On Constructing Finite, Finitely Subadditive Outer Measures, and Submodularity

Abstract

Given a nonempty abstract set 𝑋, and a covering class 𝒞, and a finite, finitely subadditive outer measure 𝜈, we construct an outer measure 𝜈 and investigate conditions for 𝜈 to be submodular. We then consider several other set functions associated with 𝜈 and obtain conditions for equality of these functions on the lattice generated by 𝒞. Lastly, we describe a construction of a finite, finitely subadditive outer measure given an arbitrary family of subsets, ℬ, of 𝑋 and a nonnegative, finite set function 𝜏 defined on ℬ.

1. Introduction

Let 𝑋 be a nonempty set, and let 𝒞 be a covering class of subsets of 𝑋 such that ∅,𝑋∈𝒞, with 𝒞 closed under finite intersections, and let 𝜈 be a finite, finitely subadditive outer measure defined for all subsets of 𝑋. In [1, 2], we considered a finite, finitely subadditive outer measure 𝜈 defined by means of 𝜈. The outer measure 𝜈 generalized results obtained by [3] for the case of a 0-1 valued, finitely additive measure 𝜇 defined on the algebra of subsets generated by a lattice of subsets ℒ of 𝑋. In investigating properties of 𝜈, the hypothesis that 𝜈 be submodular is often present. We determine conditions for 𝜈 to be submodular.

We also consider for a finite, finitely subadditive outer measure 𝜈 and covering class 𝒞 under the hypothesis that 𝐶⊂𝒮_𝜈, the finite, finitely additive measure 𝜇=𝜈/_𝒜(ℒ)∈𝑀(ℒ), where ℒ=ℒ(𝒞) is the lattice generated by 𝒞. There are several set functions associated with 𝜇 and 𝜈, namely, 𝜈^′,𝜇^′,𝜇_𝑖,𝜌,𝜌^′ and it is shown that the inequality 𝜌^′≤𝜇_𝑖≤𝜌≤𝜈≤𝜇^′≤𝜈^′ always holds for all subsets of 𝑋. We determine when an equality of all these set functions holds on ℒ. This is useful since for instance it allows one to conclude that 𝜇 is an ℒ-regular measure on 𝒜(ℒ). Once again, the condition of submodularity plays an essential role.

Lastly, we consider the general problem of constructing a finite, finitely subadditive outer measure from an arbitrary family, ℬ, of subsets of 𝑋, with ∅,𝑋∈ℬ and ℬ closed under finite unions, and a nonnegative, finite set function 𝜏 defined on ℬ, with 𝜏(∅)=0. Assuming that 𝜏 is finitely subadditive on ℬ, the function 𝜆(𝐸)=inf{𝜏(𝐵)/𝐸⊂𝐵,𝐵∈ℬ} for 𝐸⊂𝑋 is a finite, finitely subadditive outer measure defined for all subsets of 𝑋. We obtain a characterization of those sets 𝐸⊂𝑋 that are 𝜆-measurable. This provides us with a condition for the sets of ℬ to be 𝜆-measurable without requiring that 𝜆 be submodular.

2. Background and Notation

We introduce the necessary outer measure, and covering class definitions, and note some known properties that we shall need.

The definitions and notations are standard and are consistent with those found in, for example, [1, 4–6]. We collect the ones we need and some of their properties for the reader's convenience.

Throughout this section and the rest of the paper, 𝑋 will denote a nonempty abstract set. For 𝐸⊂𝑋, 𝐸^′ will denote the complement of 𝐸. The power set of 𝑋 will be denoted by 𝒫(𝑋). We will be concerned with covering classes of subsets of 𝑋. By a covering class, we will always mean a nonempty class 𝒞 of subsets of 𝑋 such that ∅,𝑋∈𝒞 and such that for any 𝐸⊂𝑋, there is a finite family of sets 𝐶₁,𝐶₂,…,𝐶_𝑛∈𝒞 with 𝐸⊂⋃^𝑛_𝑖=1𝐶_𝑖. We always assume that 𝐶 is closed under finite intersections. We define ℒ=ℒ(𝒞) to be the lattice generated by 𝒞. ℒ(𝒞) is the set of all finite unions of sets from 𝒞. The algebra generated by ℒ is 𝒜(ℒ). It is known that 𝒜(ℒ)=𝒜(𝒞), the algebra generated by 𝒞. The set of complements of the members of 𝒞 is denoted by 𝒞^′.

Definition 2.1. A covering class 𝒞 is said to coallocate itself if whenever 𝐶,𝐶₁,𝐶₂∈𝒞, and 𝐶⊂𝐶^′₁∪𝐶^′₂, there are sets 𝐴,𝐵∈𝒞 with 𝐶=𝐴∪𝐵 and 𝐴⊂𝐶^′₁,𝐵⊂𝐶^′₂.

Definition 2.2. A covering class 𝒞 is said to be normal if for any 𝐴,𝐵∈𝒞 with 𝐴∩𝐵=∅, there exist 𝐶,𝐷∈𝒞 with 𝐴⊂𝐶^′,𝐵⊂𝐷^′, and 𝐶^′∩𝐷^′=∅.

Definition 2.3. Let 𝒞₁ and 𝒞₂ be covering classes with 𝒞₁⊂𝒞₂. 𝒞₁ is said to coseparate𝒞₂ if whenever 𝐵₁,𝐵₂∈𝒞₂,𝐵₁∩𝐵₂=∅, there exist 𝐴₁,𝐴₂∈𝒞₁ with 𝐵₁⊂𝐴^′₁,𝐵₂⊂𝐴^′₂, and 𝐴^′₁∩𝐴^′₂=∅.

It is known that if a covering class coallocates itself, then it is a normal covering class.

