Topological Properties of the Complex Vector Lattice of Bounded Measurable Functions

Abstract

Let Σ be a σ-algebra of subsets of a nonempty set Ω. Let 𝐵(Σ) be the complex vector lattice of bounded Σ-measurable complex-valued functions on Ω and let 𝑐𝑎(Σ) be the Banach space of all bounded countably additive complex-valued measures on Ω. We study locally solid topologies on 𝐵(Σ). In particular, it is shown that the Mackey topology 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is the finest locally convex-solid σ-Lebesgue topology on 𝐵(Σ).

1. Introduction and Terminology

For terminology concerning complex vector lattices we refer the reader to [1, Chapter 12, Section  91], [2, Chapter 16], [3]. We denote by 𝜎(𝐿,𝐾) and 𝜏(𝐿,𝐾) the weak topology and the Mackey topology with respect to a dual pair ⟨𝐿,𝐾⟩. For a topological vector space (𝐿,𝜂) by (𝐿,𝜂)^′ or 𝐿^′_𝜂, we denote its topological dual.

We assume that Σ is a 𝜎-algebra of subsets of a non-empty set Ω. Let 𝐵(Σ) denote the Banach space of all bounded Σ-measurable functions 𝑓∶Ω→ℂ, provided with the uniform norm ‖⋅‖. Then, 𝐵(Σ) with the positive cone 𝐵(Σ)⁺={𝑓∈𝐵(Σ)∶𝑓(𝜔)≥0 for all 𝜔∈Ω} and a valuation 𝐵(Σ)∋𝑓↦|𝑓|∈𝐵(Σ)⁺ is a complex vector lattice (in the sense of [2, Section  16-2.2]). By 𝐵_ℝ(Σ) we denote the Dedekind 𝜎-complete Banach lattice of all bounded Σ-measurable functions 𝑢∶Ω→ℝ, with the natural ordering 𝑢≤𝑣 if 𝑢(𝜔)≤𝑣(𝜔) for all 𝜔∈Ω (see [4, Chapter 13.1]). Then, 𝐵(Σ)=𝐵_ℝ(Σ)+𝑖𝐵_ℝ(Σ), that is, 𝐵(Σ) is a complexification of 𝐵_ℝ(Σ) and 𝐵(Σ) has the dominated decomposition property, that is, if |𝑓|≤𝑢₁+𝑢₂, where 𝑓∈𝐵(Σ), 𝑢₁,𝑢₂∈𝐵(Σ)⁺, then there exist 𝑓₁,𝑓₂∈𝐵(Σ) such that 𝑓=𝑓₁+𝑓₂ and |𝑓₁|≤𝑢₁, |𝑓₂|≤𝑢₂ (see [1, Theorem 91.3], [3]). This means that 𝐵(Σ) is a breakable complex vector lattice (see [2, Section 16-3]).

Denote by 𝑏𝑎(Σ) the Banach space of all bounded finitely additive measures 𝜇∶Σ→ℂ with the norm ‖𝜇‖=|𝜇|(Ω), where |𝜇|(𝐴) denotes the variation of 𝜇 on 𝐴∈Σ. It is well known that the Banach dual 𝐵(Σ)^∗ of 𝐵(Σ) can be identified with 𝑏𝑎(Σ) through the integration mapping 𝑏𝑎(Σ)∋𝜇↦Φ_𝜇∈𝐵(Σ)^∗, where Φ_𝜇(𝑓)=∫_Ω𝑓𝑑𝜇 for all 𝑓∈𝐵(Σ) and ‖Φ_𝜇‖=|𝜇|(Ω) (see [5, Chapter 1, Theorem 13, p.6]. Let 𝑐𝑎(Σ) stand for the Banach space of all countably additive measures 𝜇∶Σ→ℂ.

It turns out that the basic notions and results from the theory of vector lattices (in particular, locally solid topologies) can be carried over to the complex vector lattices 𝐵(Σ) and 𝑏𝑎(Σ).

A subset 𝐻 of 𝐵(Σ) is said to be solid if it follows from |𝑓|≤|𝑔| and 𝑓∈𝐵(Σ), 𝑔∈𝐻 that 𝑓∈𝐻. Every subset 𝐻 of 𝐵(Σ) is included in the smallest (with respect to the inclusion) solid set, called the solid hull of 𝐻 and denoted by sol(𝐻). Clearly, sol(𝐻)=𝑔∈𝐵(Σ)∶𝑔≤𝑓forsome𝑓∈𝐻.(1)

A linear Hausdorff topology 𝜏 on 𝐵(Σ) is said to be locally solid if it has a local base at 0 consisting of solid sets in 𝐵(Σ).

In Section 2, we study locally solid topologies on 𝐵(Σ). In Section 3, we develop the duality theory of locally convex-solid topologies on 𝐵(Σ). In Section 4, it is shown that the natural Mackey topology 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is the finest locally convex-solid 𝜎-Lebesgue topology on 𝐵(Σ) (see Theorem 16 below).

2. Locally Solid Topologies on 𝐵(Σ)

Using the dominated decomposition property for 𝐵(Σ) we can easily obtain the following result (see [6, Theorem 1.11]).

