∪ i G i , where the G i are open sets, then one can select a finite number of the open sets G i1 , . . ., G im , so that If X is a topological space, a neighbourhood of x ∈ X is a subset V of X that includes an open set U such that x ∈ U. Definition 2.2.A topological space X is locally compact if every point in X has a compact neighbourhood.Definition 2.3.A subset S of a metric space X is called a totally bounded subset of X if, and only if, for each r ∈ R + , 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.

Measure theory
We will need following definitions and results from measure theory.The Lebesgue measure on R d is a regular measure (see e.g.[6]).However, not all σ-finite measures on R d are regular [12,Corollary 13.7].Also, there are some outer regular measures that are not σ-finite.For example, define It is easy to see that µ is a measure on (R, B(R)) that is not σ-finite but is outer regular.
A measure without any atoms is called non-atomic.
A measure space (E, E, µ), or the measure µ, is called purely atomic if there is a collection C of atoms of µ such that for each ).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 (E, E, µ) is localizable if there is a collection A of disjoint measurable sets of finite measure, whose union is all of X, such that for every set B ⊂ X, B is measurable if and only if B ∩ C ∈ E for all C ∈ A, and then µ(B) = C∈A µ(B ∩ C).Some examples of localisable measures are the σ-finite ones or counting measures on possibly uncountable sets.

Boolean algebra
In this section, we present basics concerning Boolean algebras of sets.For more details, see e.g.[17] or [19].
with two binary operations ∪ and ∩, a unary operation (•) c and two distinguished elements 0 and 1 such that for all A, B and C in X , ) if there exists a σ-additive strictly positive finite measure µ (i.e.µ(A) = 0 implies A = 0) defined on it.In this case, we use the notation (X , m).
On X × X we can define a relation ⊆ by setting It is easy to verify that ⊆ is a partial order relation.Definition 2.9.B.A. X is complete if for every non-empty subset C ⊆ X has its infimum and supremum.
Let (E, E, µ) be a finite measure space.We can define equivalence relation is the symmetric difference between the sets A and B (A c and B c denote the complements of A and B, respectively). Let The inverse result also holds.Namely, for each complete N.B.A. (X , m), there exists a measure space (E, E, µ) such that the N.B.A. (X , m) is isomorphic to ( E, µ) (see [17]).Therefore, further on we focus on investigating properties of ( E, µ).We generalise above setting, by letting measure µ be an arbitrary, possibly non-finite. Define It is easy to see that d µ is a metric on E possibly taking infinite values.We suppose the topology on E is generated by d µ .We are interested in topological properties of ( E, d µ ).
Remark 2.1.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 ( E, d µ ) coincides with the ordered topology (see [17]).
Denote L 2 (E, E, µ) a Hilbert space of measurable functions which are square integrable with respect to the measure µ, where functions which agree µ almost everywhere are identified.Let , E and I are isometric.Suppose that E, E, µ).Let us show that f is an indicator function of some measurable set.There exists a subsequence (see e.g.[21,Theorem 16.25]) Also, following we conclude that I is closed.Following Theorem 2.2 we can conclude that I is complete metric subspace of L 2 (E, E, µ).Therefore, we have shown that in case µ is finite ( E, d µ ) is complete metric space.Let us show that this holds in a general case when µ is not finite.
so it is complete metric subset of ( E, d µ ).However, E f in is not a B.A. since it is not e.g.closed under complements.