Let 𝜈 be a finite, finitely subadditive outer measure defined for all subsets of 𝑋 and let 𝒞 be a covering class of the subsets of 𝑋. In general, there are several set functions that we can associate with this outer measure 𝜈. For 𝐸⊂𝑋, we define 𝜌(𝐸)=𝜈(𝑋)−𝜈(𝐸^′). 𝜌 is finite valued and in general is not an inner measure. Conditions when 𝜌 is an inner measure have been thoroughly investigated in [4, 5]. We also define a set function 𝜈^′ by 𝜈^′(𝐸)=inf{𝜈(𝐶^′)/𝐸⊂𝐶^′,𝐶∈𝒞}for𝐸⊂𝑋, (2.1) where 𝜈^′ is always a finite, finitely subadditive outer measure, and so it has associated with it a 𝜌^′, which may be expressed as 𝜌^′(𝐸)=sup{𝜌(𝐶)/𝐶⊂𝐸,𝐶∈𝒞},𝐸⊂𝑋. (2.2)

It is always true that 𝜌^′≤𝜌≤𝜈≤𝜈^′ on 𝒫(𝑋). When a set function 𝜈 is an outer measure, the family of 𝜈-measurable sets is denoted by 𝒮_𝜈, that is, 𝒮_𝜈={𝑀⊂𝑋/𝜈(𝐴)=𝜈(𝐴∩𝑀)+𝜈(𝐴∩𝑀^′),∀𝐴⊂𝑋}. (2.3)

It is well known that 𝒮_𝜈 is an algebra and the restriction of 𝜈 to 𝒮_𝜈, that is, 𝜈/_𝒮𝜈 is a finite, finitely additive measure. We also define 𝒮^𝜈={𝐸⊂𝑋/𝜈(𝐸)=𝜌(𝐸)}.𝒮^𝜈 is not in general an algebra of sets, and we always have 𝒮_𝜈⊂𝒮^𝜈.

An outer measure 𝜈 satisfies Condition A: if 𝜈=𝜌^′ on 𝒞^′. 𝜈 satisfies Condition B: if 𝜈(𝐶^′)=sup{𝜈(𝐴)/𝐴⊂𝐶^′,𝐴∈𝒞}, for 𝐶∈𝒞.

Definition 2.4. Let ℰ be a collection of subsets of 𝑋 and let 𝑓 be a real-valued set function defined on ℰ.
If for all 𝐴,𝐵∈ℰ with 𝐴∪𝐵,𝐴∩𝐵∈ℰ, we have 𝑓(𝐴∪𝐵)+𝑓(𝐴∩𝐵)≤𝑓(𝐴)+𝑓(𝐵), (2.4) then 𝑓 is said to be submodular onℰ. If the reverse inequality holds, then 𝑓 is said to be supermodular onℰ. 𝑓 is said to be modular onℰ if equality holds. If 𝑓(𝐴∪𝐵)≥𝑓(𝐴)+𝑓(𝐵)whenever𝐴,𝐵∈ℰwith𝐴∩𝐵=∅, (2.5) then 𝑓 is said to be superadditive onℰ.
If ℰ=𝒫(𝑋), we usually say 𝑓 is submodular.

Let ℒ be a lattice of subsets of 𝑋 such that ∅,𝑋∈ℒ. 𝒜(ℒ) denotes the algebra generated by ℒ, and 𝑀(ℒ) denotes the set of all nontrivial, nonnegative, finitely additive measures on 𝒜(ℒ). 𝑀_𝑅(ℒ) denotes the set of all those 𝜇∈𝑀(ℒ) which are ℒ-regular, that is, those 𝜇∈𝑀(ℒ) with the property that for any 𝐴∈𝒜(ℒ)𝜇(𝐴)=sup{𝜇(𝐿)/𝐿⊂𝐴,𝐿∈ℒ}.

𝑀(ℒ), 𝑀_𝑅(ℒ), and several other subsets of 𝑀(ℒ) have been extensively studied in the literature; we cite just a few recent papers [4–8]. For 𝜇∈𝑀(ℒ), the set function 𝜇^′ defined for 𝐸⊂𝑋 by 𝜇^′(𝐸)=inf{𝜇(𝐿^′)/𝐸⊂𝐿^′,𝐿∈ℒ} (2.6) is a finite, finitely subadditive outer measure on 𝒫(𝑋). We also define the set function 𝜇_𝑖 by 𝜇_𝑖(𝐸)=𝜇^′(𝑋)−𝜇^′(𝐸^′)=sup{𝜇(𝐿)/𝐿⊂𝐸,𝐿∈ℒ}. (2.7)

3. Submodularity of the Outer Measure 𝜈

In [3], a 0-1 measure 𝜇 was used to give necessary and sufficient conditions for a lattice to be normal, by constructing a 0-1, finitely subadditive outer measure 𝜇. In [1, 2], the measure 𝜇 was replaced by a finite, finitely subadditive outer measure 𝜈, and the outer measure 𝜇 was generalized by a finite, finitely subadditive outer measure 𝜈. However, we now need to impose a separation property on the covering class, and require submodularity of the outer measures. In this section, we focus on the latter set functions, and determine conditions for 𝜈 to be submodular.

We begin by recalling the basic construction and properties of 𝜈. For emphasis, let 𝑋 be a nonempty set and let 𝒞 be a covering class of subsets of 𝑋, with ∅,𝑋∈𝒞 and 𝒞 closed under finite intersections. Let 𝜈 be a finite, finitely subadditive outer measure defined for all subsets of 𝑋. Assume that 𝒞 coallocates itself and the associated set function 𝜌 is finitely subadditive on 𝒫(𝑋). Since 𝜌 is finitely subadditive on 𝒫(𝑋), the set function 𝜌^′ is also finitely subadditive on 𝒫(𝑋) (see, e.g., [2]). For 𝐸⊂𝑋 define 𝜈(𝐸)=inf{𝜌^′(𝐶^′)/𝐸⊂𝐶^′,𝐶∈𝒞}, (3.1) where 𝜈 is a finite, finitely subadditive outer measure and

(1)𝜈=𝜌^′ on 𝒞^′,(2)𝜌=𝜈^′ on 𝒞,(3)𝜌^′≤𝜌≤𝜈≤𝜈^′ on 𝒫(𝑋),(4)𝜈 satisfies condition A, that is, 𝜈=𝜌^′ on 𝒞^′ which is equivalent to 𝜌=𝜈^′ on 𝒞.

(The functions 𝜈^′,𝜌,𝜌^′ are the usual ones associated with the outer measure 𝜈 as defined for a general finite, finitely subadditive outer measure in Section 2).

We now determine conditions for 𝜈 to be submodular. It will be shown that by requiring that 𝜈 or its associated set functions satisfy certain conditions on either 𝒞 or 𝒞^′, and 𝒞 is a lattice, the submodularity of 𝜈 will follow. The importance of 𝜈 being submodular can be seen from the following known facts [2].