Proposition 1. The convex hull conv𝐻 of a solid subset 𝐻 of 𝐵(Σ) is solid.

Proposition 2. Let 𝜏 be a locally solid topology on 𝐵(Σ). Then, the 𝜏-closure cl_𝜏𝐻 of a solid subset 𝐻of 𝐵(Σ) is solid.

Proof. Let ℬ_𝜏 be a local base at 0 for 𝜏 consisting of solid sets. Then cl_𝜏𝐻=⋂{𝐻+𝑉∶𝑉∈ℬ_𝜏}. Assume that |𝑓|≤|𝑔|, where 𝑓∈𝐵(Σ), 𝑔∈cl_𝜏𝐻, and let 𝑉₀∈ℬ_𝜏. Then, 𝑔=𝑔₁+𝑔₂, where 𝑔₁∈𝐻 and 𝑔₂∈𝑉₀. Since |𝑓|≤|𝑔|≤|𝑔₁|+|𝑔₂| (by the dominated decomposition property for 𝐵(Σ)), there exist 𝑓₁,𝑓₂∈𝐵(Σ) such that 𝑓=𝑓₁+𝑓₂ and |𝑓_𝑖|≤|𝑔_𝑖| for 𝑖=1,2. Hence, 𝑓₁∈𝐻 and 𝑓₂∈𝑉₀ because both sets 𝐻 and 𝑉 are solid. Thus, 𝑓∈𝐻+𝑉₀ for every 𝑉₀∈ℬ_𝜏, so 𝑓∈cl_𝜏𝐻. This means that cl_𝜏𝐻 is solid, as desired.

Following the notion of a modulus of an operator between vector lattices (see [7, Definition 1.8 and Theorem 1.10]), for a linear functional Φ∶𝐵(Σ)→ℂ let us put |Φ|(𝑢)∶=sup{|Φ(ℎ)|∶ℎ∈𝐵(Σ),|ℎ|≤𝑢}(2) for all 𝑢∈𝐵(Σ)⁺.

Proposition 3. For a linear functional Φ on 𝐵(Σ) the following statements are equivalent:(i)Φ∈B(Σ)^∗.(ii)|Φ|(𝑢)<∞ for  all 𝑢∈B(Σ)⁺.

Proof. (i)⇒(ii) By way of contradiction, assume that there exists Φ₀∈𝐵(Σ)^∗ such that |Φ₀|(𝑢₀)=∞ for some 𝑢₀∈𝐵(Σ)⁺. Hence, there exists a sequence (ℎ_𝑛) in 𝐵(Σ) such that |ℎ_𝑛|≤𝑢₀ and |Φ₀(ℎ_𝑛)|≥𝑛 for 𝑛∈ℕ. Since ‖(1/𝑛)ℎ_𝑛‖→0, we get (1/𝑛)|Φ₀(ℎ_𝑛)|→0, which is in contradiction with |Φ₀(ℎ_𝑛)|≥𝑛.
(ii)⇒(i) Assume that there exists a linear functional Φ₀ on 𝐵(Σ) such that |Φ₀|(𝑢)<∞ for all 𝑢∈𝐵(Σ)⁺ and Φ₀∉𝐵(Σ)^∗. Then, there exists a sequence (𝑓_𝑛) in 𝐵(Σ) such that ‖𝑓_𝑛‖=1 and |Φ₀(𝑓_𝑛)|>𝑛³ for all 𝑛∈ℕ. Since ∑^∞_𝑛=1(1/𝑛²)‖𝑓_𝑛‖<∞ and the space (𝐵(Σ),‖⋅‖) is complete, there exists 𝑢₀∈𝐵(Σ)⁺ such that ∑^∞_𝑛=1(1/𝑛²)|𝑓_𝑛|=𝑢₀. Then, (1/𝑛²)|𝑓_𝑛|≤𝑢₀, and; hence, 𝑛<|Φ₀((1/𝑛²)𝑓_𝑛)|<∞ for all 𝑛∈ℕ, which is impossible.
Thus, the proof is complete.

For Φ₁,Φ₂∈𝐵(Σ)^∗ we will write |Φ₁|≤|Φ₂| whenever |Φ₁|(𝑢)≤|Φ₂|(𝑢) for all 𝑢∈𝐵(Σ)⁺. A subset 𝒦 of 𝐵(Σ)^∗ is said to be solid if |Φ₁|≤|Φ₂|, where Φ₁∈𝐵(Σ)^∗, Φ₂∈𝒦 implies that Φ₁∈𝒦. A linear subspace 𝐼 of 𝐵(Σ)^∗ is said to be an ideal if 𝐼 is a solid set in 𝐵(Σ)^∗.

Note that, if |Φ₁|≤|Φ₂|, then ‖Φ₁‖≤‖Φ₂‖.

Proposition 4. Let 𝜏 be a locally solid topology on 𝐵(Σ). Then, (𝐵(Σ),𝜏)^′ is an ideal of 𝐵(Σ)^∗.