Main result
Before we show the main result, in order to get intuition, we first start with a motivating example.
If we consider a ball in ( E, For each n ∈ N, we can partition K into 2 2n smaller squares ], i, j = 1, . . ., 2 n .Intuitively, we pixelise the unit square by a 2 n × 2 n net. We show that for each ǫ we can pixelise the unit square fine enough so that the error of approximation of the set B would be less than ǫ.Denote by I n = {1, 2, . . ., 2 n } × {1, 2, . . ., 2 n }.Lemma 3.1.For an arbitrary B ∈ B(K) and an arbitrary ǫ > 0, there exists n ∈ N and This result follows directly from Lemma 3.2 which we prove later in the paper.
Note that the family Following Lemma 3.1, for arbitrary ǫ > 0 the collection of balls where has no finite subcover.We can conclude that ( B(K), d λ ) is not compact.
In order to show that ( B(K), d λ ) is not locally compact, we will prove that closed ball with the centre in [∅] and radius ǫ < 1 Since C ǫ covers B(K) it also covers B ([∅], ǫ) We will show that open cover C ǫ cannot be reduced to a finite subcover.For that purpose, for 0 < ǫ 2 ≤ 1 2 we define a set where where the last inequality follows from the fact that from 0 , ǫ) has no finite subcover, B ([∅], ǫ) is not compact.We conclude that [∅] does not have a compact neighbourhood, so B(K) is not locally compact.
In order to generalise these ideas on arbitrary measure space (E, E, µ) we introduce following definitions.Definition 3.1.Let (E, E, µ) be a measure space and F = (F n ) n∈N a filtration, i.e.F n ⊆ E is σ-algebra such that F n ⊆ F n+1 .Let E ′ be an arbitrary subset of E.
Measure µ is approximable on E ′ with respect to F if for each A ∈ E ′ and each ǫ > 0 there exists n ǫ ∈ N and A ′ ∈ F nǫ such that µ(A∆A ′ ) < ǫ.
Measure µ is uniformly approximable on E ′ with respect to F if for each ǫ > 0 there exists n ǫ ∈ N such that for every A ∈ E ′ there exists A ′ ∈ F nǫ so that µ(A∆A ′ ) < ǫ.
For arbitrary Theorem 3.2.E ′ is totally bounded in ( E, d µ ) if and only if there exist exists a filtration F = (F n ) n∈N such that µ is uniformly approximable on E ′ with respect to F and |F n | is finite for each n ∈ N.
Proof.Suppose that E ′ is totally bounded.Then for each ǫ > 0 there exists a finite family of sets i ], ǫ), so there exists we get that µ is uniformly approximable on E ′ with respect to (F n ).
Conversely, if there exists (F n ) where |F n | is finite for each n ∈ N and µ is uniformly approximable on E ′ with respect to (F n ).For each ǫ > 0 there exists n such that for each B ∈ E ′ there exists Intuitively speaking, we can imagine a finite filtration (F n ) 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, ( E, d µ ) is separable if for each measurable set we can find a level of pixelisation such that the error is smaller than arbitrary ǫ > 0 and ( E, d µ ) is compact if for each ǫ > 0 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 ǫ for each measurable set. Let The last equality follows from continuity of the measure µ from below with respect to the increasing sequence (j1,...,j d ) can be represented as finite union of disjointed d-intervals A (m2) (j ′ 1 ,...,j ′ d ) .We take n 0 = max{n 1 , n 2 } and define where µ is an outer regular measure.Then ( E f in , d µ ) is separable.
d is finite (see Figure 2 for visualisation).Denote by A can be written as a union of sets from A n } ∪ {∅}.
Note that since A n forms a finite partition of R d , is a Polish space (i.e.complete separable metric space).
we can also conclude that in case of outer regular measure ) is a Polish space (i.e.complete separable metric space).
Note that Corollary 3.3 could be proven using separability of set indicators in As it has been already mentioned in Introduction, if E is countably generated and µ is a σ-finite measure then L 2 (E, E, µ) and set of indicators in L 2 (E, E, µ) are separable.For a finite µ, since ( E f in , d µ ) = ( E, d µ ) and ( E f in , d µ ) is homeomorfic to set of indicators, we can conclude ( E, d µ ) is separable.We provide an alternative proof of this fact using the notion of approximability.Theorem 3.4.Suppose that there exists C a countable family of subsets of E such that E = σ(C) and (E, E, µ) is a finite measure space.Then ( E, d µ ) is separable.
Proof.Without loss of generality, we can suppose that the family C is a family of disjointed sets that cover E and C = {C n : n ∈ N}.We define F n = σ(C 1 , . . ., C n ).For an arbitrary A ∈ E there exists {C k : k ∈ I} such that C k ∈ C and I is at most countable and µ(C kn ) < ǫ.So, µ is approximable on E with respect to finite (F n ) and therefore ( E, d µ ) is separable.
Theorem 3.5.Suppose µ = µ 1 + µ 2 and suppose that µ 2 is not (uniformly) approximable on E ′ with respect to a finite filtration, then µ is not (uniformly) approximable on E ′ 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 µ 2 is not uniformly approximable on E ′ with respect to any finite filtration, for arbitrary (F n ) and ǫ > 0 there exists B ∈ E ′ such that µ 2 (A∆B) ≥ ǫ for all A ∈ ∪ n∈N F n .But then µ(A∆B) = µ 1 (A∆B) + µ 2 (A∆B) ≥ ǫ, from which follows that µ is not uniformly approximable approximable on E ′ with respect to finite filtration.
Localizable measures can be decomposed into a non-atomic part and purely atomic part (Theorem 2.7).Following Theorem 3.5, measure µ is (uniformly) approximable if its non-atomic and purely atomic part are (uniformly) approximable.In other words, ( E, d µ ) is separable (compact) if non-atomic and purely atomic part of µ are (uniformly) approximable.Therefore, we focus on separability and compactness properties of ( E, d µ ), 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 µ(E) = ∞, then ( E, µ) is not separable.
Proof.Suppose that ( E, µ) is separable.Then there exists a filtration (F n ), |F n | < ∞ on (E, E, µ) such that µ is approximable on E with respect to (F n ).
We can suppose that , for each n we can find i n , For each n ∈ N, and an arbitrary So, µ(B∆A ′ ) ≥ 1 for each n ∈ N which contradicts the assumption of approximability.
Theorem 3.7.If measure µ on (E, E) is non-atomic than ( E, d µ ) is not compact or locally compact.
Proof.Suppose that µ is non-atomic, then for each B ∈ E such that µ(B) > 0, there exists A ∈ E such that A ⊆ B and 0 < µ(A) < µ(B).So for each B ∈ E and for each r ∈ R, 0 ≤ r < µ(B) there exists a measurable set A ⊂ B such that µ(A) = r.
Let (F n ) n∈N be an arbitrary filtration on E, such that F n is finite for each n ∈ N. In this case, we can assume that F n = σ{A ǫ ⊂ E with respect to any filtration containing finite σ-algebras.We conclude that ( E, d µ ) is not compact, and also each closed ball This is contradiction to a fact that µ is approximable on E f in with respect to (F n ) and therefore If E ∞ is infinite, its partitive set is also infinite.So, for an arbitrary filtration Theorem 3.9.Let µ be purely atomic measure on (E, E) where all the atoms are singletons.
Proof.First, let us prove that