It is known that if 𝜈 is submodular then

(1)𝒞⊂𝒮_𝜈=𝒮^𝜈,(2)𝜈 is 𝒞 regular on ℒ^′, where ℒ=ℒ(𝒞),(3)𝜈/_𝒜(ℒ)∈𝑀_𝑅(ℒ),(4)𝜈≤𝜈 on 𝒞.

Theorem 3.1. If 𝒞 is a lattice, and if 𝜈^′ is supermodular on 𝒞, then 𝜌^′ is submodular on 𝒞^′ and so 𝜈 is submodular on 𝒞^′.

Proof. Let 𝐶₁,𝐶₂∈𝒞. Since 𝜈^′ is supermodular on the lattice 𝒞, 𝜈^′(𝐶₁∪𝐶₂)+𝜈^′(𝐶₁∩𝐶₂)≥𝜈^′(𝐶₁)+𝜈^′(𝐶₂). (3.2) Therefore, 𝜈^′(𝑋)−𝜈^′(𝐶₁∪𝐶₂)+𝜈^′(𝑋)−𝜈^′(𝐶₁∩𝐶₂)≤𝜈^′(𝑋)−𝜈^′(𝐶₁)+𝜈^′(𝑋)−𝜈^′(𝐶₂), (3.3) and so 𝜌^′(𝐶^′₁∩𝐶^′₂)+𝜌^′(𝐶^′₁∪𝐶^′₂)≤𝜌^′(𝐶^′₁)+𝜌^′(𝐶^′₂). (3.4) Since 𝜈=𝜌^′ on 𝒞^′, the above inequality shows 𝜈(𝐶^′₁∩𝐶^′₂)+𝜈(𝐶^′₁∪𝐶^′₂)≤𝜈(𝐶^′₁)+𝜈(𝐶^′₂). (3.5) Since 𝐶₁,𝐶₂∈𝒞 are arbitrary, this shows that 𝜌^′ is submodular on 𝒞^′ and 𝜈 is submodular on 𝒞^′.

Theorem 3.2. Suppose 𝒞 is a lattice. If 𝜈 is submodular on 𝒞^′, then 𝜈 is submodular.

Proof. Let 𝐴,𝐵⊂𝑋 and let 𝜖>0 be given and arbitrary. By definition of 𝜈 there exist 𝐶₁,𝐶₂∈𝒞 such that 𝐴⊂𝐶^′₁,𝐵⊂𝐶^′₂ and 𝜈(𝐴)≤𝜌^′(𝐶^′₁)<𝜖𝜈(𝐴)+2,𝜈(𝐵)≤𝜌^′(𝐶^′₂)<𝜖𝜈(𝐵)+2. (3.6) Now 𝐴∪𝐵⊂𝐶^′₁∪𝐶^′₂,𝐴∩𝐵⊂𝐶^′₁∩𝐶^′₂, 𝜈(𝐴∪𝐵)+𝜈(𝐴∩𝐵)≤𝜈(𝐶^′₁∪𝐶^′₂)+𝜈(𝐶^′₁∩𝐶^′₂). (3.7) Since 𝜈 is submodular on 𝒞^′, 𝜈(𝐶^′₁∪𝐶^′₂)+𝜈(𝐶^′₁∩𝐶^′₂)≤𝜈(𝐶^′₁)+𝜈(𝐶^′₂)<𝜌^′(𝐶^′₁)+𝜌^′(𝐶^′₂). (3.8) By (3.7), (3.8), (3.6), we have 𝜈(𝐴∪𝐵)+𝜈(𝐴∩𝐵)<𝜈(𝐴)+𝜈(𝐵)+𝜖. (3.9) Since 𝜖>0 is arbitrary, (3.9) shows 𝜈(𝐴∪𝐵)+𝜈(𝐴∩𝐵)≤𝜈(𝐴)+𝜈(𝐵), (3.10) and so 𝜈 is submodular.

Combining Theorems 3.1 and 3.2, we have the following theorem.

Theorem 3.3. If 𝒞 is a lattice, and if 𝜈^′ is supermodular on 𝒞 then 𝜈 is submodular.

We recall that given an outer measure 𝜈, a measurable cover for a set 𝐸⊂𝑋 is a set 𝑀∈𝒮_𝜈 such that 𝜈(𝐸)=𝜈(𝑀).

Definition 3.4. Let 𝜈 be a finite, finitely subadditive outer measure, and let 𝒮_𝜈 be the 𝜈-measurable sets. We define for 𝐸⊂𝑋𝜈⁰(𝐸)=inf{𝜈(𝑀)/𝐸⊂𝑀∈𝒮_𝜈}, (3.11) and say that 𝜈 is approximately regular if 𝜈=𝜈⁰.
𝜈⁰ is a finite, finitely subadditive, submodular outer measure, and 𝜈⁰(𝑀)=𝜈(𝑀) for 𝑀∈𝒮_𝜈.

It is known that when a finite, finitely subadditive outer measure 𝜈 is approximately regular then 𝜈 is submodular [5].

Theorem 3.5. Let 𝒞 be a covering class and let 𝜈 be a finite, finitely subadditive outer measure. If every 𝐶^′∈𝒞^′ has a measurable cover with respect to 𝜈, then 𝜈 is approximately regular and therefore submodular.