Proof. To show that (𝐵(Σ),𝜏)^′⊂𝐵(Σ)^∗, by the way of contradiction, assume that for some Φ₀∈(𝐵(Σ),𝜏)^′ we have Φ₀∉𝐵(Σ)^∗, so in view of Proposition 3, we obtain that |Φ₀|(𝑢₀)=∞ for some 𝑢₀∈𝐵(Σ)⁺. Hence, there exists a sequence (ℎ_𝑛) in 𝐵(Σ) such that |ℎ_𝑛|≤𝑢₀ and |Φ₀(ℎ_𝑛)|≥𝑛 for all 𝑛∈ℕ. Since (1/𝑛)𝑢₀→0 for 𝜏 and 𝜏 is locally solid, we get (1/𝑛)ℎ_𝑛→0 for 𝜏. Hence, Φ₀((1/𝑛)ℎ_𝑛)→0, which is in contradiction with |Φ₀(ℎ_𝑛)|≥𝑛 for all 𝑛∈ℕ.
To see that (𝐵(Σ),𝜏)^′ is an ideal of 𝐵(Σ)^∗, assume that |Φ₁|≤|Φ₂|, where Φ₁∈𝐵(Σ)^∗ and Φ₂∈(𝐵(Σ),𝜏)^′. Let 𝑓_𝛼→0 for 𝜏 and 𝜀>0 be given. Then, there exists a net (ℎ_𝛼) in 𝐵(Σ)such that |ℎ_𝛼|≤|𝑓_𝛼|for each 𝛼and |Φ₂|(|𝑓_𝛼|)≤|Φ₂(ℎ_𝛼)|+𝜀. Clearly, ℎ_𝛼→0for 𝜏 because 𝜏 is locally solid, so Φ₂(ℎ_𝛼)→0. Since |Φ₁(𝑓_𝛼)|≤|Φ₁|(|𝑓_𝛼|)≤|Φ₂|(|𝑓_𝛼|)≤|Φ₂(ℎ_𝛼)|+𝜀, we get Φ₁(𝑓_𝛼)→0, so Φ₁∈(𝐵(Σ),𝜏)^′, as desired.

3. Duality of the Space 𝐵(Σ)

A locally convex Hausdorff topology 𝜏 on 𝐵(Σ) is said to be locally convex-solid if it has a local base at 0 consisting of convex and solid sets in 𝐵(Σ). In view of Propositions 1 and 2, we see that for a locally convex-solid topology 𝜏 on 𝐵(Σ) the collection of all 𝜏-closed convex and solid 𝜏-neighborhoods of 0 forms a local base at 0 for 𝜏.

In this section, we develop the duality theory of locally convex-solid topologies on 𝐵(Σ). We start with the following lemma.

Lemma 5. Let 𝜇∈𝑏𝑎(Σ). Then, for each 𝑢∈𝐵(Σ)⁺ we have Φ_𝜇(𝑢)=_Ω𝑢𝑑𝜇=sup_Ωℎ𝑑𝜇∶ℎ∈𝐵(Σ),|ℎ|≤𝑢.(3) In particular, we have for 𝐴∈Σ𝜇(𝐴)=sup_𝐴ℎ𝑑𝜇∶ℎ∈𝐵(Σ),‖ℎ‖≤1.(4)

Proof. Let 𝑢∈𝐵(Σ)⁺ and ℎ∈𝐵(Σ) with |ℎ|≤𝑢. Then, we have ℎ𝑑𝜇≤_ΩΩ|ℎ|𝑑𝜇≤_Ω𝑢𝑑𝜇,(5) so |Φ_𝜇|(𝑢)≤∫_Ω𝑢𝑑|𝜇|. Now we shall show that ∫_Ω𝑢𝑑|𝜇|≤|Φ_𝜇|(𝑢). Indeed, let 𝜀>0 be given. Then, there exists a Σ-simple function 𝑠=∑^𝑘_𝑖=1𝛼_𝑖𝟙_𝐴𝑖 such that 𝛼_𝑖>0, 𝑠≤𝑢 and 𝑢𝑑𝜇_Ω−^𝑘_𝑖=1𝛼_𝑖𝜇𝐴_𝑖≤𝜀2.(6) Let 𝛼=∑^𝑘_𝑖=1𝛼_𝑖. Then for each 𝑖=1,2,…,𝑘 one can choose a finite Σ-partition {𝐴_𝑖𝑗∶𝑗=1,…,𝑘_𝑖} of 𝐴_𝑖 and 𝜀_𝑖𝑗∈ℂ with |𝜀_𝑖𝑗|=1, 𝜀_𝑖𝑗𝜇(𝐴_𝑖𝑗)=|𝜇(𝐴_𝑖𝑗)|for 𝑗=1,…,𝑘_𝑖 such that 𝜇𝐴_𝑖≤^𝑘𝑖_𝑖=1𝜀_𝑖𝑗𝜇𝐴_𝑖𝑗+𝜀.2𝛼(7) Let 𝑠₀=∑^𝑘_𝑖=1∑^𝑘𝑖_𝑗=1𝛼_𝑖𝜀_𝑖𝑗𝟙_𝐴𝑖𝑗. Then |𝑠₀|=𝑠≤𝑢 and we have 𝑢𝑑𝜇_Ω−_Ω𝑠₀≤𝑑𝜇_Ω𝑢𝑑𝜇−^𝑘_𝑖=1𝛼_𝑖𝜇𝐴_𝑖+^𝑘_𝑖=1𝛼_𝑖𝜇𝐴_𝑖−^𝑘𝑘_𝑖=1^𝑖_𝑗=1𝛼_𝑖𝜀_𝑖𝑗𝜇𝐴_𝑖𝑗≤𝜀2+^𝑘_𝑖=1𝛼_𝑖𝜇𝐴_𝑖−^𝑘𝑖_𝑗=1𝜀_𝑖𝑗𝜇𝐴_𝑖𝑗≤𝜀2+^𝑘_𝑖=1𝛼_𝑖𝜀=𝜀2𝛼2+𝜀2=𝜀.(8) Hence, ∫_Ω𝑢𝑑|𝜇|≤|∫_Ω𝑠₀𝑑𝜇|+𝜀, and it follows that ∫_Ω𝑢𝑑|𝜇|≤|Φ_𝜇|(𝑢).