Discussion
Using newly defined terms of approximability and uniformly approximability, the conditions for separability and compactness of ( E, d µ ) and ( E f in , d µ ) can be summarised in a Table 1.Table 2 provides the topological properties of ( E, d µ ) and ( E f in , d µ ) based on finiteness and atomicity properties of the corresponding measure space (E, E, µ).
Separable for E countably generated if and only if µ is finite (Theorem 3.4 and Theorem 3.7).
• for

Theorem 2 . 5 (
[11]).Every open subset U of R d , d ≥ 1, can be written as a countable union of disjoint half-open cubes of form A

Definition 2 . 5 .
Let A be a σ-algebra on R d that includes the σ-algebra B R d of Borel sets.A measure µ on R d , A is regular if (a) (locally finite) each compact subset K of R d satisfies µ(K) < +∞, (b) (outer regular) each set A in A satisfies µ(A) = inf{µ(U ) : U is open and A ⊆ U }, and (c) (inner regular) each open subset U of R d satisfies µ(U ) = sup{µ(K) : K is compact and K ⊆ U }.

1 √ 2
j) for some I ⊂ I n .}(3.1) is an infinite (countable) open cover of ( B(K), d λ ).(Since for every ǫ > 0 and arbitrary B ∈ B(K) there exists A in form A =(i,j)∈I A (n) (i,j) for some I ⊂ I n such that [B] ∈ B([A], ǫ) ) Suppose that ( B(K), d λ ) is compact,therefore the open cover (3.1) should have a finite subcover.It means that there exists m ∈ N such that the collection of open balls C m ǫ,f in = {B([A], ǫ) : A = (i,j)∈I A (m) (i,j) for some I ⊂ I m .}covers ( B(K), d λ ).However, if we take ǫ = and define a set

FurtherLemma 3 . 2 .
j − 1)/2 n , i j /2 n , i 1 , . . ., i d ∈ Z, a d-dimensional half-open interval in R d .We prove that an arbitrary Borel set in R d 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.If µ is a outer regular measure on R d then for an arbitrary B ∈ B(R d ) such that µ(B) < ∞ and an arbitrary ǫ > 0, there exists n 0 ∈ N and finite I ⊆ { − n 0 2 n0 + 1 . . ., 0, . . ., n 0 2 n0 } d such that µ Let us take an arbitrary B ∈ B(R d ), µ(B) < ∞ and an arbitrary ǫ > 0. Space R d can be represented as a decreasing union of the half-open d-intervals [−n, n d , n ∈ N. It holds that

Figure 2 :
Figure 2: Visualisation of sets in family {A x∈E f in µ({x}) < ∞ implies that ( E f in , d µ ) is compact.Since
Every metric subspace of separable metric space is separable.
which shows that E ′ is totally bounded.Corollary 3.1.(a) ( E, d µ ) is is separable if and only if there exist F = (F n ) n∈N a filtration for which |F n | is finite for each n ∈ N such that µ is approximable on E with respect to F .(b) ( E, d µ ) is is compact if and only if there exist F = (F n ) n∈N a filtration for which |F n | is finite for each n ∈ N such that µ is approximable on E with respect to F .
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.

Table 1 :
in , if µ is outer regular (Theorem 3.3),• for µ purely atomic measure where all the atoms are singletons, if and only if the set of atoms with finite measure E f in is countable (Theorem 3.8).Condition on (E, E, µ) for separability and compactness of ( E, d µ ) and ( E f in , d µ )