Proof. We first show that the hypothesis that each 𝐶^′∈𝒞^′ has a measurable cover with respect to 𝜈 implies that 𝜈=𝜈⁰ on 𝒞^′.
Let 𝐶^′∈𝒞^′. We always have for any 𝑀∈𝒮_𝜈 with 𝐶^′⊂𝑀,𝜈(𝐶^′)≤𝜈(𝑀). Therefore, by the definition of 𝜈⁰(𝐶^′),𝜈(𝐶^′)≤𝜈⁰(𝐶^′). (3.12) Now by hypothesis, there exists an 𝑀₀∈𝒮_𝜈 such that 𝐶^′⊂𝑀₀ and 𝜈(𝐶^′)=𝜈(𝑀₀). By monotonicity of 𝜈⁰, 𝜈⁰(𝐶^′)≤𝜈(𝑀₀)=𝜈(𝐶^′). (3.13)
By (3.12) and (3.13), 𝜈⁰(𝐶^′)=𝜈(𝐶^′). Since 𝐶^′∈𝒞^′ is arbitrary, 𝜈=𝜈⁰ on 𝒞^′, so that 𝜈 is approximately regular on 𝒞^′.
We next show that when 𝜈 is approximately regular on 𝒞^′ implies that 𝜈 is approximately regular.
Let 𝐸⊂𝑋. Let 𝑀∈𝒮_𝜈 with 𝐸⊂𝑀. By monotonicity of 𝜈,𝜈(𝐸)≤𝜈(𝑀). It follows from the definition that 𝜈(𝐸)≤𝜈⁰(𝐸). (3.14) Suppose now that 𝐶^′∈𝒞^′ and 𝐸⊂𝐶^′. By monotonicity of 𝜈⁰,𝜈⁰(𝐸)≤𝜈⁰(𝐶^′)=𝜈(𝐶^′). (3.15) We always have 𝜈=𝜌^′ on 𝒞^′, so (3.15) shows that 𝜈⁰(𝐸)≤𝜌^′(𝐶^′). (3.16) Since 𝒞^′ is any set from 𝒞^′ with 𝐸⊂𝐶^′, (3.16) shows that 𝜈⁰(𝐸)≤𝜈(𝐸). (3.17) By (3.14) and (3.17), we have 𝜈(𝐸)=𝜈⁰(𝐸) for any 𝐸⊂𝑋. Therefore, 𝜈=𝜈⁰, so that 𝜈 is approximately regular. It follows from a result in [5] that 𝜈 is submodular.

An important question to consider is whether or not 𝒞⊂𝒮_𝜈 without requiring that 𝜈 is submodular. If this is true, then since 𝒮_𝜈 is an algebra, 𝒜(ℒ(𝒞))=𝒜(ℒ)⊂𝒮_𝜈 and thus 𝜈/_𝒜(ℒ)∈𝑀(ℒ). Our next theorem shows that this is indeed the case when 𝒞 is a lattice and 𝜌 is superadditive on 𝒞.

Theorem 3.6. If 𝒞 is a lattice, and if 𝜌 is superadditive on 𝒞, then 𝒞⊂𝒮_𝜈.

Proof. It suffices to show by a theorem in [2] that each 𝐶∈𝒞 splits all sets of 𝒞^′ additively with respect to 𝜈.
Let 𝐶∈𝒞 and 𝐴^′∈𝒞^′. We always have 𝜈(𝐴^′)≤𝜈(𝐴^′∩𝐶)+𝜈(𝐴^′∩𝐶^′). For the reverse inequality, we proceed as follows. Let 𝜖>0 be given and arbitrary. There exists 𝐵,𝐷∈𝒞 such that
𝐵⊂𝐴^′∩𝐶^′,𝜌^′(𝐴^′∩𝐶^′𝜖)−2<𝜌(𝐵),𝐷⊂𝐴^′∩𝐵^′,𝜌^′(𝐴^′∩𝐵^′𝜖)−2<𝜌(𝐷). (3.18)
Now 𝐵∩𝐷⊂𝐵∩(𝐴^′∩𝐵^′)=∅, and 𝐵∪𝐷⊂(𝐴^′∩𝐶^′)∪(𝐴^′∩𝐵^′)⊂𝐴^′. Since 𝒞 is a lattice, 𝐵∪𝐷∈𝒞. We have 𝜈(𝐴^′)=𝜌^′(𝐴^′), so by the definition of 𝜌^′(𝐴^′), we have 𝜈(𝐴^′)>𝜌^′(𝐴^′)≥𝜌(𝐵∪𝐷). (3.19) By the superadditivity of 𝜌 on 𝒞, 𝜌(𝐵∪𝐷)≥𝜌(𝐵)+𝜌(𝐷). (3.20) Using (3.19), (3.20), and (3.18), we have 𝜈(𝐴^′)>𝜌^′(𝐴^′∩𝐶^′)+𝜌^′(𝐴^′∩𝐵^′)−𝜖. (3.21) Again, since 𝒞 is a lattice and 𝐴,𝐵,𝐶∈𝒞 and 𝜈=𝜌^′ on 𝒞^′, we have by (3.21) 𝜈(𝐴^′)>𝜈(𝐴^′∩𝐶^′)+𝜈(𝐴^′∩𝐵^′)−𝜖. (3.22) Since 𝐴^′∩𝐵^′⊃𝐴^′∩(𝐴∪𝐶)=𝐴^′∩𝐶, by monotonicity of 𝜈, 𝜈(𝐴^′∩𝐵^′)>𝜈(𝐴^′∩𝐶). (3.23)
Therefore, by (3.22) and (3.23), we get 𝜈(𝐴^′)>𝜈(𝐴^′∩𝐶^′)+𝜈(𝐴^′∩𝐶)−𝜖. Since 𝜖>0 is arbitrary, we get 𝜈(𝐴^′)≥𝜈(𝐴^′∩𝐶^′)+𝜈(𝐴^′∩𝐶). Therefore, 𝜈(𝐴^′)=𝜈(𝐴^′∩𝐶)+𝜈(𝐴^′∩𝐶^′), and 𝐶 splits 𝐴^′ additively with respect to 𝜈. Since 𝐴^′∈𝒞^′ is arbitrary, it follows that 𝐶 splits all sets of 𝒞^′ additively with respect to 𝜈 and so 𝐶∈𝒮_𝜈. The arbitrariness of 𝐶∈𝒞 shows that 𝒞⊂𝒮_𝜈.
Consequently, 𝒜(ℒ)⊂𝒮_𝜈 and so 𝜈/_𝒜(ℒ)∈𝑀(ℒ).

4. Equality of 𝜈 and Its Associated Set Functions on ℒ

In this section, we again consider a covering class 𝒞 of subsets of a nonempty set 𝑋, 𝒞 being closed under finite intersections, and such that ∅,𝑋∈𝒞, and 𝜈 a finite, finitely subadditive outer measure defined for all subsets of 𝑋. We assume that 𝒞⊂𝒮_𝜈, the set of 𝜈-measurable sets. Since 𝒮_𝜈 is an algebra of sets, ℒ=ℒ(𝒞)⊂𝒮_𝜈, hence 𝒜(ℒ)⊂𝒮_𝜈. Therefore, 𝜇=𝜈/_𝒜(ℒ)∈𝑀(ℒ). We consider the following set functions obtained from 𝜈, which are defined in Section 2: 𝜇^′,𝜇_𝑖,𝜌,𝜈^′,𝜌^′. We obtain an inequality between all these set functions that holds for all subsets of 𝑋, and determine conditions for equality of these functions on ℒ. Equality on ℒ becomes useful, since for instance we will then have 𝜇∈𝑀_𝑅(ℒ).