Since |Φ_𝜇|∶𝐵(Σ)⁺→ℝ⁺ is additive, |Φ_𝜇| can be uniquely extended to a positive linear functional (denoted by |Φ_𝜇| again) |Φ_𝜇|∶𝐵(Σ)→ℂ (see [2, Theorem 16-4.7]). Then, for 𝑓∈𝐵(Σ) we have Φ_𝜇𝑓=Φ_𝜇Re𝑓⁺−Φ_𝜇Re𝑓⁻+𝑖Φ_𝜇Im𝑓⁺−𝑖Φ_𝜇Im𝑓⁻=_ΩRe𝑓⁺𝑑𝜇−_ΩRe𝑓⁻𝑑𝜇+𝑖_ΩIm𝑓⁺𝑑𝜇−𝑖_ΩIm𝑓⁻𝑑𝜇=_Ω𝑓𝑑𝜇=Φ_|𝜇|𝑓.(9) Hence, |Φ_𝜇|=Φ_|𝜇|.

Proposition 6. For 𝜇,𝜈∈𝑏𝑎(Σ) the following statements are equivalent: (i)|𝜇|≤|𝜈|.(ii)|Φ_𝜇|≤|Φ_𝜈|.

Proof. (i)⇒(ii) Assume that |𝜇|≤|𝜈|. Then, by Lemma 5, for 𝑢∈𝐵(Σ)⁺, |Φ_𝜇|(𝑢)=∫_Ω𝑢𝑑|𝜇|≤∫_Ω𝑢𝑑|𝜈|=|Φ_𝜈|(𝑢), that is, |Φ_𝜇|≤|Φ_𝜈|.
(ii)⇒(i) Assume that |Φ_𝜇|≤|Φ_𝜈|. Then, for 𝐴∈Σ we have |𝜇|(𝐴)=|Φ_𝜇|(𝟙_𝐴)≤|Φ_𝜈|(𝟙_𝐴)=|𝜈|(𝐴).

Now we consider the concept of solidness in the space 𝑏𝑎(Σ). A subset ℳ of 𝑏𝑎(Σ) is said to be solid if |𝜈|≤|𝜇|, where 𝜈∈𝑏𝑎(Σ), 𝜇∈ℳ implies that 𝜈∈ℳ.

For a subset ℳ of 𝑏𝑎(Σ) let sol(ℳ)=𝜈∈𝑏𝑎(Σ)∶|𝜈|≤𝜇forsome𝜇∈ℳ.(10) Then, sol(ℳ) is the solid hull of ℳ, that is, sol(ℳ) is the smallest solid set in ℳ that contains ℳ.

A linear subspace 𝐼 of 𝑏𝑎(Σ) is called an ideal in 𝑏𝑎(Σ) if 𝐼 is solid. Note that 𝑐𝑎(Σ) is an ideal of 𝑏𝑎(Σ). In view of Proposition 6, we see that a subset ℳ of 𝑏𝑎(Σ) is solid if and only if the set {Φ_𝜇∶𝜇∈ℳ} is solid in 𝐵(Σ)^∗.

Corollary 7. Let 𝜇∈𝑏𝑎(Σ). Then, for 𝑓∈𝐵(Σ) we have Φ_𝜇𝑓=_Ω𝑓𝑑𝜇=sup_Ω(Σ),|𝜈|≤𝜇.(∗)𝑓𝑑𝜈∶𝜈∈𝑏𝑎 Moreover, for a subset ℳ of 𝑏𝑎(Σ) and each 𝑓∈𝐵(Σ) we have supΦ_𝜇𝑓∶𝜇∈ℳ=sup_Ω𝑓𝑑𝜇∶𝜇∈ℳ=sup_Ω𝑓𝑑𝜈∶𝜈∈sol(ℳ)=sup_Ω((ℳ.()𝑓𝑑𝜈∶𝜈∈convsol))∗∗

Proof. In view of Proposition 6 and [2, Theorem 16-5.6] we get Φ_𝜇𝑓=sup_Ω𝑓𝑑𝜈∶𝜈∈𝑏𝑎(Σ),Φ_𝜈≤Φ_𝜇=sup_Ω𝑓𝑑𝜈∶𝜈∈𝑏𝑎(Σ),|𝜈|≤𝜇.(11) Next, we shall show that for each 𝑓∈𝐵(Σ), supΦ_𝜇𝑓∶𝜇∈ℳ≤sup_Ω𝑓𝑑𝜈∶𝜈∈sol(ℳ).(12) Indeed, let 𝑓∈𝐵(Σ) be given. Then, for 𝜇∈ℳ and 𝜀>0 by (∗) there exists 𝜈∈𝑏𝑎(Σ) with |𝜈|≤|𝜇| and such that |Φ_𝜇|(|𝑓|)≤|∫_Ω𝑓𝑑𝜈|+𝜀. Since 𝜈∈sol(ℳ), we get |Φ_𝜇|(|𝑓|)≤sup{|∫_Ω𝑓𝑑𝜈|∶𝜈∈sol(ℳ)}+𝜀, and it follows that (12) holds.
It is enough to show that for each 𝑓∈𝐵(Σ)we have sup_Ω((ℳΦ𝑓𝑑𝜈∶𝜈∈convsol))≤sup_𝜇𝑓∶𝜇∈ℳ.(13) Indeed, let 𝜈∈conv(sol(ℳ)). Then, 𝜈=∑^𝑛_𝑖=1𝛼_𝑖𝜈_𝑖 with 𝜈_𝑖∈sol(ℳ), 𝛼_𝑖≥0,𝑖=1,…,𝑛 and ∑^𝑛_𝑖=1𝛼_𝑖=1. Hence, |𝜈_𝑖|≤|𝜇_𝑖| for some 𝜇_𝑖∈ℳ, 𝑖=1,2,…,𝑛. Hence, for 𝑓∈𝐵(Σ) we get 𝑓𝑑𝜈≤_ΩΩ𝑓𝑑|𝜈|≤^𝑛_𝑖=1𝛼_𝑖Ω𝑓𝑑𝜈_𝑖≤^𝑛_𝑖=1𝛼_𝑖Ω𝑓𝑑𝜇_𝑖≤^𝑛_𝑖=1𝛼_𝑖sup_Ω𝑓𝑑𝜇Φ∶𝜇∈ℳ=sup_𝜇𝑓∶𝜇∈ℳ.(14) It follows that (13) holds, as desired.

Let 𝐼 be an ideal of 𝑏𝑎(Σ) separating the points of 𝐵(Σ). Then, the pair ⟨𝐵(Σ),𝐼⟩, under its natural duality 𝑓,𝜇∶=_Ω𝑓𝑑𝜇for𝑓∈𝐵(Σ),𝜇∈𝐼(15) will be referred to as a solid dual system (see [7, Definition 11.7]). For subsets 𝐻 of 𝐵(Σ) and ℳ of 𝐼, let us put 𝐻⁰=𝜇∈𝐼∶_Ω𝑓𝑑𝜇≤1∀𝑓∈𝐻₀,ℳ=𝑓∈𝐵(Σ)∶_Ω𝑓𝑑𝜇≤1∀𝜇∈ℳ.(16)

Proposition 8. Let ⟨𝐵(Σ),𝐼⟩ be a solid dual system.(i)If 𝐻 is a solid subset of 𝐵(Σ), then 𝐻⁰ is a solid subset of 𝐼.(ii)If ℳ is a solid subset of 𝐼, then ⁰ℳ is a solid subset of 𝐵(Σ).

Proof. (i) Let |𝜈|≤|𝜇| with 𝜈∈𝐼 and 𝜇∈𝐻⁰. Assume that 𝑓∈𝐻 and let ℎ∈𝐵(Σ) with |ℎ|≤|𝑓|. Then, ℎ∈𝐻because 𝐻is solid, so |∫_Ωℎ𝑑𝜇|≤1. Hence ∫_Ω|𝑓|𝑑|𝜇|=|Φ_𝜇|(|𝑓|)|≤1(see Lemma 5). Thus |∫_Ω𝑓𝑑𝜈|≤∫_Ω|𝑓|𝑑|𝜈|≤∫_Ω|𝑓|𝑑|𝜇|≤1, that is, 𝜈∈𝐻⁰. This means that 𝐻⁰ is a solid subset of 𝐼.
(ii) Let |𝑓|≤|𝑔|, where 𝑓∈𝐵(Σ) and 𝑔∈⁰ℳ. Since ℳ is a solid set in 𝑏𝑎(Σ), by Corollary 7 for 𝜇∈ℳ, we have _Ω𝑔𝑑𝜇=sup_Ω𝑔𝑑𝜈∶𝜈∈ℳ,|𝜈|≤𝜇.(17) Then, for every 𝜈∈ℳ with |𝜈|≤|𝜇| we have |∫_Ω𝑔𝑑𝜈|≤1, so ∫_Ω|𝑔|𝑑|𝜇|≤1. It follows that 𝑓𝑑𝜇≤_ΩΩ𝑓𝑑𝜇≤_Ω𝑔𝑑𝜇≤1,(18) that is, 𝑓∈⁰ℳ.