Theorem 4.1. 𝜌^′≤𝜇_𝑖≤𝜌≤𝜈≤𝜇^′≤𝜈^′ on 𝒫(𝑋).

Proof. Let 𝐸⊂𝑋 be fixed but arbitrary. We always have 𝜌(𝐸)≤𝜈(𝐸), (4.1)so it is the other inequalities that we must establish.
Let 𝐿∈ℒ with 𝐸⊂𝐿^′. The definition of 𝜇^′(𝐸) shows that 𝜇^′(𝐸)≤𝜇^′(𝐿^′)=𝜈(𝐿^′). By monotonicity of 𝜈, we have 𝜈(𝐸)≤𝜈(𝐿^′)=𝜇(𝐿^′) whenever 𝐸⊂𝐿^′,𝐿∈ℒ. Thus, 𝜈(𝐸) is a lower bound for the set {𝜇(𝐿^′)/𝐸⊂𝐿^′,𝐿∈ℒ}. Therefore, 𝜈(𝐸)≤𝜇^′(𝐸). (4.2)
Now suppose 𝐶∈𝒞 and 𝐸⊂𝐶^′. Since 𝐶∈ℒ, 𝜇^′(𝐸)≤𝜇(𝐶^′)=𝜈(𝐶^′). Therefore, 𝜇^′(𝐸) is a lower bound for the set {𝜈(𝐶^′)/𝐸⊂𝐶^′,𝐶∈𝒞}. The definition of 𝜈^′(𝐸) gives 𝜇^′(𝐸)≤𝜈^′(𝐸). (4.3)
Next, consider any 𝐶∈𝒞 with 𝐶⊂𝐸. By the definition of 𝜇_𝑖(𝐸), we have 𝜇(𝐶)≤𝜇_𝑖(𝐸). Since 𝜇(𝐶)=𝜈(𝐶), we have 𝜈(𝐶)≤𝜇_𝑖(𝐸) for all 𝐶⊂𝐸 with 𝐶∈𝒞. Since 𝜌(𝐶)≤𝜈(𝐶), we have 𝜌(𝐶)≤𝜇_𝑖(𝐸) for all 𝐶∈𝒞, with 𝐶⊂𝐸. Therefore, 𝜇_𝑖(𝐸) is an upper bound for the set {𝜌(𝐶)/𝐶⊂𝐸,𝐶∈𝒞}. The definition of 𝜌^′(𝐸) thus gives 𝜌^′(𝐸)≤𝜇_𝑖(𝐸). (4.4)
To show that 𝜇_𝑖(𝐸)≤𝜌(𝐸), we argue by contradiction. Thus suppose that 𝜌(𝐸)<𝜇_𝑖(𝐸). Then there exists an 𝐿₀∈ℒ such that 𝐿₀⊂𝐸 and 𝜌(𝐸)<𝜇(𝐿₀)≤𝜇_𝑖(𝐸). By monotonicity of 𝜌,𝜌(𝐿₀)≤𝜌(𝐸), so we have 𝜌(𝐿₀)<𝜇(𝐿₀). Now 𝜌(𝐿₀)=𝜈(𝑋)−𝜈(𝐿^′₀)=𝜇(𝑋)−𝜇(𝐿^′₀), hence 𝜇(𝑋)<𝜇(𝐿₀)+𝜇(𝐿^′₀). This contradicts the finite additivity of 𝜇. Therefore, it must be 𝜇_𝑖(𝐸)≤𝜌(𝐸). (4.5)
Combining inequalities (4.1) through (4.5), we have 𝜌^′(𝐸)≤𝜇_𝑖(𝐸)≤𝜌(𝐸)≤𝜈(𝐸)≤𝜇^′(𝐸)≤𝜈^′(𝐸). (4.6)
Since 𝐸⊂𝑋 was arbitrary, we have shown that 𝜌^′≤𝜇_𝑖≤𝜌≤𝜈≤𝜇^′≤𝜈^′ on 𝒫(𝑋).

We now determine conditions when the inequality of Theorem 4.1 will be an equality on ℒ. Before doing so, we indicate what happens when this is true. We always have that 𝜇^′ is submodular. Assuming 𝜌^′=𝜇_𝑖=𝜌=𝜈=𝜇^′=𝜈^′ on ℒ, suppose 𝐿∈ℒ. Then 𝜇_𝑖(𝐿^′)=𝜇^′(𝑋)−𝜇^′(𝐿)=𝜇(𝑋)−𝜇(𝐿)=𝜇(𝐿^′) since 𝜇∈𝑀(ℒ). Now 𝜇_𝑖(𝐿^′)=sup{𝜇(𝐾)/𝐾⊂𝐿^′,𝐾∈ℒ} so we have 𝜇(𝐿^′)=sup{𝜇(𝐾)/𝐾⊂𝐿^′,𝐾∈ℒ}. By [5], this shows since 𝐿∈ℒ is arbitrary that 𝜇∈𝑀_𝑅(ℒ). It follows from this that ℒ⊂𝒮_𝜇′, and so (𝜇^′)⁰=𝜇,^′ so that 𝜇^′ is approximately regular, hence submodular.

Theorem 4.2. If 𝒞 is a lattice, 𝜈 satisfies condition A, and 𝜈 is submodular on 𝒞^′, then 𝜌^′=𝜇_𝑖=𝜌=𝜈=𝜇^′=𝜈^′ on ℒ.