Observe that for each 𝜇∈𝑏𝑎(Σ),  by Corollary 7 the functional 𝜚_𝜇∶𝐵(Σ)→[0,∞) defined by 𝜚_𝜇𝑓=Φ_𝜇𝑓=sup_Ωℎ𝑑𝜇∶ℎ∈𝐵(Σ),|ℎ|≤𝑓=sup_Ω𝑓𝑑𝜈∶𝜈∈𝑏𝑎(Σ),|𝜈|≤𝜇,(19) is a solid seminorm on 𝐵(Σ), that is, 𝜚_𝜇(𝑓₁)≤𝜚_𝜇(𝑓₂) if |𝑓₁|≤|𝑓₂|.

Let 𝐼 be an ideal of 𝑏𝑎(Σ) separating the points of 𝐵(Σ). The absolute weak topology |𝜎|(𝐵(Σ),𝐼) generated by 𝐼 on 𝐵(Σ) is the locally convex-solid topology on 𝐵(Σ) generated by the family of solid seminorms {𝜚_𝜇∶𝜇∈𝐼}. Note that |𝜎|(𝐵(Σ),𝐼) is the topology of uniform convergence on all sets 𝐼_𝜇={𝜈∈𝐼∶|𝜈|≤|𝜇|}, where 𝜇∈𝐼.

Proposition 9. For each 𝜇∈𝐼, the set 𝐼_𝜇 is convex, solid, and 𝜎(𝐼,𝐵(Σ))-compact in 𝐼.

Proof. Clearly, 𝐼_𝜇 is a convex and solid subset of 𝐼. We shall show that 𝐼_𝜇 is 𝜎(𝑏𝑎(Σ),𝐵(Σ))-closed subset of 𝑏𝑎(Σ). To this end, assume that (𝜈_𝛼) is a net in 𝐼_𝜇 such that ∫_Ω𝑓𝑑𝜈_𝛼→∫_Ω𝑓𝑑𝜈 for all 𝑓∈𝐵(Σ), where 𝜈∈𝑏𝑎(Σ). Then _Ω𝑓𝑑𝜈_𝛼⟶_Ω𝑓𝑑𝜈_Ω,𝑓𝑑𝜈_𝛼≤_Ω𝑓𝑑𝜈_𝛼≤_Ω𝑓𝑑𝜇.(20) Hence, for each 𝐴∈Σ, we get |∫_𝐴𝑓𝑑𝜈|≤∫_𝐴|𝑓|𝑑|𝜇|for all 𝑓∈𝐵(Σ), and by Lemma 5, |𝜈|(𝐴)=sup_𝐴(Σ),𝑓≤𝑓𝑑𝜈∶𝑓∈𝐵≤1_𝐴𝟙_Ω𝑑𝜇=𝜇(𝐴),(21) that is, |𝜈|≤|𝜇|. Thus, 𝜈∈𝐼_𝜇, as desired.
In view of the Banach-Alaoglu theorem, 𝐼_𝜇 is a relatively 𝜎(𝑏𝑎(Σ),𝐵(Σ))-compact subset of 𝑏𝑎(Σ). Since 𝐼_𝜇 is 𝜎(𝑏𝑎(Σ),𝐵(Σ))-closed in 𝑏𝑎(Σ), 𝐼_𝜇 is 𝜎(𝑏𝑎(Σ),𝐵(Σ))-compact, and it follows that 𝐼_𝜇 is 𝜎(𝐼,𝐵(Σ))-compact.

We define the absolute Mackey topology  |𝜏|(𝐵(Σ),𝐼) as the topology of uniform convergence on all convex, solid and 𝜎(𝐼,𝐵(Σ))-compact subsets of 𝑏𝑎(Σ). By Proposition 8  |𝜏|(𝐵(Σ),𝐼) is a locally convex-solid topology on 𝐵(Σ). Thus, in view of Proposition 9 we have 𝜎(𝐵(Σ))⊂|𝜎𝐵(Σ))⊂|𝜏𝐵(Σ))(𝐵(Σ)).,𝐼|(,𝐼|(,𝐼⊂𝜏,𝐼(22)

As a consequence, we obtain the following Mackey-Arens type theorem for locally convex-solid topologies on 𝐵(Σ).

Proposition 10. Let 𝜏 be a locally convex-solid topology on 𝐵(Σ). Then, |𝜎|(𝐵(Σ),𝐵(Σ)^′_𝜏)⊂𝜏⊂|𝜏|(𝐵(Σ),𝐵(Σ)^′_𝜏).

Proof. Note that the family ℬ_𝜏 of all convex, solid and 𝜏-closed 𝜏-neighborhoods of 0 forms a local base at 0 for 𝜏. Then, 𝜏 is the topology of uniform convergence on the family {𝑉⁰∶𝑉∈ℬ_𝜏}. By Proposition 8 and the Banach-Alaoglu theorem, for each 𝑉∈ℬ_𝜏, 𝑉⁰ is convex, solid, and 𝜎(𝐵(Σ)^′_𝜏,𝐵(Σ))-compact in 𝑏𝑎(Σ). It follows that 𝜏⊂|𝜏|(𝐵(Σ),𝐵(Σ)^′_𝜏).
To show that |𝜎|(𝐵(Σ),𝐵(Σ)^′_𝜏)⊂𝜏, assume that 𝑓_𝛼→0 in 𝐵(Σ) for 𝜏. Then, |𝑓_𝛼|→0 for 𝜏 because 𝜏 is locally solid. Let 𝜇∈𝑏𝑎(Σ) be such that Φ_𝜇∈𝐵(Σ)^′_𝜏. Then, |Φ_𝜇|∈𝐵(Σ)^′_𝜏 because 𝐵(Σ)^′_𝜏 is an ideal of 𝐵(Σ)^∗ (see Proposition 4). Hence, 𝜌_𝜇(𝑓_𝛼)=|Φ_𝜇|(|𝑓_𝛼|)→0, and it follows that 𝑓_𝛼→0 for |𝜎|(𝐵(Σ),𝐵(Σ)^′_𝜏), as desired.

4. The Mackey Topology 𝜏(𝐵(Σ),𝑐𝑎(Σ))

In this section we study the Mackey topology 𝜏(𝐵(Σ),𝑐𝑎(Σ)) on 𝐵(Σ). It is known that (𝐵(Σ),𝜏(𝐵(Σ),𝑐𝑎(Σ))) is a generalized DF-space, that is, 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is the finest locally convex topology agreeing with itself on norm-bounded sets in 𝐵(Σ) (see [8, Corollary 11.8], [9, Section  4], [10, Theorem 2]).

Let 𝑐𝑎_ℝ(Σ) stand for the Dedekind complete Banach lattice of all countably additive measures 𝜇∶Σ→ℝ. Then, 𝑐𝑎_ℝ(Σ) is a band of 𝑏𝑎_ℝ(Σ). Moreover, the 𝜎-order continuous dual 𝐵_ℝ(Σ)^∗_𝑐 of 𝐵_ℝ(Σ) can be identified with 𝑐𝑎_ℝ(Σ) through the integration mapping 𝑐𝑎_ℝ(Σ)∋𝜇↦𝜑_𝜇∈𝐵_ℝ(Σ)^∗_𝑐, where 𝜑_𝜇(𝑢)=∫_Ω𝑢𝑑𝜇 for all 𝑢∈𝐵_ℝ(Σ) (see [4, Theorem 13.5]).

Recall that a sequence (𝑢_𝑛) in 𝐵_ℝ(Σ) is said to be order convergent to 𝑢∈𝐵_ℝ(Σ) (in symbols,  𝑢−−→_𝑛^(o)𝑢) if there exists a sequence (𝑣_𝑛) in 𝐵_ℝ(Σ)⁺ such that |𝑢_𝑛−𝑢|≤𝑣_𝑛↓0 (in the vector lattice sense; see [6, Definition  1.12]).

Definition 11. A linear functional Φ on 𝐵(Σ) is said to be 𝜎-order continuous if for each sequence (𝑓_𝑛) in 𝐵(Σ), |𝑓_𝑛|⟶^(o)0 in 𝐵_ℝ(Σ) implies Φ(𝑓_𝑛)→0.

Proposition 12. For 𝜇∈𝑏𝑎(Σ) the following statements are equivalent:(i)Φ_𝜇 is 𝜎-order continuous. (ii)Φ_𝜇(𝑓_𝑛)→0 for each uniformly bounded sequence (𝑓_𝑛) in 𝐵(Σ) such that 𝑓_𝑛(𝜔)→0 for all 𝜔∈Ω. (iii)Φ_𝜇(𝑢_𝑛)→0 for each sequence (𝑢_𝑛) in 𝐵(Σ)⁺ such that 𝑢_𝑛(𝜔)↓0 for all 𝜔∈Ω. (iv)𝜇∈𝑐𝑎(Σ).