Proof. We show that 𝜈^′ is submodular. Since 𝜈 satisfies condition A, 𝒞⊂𝒮^𝜈′. Let 𝐸,𝐹⊂𝑋 and let 𝜖>0 be arbitrary. By the definition of 𝜈^′, there exist 𝐶₁,𝐶₂∈𝒞 such that 𝐸⊂𝐶^′₁ and 𝜈^′(𝐸)≤𝜈(𝐶^′₁)<𝜈^′(𝐸)+𝜖/2, and 𝐹⊂𝐶^′₂ and 𝜈^′(𝐹)≤𝜈(𝐶^′₂)<𝜈^′𝜖(𝐹)+2. (4.7)
Now 𝐸∪𝐹⊂𝐶^′₁∪𝐶^′₂, 𝐸∩𝐹⊂𝐶^′₁∩𝐶^′₂, where 𝐶₁∩𝐶₂ and 𝐶₁∪𝐶₂ belong to the lattice 𝒞. Therefore, 𝜈^′(𝐸∪𝐹)+𝜈^′(𝐸∩𝐹)≤𝜈(𝐶^′₁∪𝐶^′₂)+𝜈(𝐶^′₁∩𝐶^′₂). (4.8)
Since 𝜈 is submodular on 𝒞^′, 𝜈(𝐶^′₁∪𝐶^′₂)+𝜈(𝐶^′₁∩𝐶^′₂)≤𝜈(𝐶^′₁)+𝜈(𝐶^′₂). Combining this last inequality with (4.8) and (4.7), we have 𝜈^′(𝐸∪𝐹)+𝜈^′(𝐸∩𝐹)<𝜈^′(𝐸)+𝜈^′(𝐹)+𝜖. (4.9)
Since 𝜖>0 is arbitrary, this shows that 𝜈^′(𝐸∪𝐹)+𝜈^′(𝐸∩𝐹)≤𝜈^′(𝐸)+𝜈^′(𝐹), where 𝐸,𝐹⊂𝑋 are arbitrary. Therefore, 𝜈^′ is submodular. Hence, 𝒮^𝜈′=𝒮_𝜈′ where 𝒮_𝜈′ is an algebra. By our initial observation, 𝒞⊂𝒮^𝜈′=𝒮_𝜈′, and so ℒ⊂𝒮^𝜈′. Thus for all 𝐿∈ℒ, 𝜈^′(𝐿)=𝜌^′(𝐿), and so by Theorem 4.1, we have 𝜌^′=𝜇_𝑖=𝜌=𝜈=𝜇^′=𝜈^′ on ℒ.

Theorem 4.3. If 𝜈=𝜈⁰ on 𝒞^′, 𝒞 coseparates ℒ(𝒞), and 𝜈 satisfies condition B, then 𝜌^′=𝜇_𝑖=𝜌=𝜈=𝜇^′=𝜈^′ on ℒ.

Proof. The hypothesis that 𝜈=𝜈⁰ on 𝒞^′ shown by [6] that 𝜈^′ is submodular and so 𝒮_𝜈′=𝒮^𝜈′. We show that the hypotheses that 𝒞 coseparates ℒ(𝒞) and 𝜈 satisfies condition B give 𝜌^′=𝜇_𝑖=𝜌=𝜈=𝜇^′=𝜈^′ on 𝒞.
Since 𝒞⊂𝒮_𝜈⊂𝒮^𝜈,  𝜈(𝐶)=𝜌(𝐶) for all 𝐶∈𝒞. Since for any 𝐶∈𝒞, 𝜌^′(𝐶)=𝜌(𝐶)=𝜇_𝑖(𝐶), we have 𝜌^′=𝜇_𝑖=𝜌=𝜈on𝒞. (4.10)
We next show 𝜇^′=𝜈^′ on 𝒞. To do so, we argue by contradiction. Suppose there is a 𝐶₀∈𝒞 such that 𝜇^′(𝐶₀)<𝜈^′(𝐶₀). Then there is an 𝐿₀∈ℒ such that 𝐶₀⊂𝐿^′₀ and 𝜇^′(𝐶₀)≤𝜇(𝐿^′₀)<𝜈^′(𝐶₀). Since 𝐶₀∩𝐿₀=∅ with 𝐶₀,𝐿₀∈ℒ and 𝒞 coseparates ℒ, there exist 𝐴,𝐵∈𝒞 with 𝐶₀⊂𝐴^′,𝐿₀⊂𝐵^′ and 𝐴^′∩𝐵^′=∅. Therefore, 𝐶₀⊂𝐴^′⊂𝐵⊂𝐿^′₀ and 𝜇^′(𝐶₀)≤𝜇^′(𝐴^′)=𝜇(𝐴^′)≤𝜇(𝐵)≤𝜇(𝐿^′₀)<𝜈^′(𝐶₀). (4.11)
We also have 𝜇(𝐴^′)=𝜈(𝐴^′). Thus, we have an 𝐴∈𝒞⊂ℒ, with 𝐶₀⊂𝐴^′ and 𝜈(𝐴^′)<𝜈^′(𝐶₀). This contradicts the definition of 𝜈^′(𝐶₀). Therefore, we must have 𝜇^′=𝜈^′ on 𝒞.
We next show 𝜈=𝜇^′ on 𝒞. We reason by contradiction. Suppose there is a 𝐶∈𝒞 with 𝜈(𝐶)<𝜇^′(𝐶). Therefore, 𝜈(𝑋)−𝜈(𝐶)>𝜈(𝑋)−𝜇^′(𝐶). But we also have 𝜈(𝑋)=𝜈(𝐶)+𝜈(𝐶^′) so 𝜈(𝐶^′)=𝜈(𝑋)−𝜈(𝐶) and 𝜈(𝑋)=𝜇(𝑋)=𝜇^′(𝑋). Therefore, 𝜈(𝐶^′)>𝜇^′(𝑋)−𝜇^′(𝐶)=𝜇_𝑖(𝐶^′). Since 𝜈 satisfies condition B, 𝜈(𝐶^′)=sup{𝜈(𝐴)/𝐴⊂𝐶^′,𝐴∈𝒞}>𝜇_𝑖(𝐶^′). Thus, there is an 𝐴₀∈𝒞 with 𝐴₀⊂𝐶^′ and 𝜇_𝑖(𝐶^′)<𝜈(𝐴₀)≤𝜈(𝐶^′), and 𝐴₀∩𝐶=∅. Since 𝒞 coseparates ℒ, there exist 𝐵,𝐷∈𝒞 such that 𝐴₀⊂𝐵^′,𝐶⊂𝐷^′ and 𝐵^′∩𝐷^′=∅. Therefore, 𝐴₀⊂𝐵^′⊂𝐷⊂𝐶^′. This gives 𝜇_𝑖(𝐶^′)<𝜈(𝐴)≤𝜈(𝐵^′)≤𝜈(𝐷)≤𝜈(𝐶^′). (4.12)
But 𝜈(𝐷)=𝜇(𝐷), so we have a 𝐷∈𝒞⊂ℒ with 𝐷⊂𝐶^′ and 𝜇_𝑖(𝐶^′)<𝜇(𝐷). This contradicts the definition of 𝜇_𝑖(𝐶^′). Therefore, we must have 𝜈=𝜇^′ on 𝒞. Thus we have 𝜌^′=𝜇_𝑖=𝜌=𝜈=𝜇^′=𝜈^′ on 𝒞. Since 𝜌^′=𝜈^′ on 𝒞, 𝒞⊂𝒮^𝜈′. By the submodularity of 𝜈^′, 𝒮^𝜈′=𝒮_𝜈′. Therefore, 𝒞⊂𝒮_𝜈′ and so ℒ⊂𝒮_𝜈′. Therefore, for all 𝐿∈ℒ, 𝜌^′(𝐿)=𝜈^′(𝐿) and so 𝜌^′=𝜇_𝑖=𝜌=𝜈=𝜇^′=𝜈^′ on ℒ.

Thus far, in order to have the equality 𝜌^′=𝜇_𝑖=𝜌=𝜈=𝜇^′=𝜈^′onℒ, (∗) the property of submodularity of 𝜈 or 𝜈^′ on 𝒞 or 𝒞^′ played a major role. A natural question to ask is whether this equality can be achieved without submodularity. To obtain some insight into how one may proceed, we make the following observation. For the outer measure 𝜈^′, it is always true that 𝒮_𝜈′⊂𝒮^𝜈′, where 𝒮_𝜈′ is an algebra of sets. If 𝒞⊂𝒮_𝜈′, then ℒ=ℒ(𝒞)⊂𝒮_𝜈′, and consequently 𝒞⊂𝒮^𝜈′, whence 𝜈^′(𝐿)=𝜌^′(𝐿) for all 𝐿∈ℒ. It follows from this that the equality (*) holds on ℒ. It seems reasonable then to seek conditions for 𝒞⊂𝒮_𝜈′.

We have the following theorem.

Theorem 4.4. Suppose each 𝐶∈𝒞 splits all sets of 𝒞^′ additively with respect to 𝜈^′. Then 𝒞⊂𝒮_𝜈′ and 𝜌^′=𝜇_𝑖=𝜌=𝜈=𝜇^′=𝜈^′ on ℒ.

The result follows from [2].

The results of this section prove useful in studying the restriction of a finite, finitely subadditive outer measure to the algebra generated by the covering class, when the sets of the covering class are measurable, which will be investigated in a subsequent paper. Although Theorem 4.4 may be difficult to implement in practice, it leads naturally to the content of the next section.

5. An Outer Measure Construction

In this section, we consider a nonempty set 𝑋 and the construction of a finite, finitely subadditive outer measure given an arbitrary family of subsets ℬ of 𝑋, and a finite, nonnegative set function 𝜏 on ℬ. By the standard method, we construct a finite, finitely subadditive outer measure 𝜆 from 𝜏 and ℬ. We seek a characterization of those sets that split all sets of ℬ additively with respect to 𝜆.

For emphasis, let 𝑋 be a nonempty set and let ℬ be an arbitrary family of subsets of 𝑋 with ∅,𝑋∈ℬ and ℬ is closed under finite unions. Let 𝜏 be a finite, nonnegative set function defined on ℬ with 𝜏(∅)=0, and 𝜏 subadditive on ℬ. For 𝐸⊂𝑋 let 𝜆(𝐸)=inf{𝜏(𝐵)/𝐸⊂𝐵,𝐵∈ℬ}. (5.1)

It follows by a standard argument that 𝜆 is a finite, finitely subadditive outer measure defined for all subsets of 𝑋. Let 𝒮_𝜆 be the algebra of 𝜆-measurable subsets of 𝑋. We determine conditions when a set or a family of sets splits all sets of ℬ additively with respect to 𝜆, this set or family will split all subsets of 𝑋 additively with respect to 𝜆. The importance of this was hinted at in the last section.

We observe that if 𝐵∈ℬ, then 𝜆(𝐵)≤𝜏(𝐵). Also, if 𝜏 is monotone on ℬ, then 𝜆(𝐵)=𝜏(𝐵) for all 𝐵∈ℬ.

The following theorem is known, and its proof can be found in [2], for instance.

Theorem 5.1. If 𝐸⊂𝑋 splits all sets of ℬ additively with respect to 𝜆, then 𝐸∈𝒮_𝜆.

Our motivation for what follows comes from the characterization of the measurable sets for a 𝜎-finite measure which can be found in [8, 9]. We begin with the observation that there are always at least two sets in ℬ that split all sets of ℬ additively with respect to 𝜆, namely, ∅ and 𝑋 (they may be the only ones). We seek a characterization of such sets.

Theorem 5.2. Let 𝐵₀∈ℬ split all sets of ℬ additively with respect to 𝜆. Let 𝑁∈ℬ be such that 𝜏(𝑁)=0. Then 𝐸=𝐵₀−𝑁 splits all sets of ℬ additively with respect to 𝜆, and so 𝐸∈𝒮_𝜆.

Proof. Let 𝐵∈ℬ be arbitrary. We can write 𝐵=(𝐵∩𝐸)∪(𝐵∩𝐸^′). Now, 𝐵∩𝐸=𝐵∩(𝐵₀∩𝑁^′)⊂𝐵∩𝐵₀ and 𝐵∩𝐸^′=𝐵∩(𝐵^′₀∪𝑁)⊂(𝐵∩𝐵^′₀)∪𝑁. By the monotonicity and finite subadditivity of 𝜆,𝜆(𝐵∩𝐸)+𝜆(𝐵∩𝐸^′)≤𝜆(𝐵∩𝐵₀)+𝜆(𝐵∩𝐵^′₀)+𝜆(𝑁). (5.2)
Since 𝑁∈ℬ, 𝜆(𝑁)≤𝜏(𝑁)=0, so 𝜆(𝑁)=0 and (5.2) becomes 𝜆(𝐵∩𝐸)+𝜆(𝐵∩𝐸^′)≤𝜆(𝐵∩𝐵₀)+𝜆(𝐵∩𝐵^′₀). (5.3)
The hypothesis on 𝐵₀ gives 𝜆(𝐵∩𝐵₀)+𝜆(𝐵∩𝐵^′₀)=𝜆(𝐵) so by (5.3), 𝜆(𝐵∩𝐸)+𝜆(𝐵∩𝐸^′)≤𝜆(𝐵). The reverse inequality is always true by finite subadditivity of 𝜆. Thus by Theorem 5.1, 𝐸∈𝒮_𝜆. (5.4)

Thus we see that a sufficient condition for a set 𝐸 to be 𝜆-measurable is: 𝐸 must be expressible as the difference of a set from ℬ that splits all sets of ℬ additively with respect to 𝜆 and a subset of 𝑋 for which 𝜏 is 0.

An example may help to illustrate the theorem.

Example 5.3. Let 𝑋={𝑎,𝑏,𝑐} and take ℬ={∅,𝑋,{𝑎},{𝑏},{𝑎,𝑏}}. Define 𝜏 on ℬ by 𝜏(∅)=𝜏({𝑎})=0,𝜏({𝑏})=𝜏({𝑎,𝑏})=1,𝜏(𝑋)=2. The outer measure 𝜆 determined by ℬ and 𝜏 is 𝜆(∅)=𝜆({𝑎})=0, 𝜆({𝑏})=𝜆({𝑎,𝑏})=1, 𝜆({𝑐})=𝜆({𝑎,𝑐})=𝜆({𝑏,𝑐})=𝜆(𝑋)=2. Now 𝒮_𝜆={∅,𝑋,{𝑎},{𝑏,𝑐}}. We see that ∅=𝑋−∅, 𝑋=𝑋−∅, {𝑎}={𝑎}−∅,{𝑏,𝑐}=𝑋−{𝑎}.

We return to characterizing those sets 𝐸 that split all sets of ℬ additively with respect to 𝜆. In the style of Munroe [10], we have the following theorem.

Theorem 5.4. Let 𝐸⊂𝑋 be any set. There is a sequence of sets {𝐵_𝑛} from ℬ such that 𝐸⊂𝐵_𝑛 all n and 𝜆(𝐸)=𝜆(⋂^∞_𝑛=1𝐵_𝑛).

Proof. Let 𝐸⊂𝑋 be arbitrary. By the definition of 𝜆(𝐸), for each 𝑛∈ℕ there is a 𝐵_𝑛∈ℬ such that 𝐸⊂𝐵_𝑛 and 𝜆(𝐸)≤𝜏(𝐵_𝑛1)<𝜆(𝐸)+𝑛. (5.5)
Let 𝐵=⋂^∞_𝑛=1𝐵_𝑛. Then 𝐸⊂𝐵⊂𝐵_𝑛 all n. By monotonicity of 𝜆, 𝜆(𝐸)≤𝜆(𝐵)≤𝜆(𝐵_𝑛)all𝑛. (5.6)
Now 𝜆(𝐵_𝑛)≤𝜏(𝐵_𝑛) all n, so by (5.6) and (5.5) we get 𝜆(𝐸)≤𝜆(𝐵)<𝜆(𝐸)+1/𝑛 for any 𝑛∈ℕ. Thus for any given 𝜖>0, we can find an 𝑛∈𝑁 large enough so that 0<1/𝑛<𝜖. For any such 𝑛,𝜆(𝐸)≤𝜆(𝐵)<𝜆(𝐸)+𝜖. Since 𝜖>0 is arbitrary, it follows that 𝜆(𝐸)=𝜆(𝐵).

Observation. Suppose 𝐸⊂𝑋 splits all sets 𝐴⊂𝑋 additively with respect to 𝜆. By Theorem 5.4, there exist 𝐵_𝑛∈ℬ,𝑛∈ℕ with 𝐸⊂⋂^∞_𝑛=1𝐵_𝑛 and 𝜆(𝐸)=𝜆(⋂^∞_𝑛=1𝐵_𝑛). Let 𝐵=⋂^∞_𝑛=1𝐵_𝑛. We can write 𝐵=𝐸∪(𝐵∩𝐸^′). Since 𝐸∈𝒮_𝜆,𝜆(𝐵)=𝜆(𝐸)+𝜆(𝐵∩𝐸^′), and since all quantities are finite, 𝜆(𝐵∩𝐸^′)=𝜆(𝐵)−𝜆(𝐸)=0. Since 𝐸=𝐵−(𝐵−𝐸^′), let 𝑁=𝐵−𝐸 so that 𝜆(𝑁)=0.
This observation establishes the following theorem.

Theorem 5.5. If 𝐸∈𝒮_𝜆, then there is a sequence of sets 𝐵_𝑛∈ℬ, and a set 𝑁⊂𝑋 such that 𝐸⊂𝐵_𝑛 all n, and 𝐸=(⋂^∞_𝑛=1𝐵_𝑛)−𝑁, where 𝜆(𝑁)=0.

We see that our result is analogous to what happens in the case of constructing an outer measure from a measure in the general case, see for instance Munroe [10].

In summary, to obtain a set 𝐸 that splits all sets of ℬ additively with respect to 𝜆: let 𝐵₀∈ℬ be a set that splits all sets of ℬ additively with respect to 𝜆 (for instance 𝑋) and let 𝑁⊂𝑋 with 𝜏(𝑁)=0. Then 𝐸=𝐵₀−𝑁 will split all sets of ℬ additively with respect to 𝜆, so that 𝐸∈𝒮_𝜆.

If 𝐸∈𝒮_𝜆, then there is a sequence of sets {𝐵_𝑛} from ℬ such that 𝐸⊂𝐵_𝑛 all n , and a set 𝑁⊂𝑋 such that 𝜆(𝑁)=0 and 𝐸=(⋂^∞_𝑛=1𝐵_𝑛)−𝑁.

We note that if ℬ is a 𝛿-lattice that is, a lattice closed under countable intersections, then the set 𝐵=⋃^∞_𝑛=1𝐵_𝑛∈ℬ.

Acknowledgments

The author wishes to thank the referee for several useful suggestions leading to the improved clarity of the paper, and to the editor for his courtesy. The questions considered for this paper were presented to the author by the late Dr. George Bachman, the author’s teacher and friend.