Proof. (i)⇒(ii) Assume that Φ_𝜇 is 𝜎-order continuous and let 𝑓_𝑛(𝜔)→0 for all 𝜔∈Ω and sup_𝑛‖𝑓_𝑛‖<∞. Let 𝑢_𝑛(𝜔)=sup_𝑘≥𝑛|𝑓_𝑘(𝜔)| for 𝜔∈Ω, 𝑛∈ℕ. Then, 𝑢_𝑛∈𝐵_ℝ(Σ) and |𝑓_𝑛(𝜔)|≤𝑢_𝑛(𝜔)↓0 for all 𝜔∈Ω and 𝑛∈ℕ. It follows that |𝑓_𝑛|−−→^(o)0 in 𝐵_ℝ(Σ), and hence, ∫_Ω𝑓_𝑛𝑑𝜇→0.
(ii)⇒(iii) It is obvious.
(iii)⇒(iv) Assume that (iii) holds, and let 𝐴_𝑛↓∅, (𝐴_𝑛)⊂Σ. Then, 𝟙_𝐴𝑛(𝜔)↓0 for all 𝜔∈Ω. Hence, 𝜇(𝐴_𝑛)=∫_Ω𝟙_𝐴𝑛𝑑𝜇→0.
(iv)⇒(i) Assume that 𝜇∈𝑐𝑎(Σ). Let (𝑓_𝑛) be a sequence in 𝐵(Σ) such that |𝑓_𝑛|→^(𝑜)0 in 𝐵_ℝ(Σ), that is, |𝑓_𝑛|≤𝑢_𝑛↓0 for some sequence (𝑢_𝑛) in 𝐵(Σ)⁺. Then Φ_𝜇𝑓_𝑛=_Ω𝑓_𝑛𝑑𝜇≤_Ω𝑓_𝑛𝑑𝜇≤_Ω𝑢_𝑛𝑑𝜇.(23) Note that 𝑢_𝑛(𝜔)↓0 for |𝜇|-almost all 𝜔∈Ω. Indeed, let 𝑢_𝑛(𝜔)↓𝑢(𝜔) for each 𝜔∈Ω. Then, 𝑢∈𝐵_ℝ(Σ), so 𝐴={𝜔∈Ω∶𝑢(𝜔)>0}∈Σ. If 𝜇(𝐴)>0, then there exists 𝜀>0 such that 𝐴_𝜀={𝜔∈Ω∶𝑢(𝜔)≥𝜀}∈Σ satisfies |𝜇|(𝐴_𝜀)>0. But then 𝑢_𝑛≥𝜀𝟙_𝐴𝜀 for each 𝑛∈ℕ, contradicting 𝑢_𝑛↓0 in 𝐵_ℝ(Σ). Hence, by Lebesgue Dominated Convergence theorem, we get ∫_Ω𝑢_𝑛𝑑|𝜇|→0, so ∫_Ω𝑓_𝑛𝑑𝜇→0.

By 𝐵(Σ)^∗_𝑐 we will denote the space of all 𝜎-order continuous functionals on 𝐵(Σ). For 𝜔∈Ω let 𝛿_𝜔 denote the evaluation functional on 𝐵(Σ), that is, 𝛿_𝜔(𝑓)=𝑓(𝜔) for all 𝑓∈𝐵(Σ). Then 𝛿_𝜔∈𝐵(Σ)^∗_𝑐 and it follows that 𝐵(Σ)^∗_𝑐 separates the points of 𝐵(Σ).

From Proposition 12 and Lemma 5 if follows that for 𝜇∈𝑏𝑎(Σ), Φ_𝜇∈𝐵(Σ)^∗_𝑐 if and only if |Φ_𝜇|∈𝐵(Σ)^∗_𝑐. Since 𝑐𝑎(Σ) is an ideal of 𝑏𝑎(Σ), we see that 𝐵(Σ)^∗_𝑐 is an ideal of 𝐵(Σ)^∗. For a subset ℳ of 𝑏𝑎(Σ) let |ℳ|=𝜇∶𝜇∈ℳ.(24)

Proposition 13. For a subset ℳ of 𝑐𝑎(Σ) the following statements are equivalent:(i)ℳ is relatively 𝜎(𝑐𝑎(Σ),𝐵(Σ))-compact. (ii)ℳ is relatively 𝜎(𝑐𝑎(Σ),𝐵(Σ))-countably compact. (iii)sup_𝜇∈ℳ‖𝜇‖<∞ and ℳ is uniformly countably additive. (iv)sup_{𝜈∈conv(sol(ℳ))}‖𝜈‖<∞ and conv(sol(ℳ)) is uniformly countably additive. (v)conv(sol(ℳ)) is relatively weakly compact. (vi)ℳ is relatively weakly compact.

Proof. (i)⇒(ii) It is obvious.
(ii)⇒(iii) Assume that ℳ is relatively 𝜎(𝑐𝑎(Σ),𝐵(Σ))-countably compact. It follows that ℳ is 𝜎(𝑐𝑎(Σ),𝐵(Σ))-bounded, and since {Φ_𝜇∶𝜇∈ℳ}⊂𝐵(Σ)^∗, by the uniform boundedness theorem, sup_𝜇∈ℳ‖Φ_𝜇‖<∞, that is, sup_𝜇∈ℳ‖𝜇‖<∞.
Assume, on the contrary, that ℳ is not uniformly countably additive. Then there exist a sequence (𝐴_𝑛) in Σ, a sequence (𝜇_𝑛) in ℳ and 𝛿>0 such that 𝐴_𝑛↓∅ and |𝜇_𝑛(𝐴_𝑛)|≥𝛿 for all 𝑛∈ℕ. Let 𝒞={𝜇_𝑛∶𝑛∈ℕ}. Putting 𝜆=∑^∞_𝑛=1(1/2^𝑛)(|𝜇_𝑛|/(1+‖𝜇_𝑛‖)), we see that 𝜆∈𝑐𝑎(Σ)⁺ and 𝜇_𝑛≪𝜆 for all 𝑛∈ℕ. Hence, 𝒞 is a relatively 𝜎(𝑐𝑎_𝜆(Σ),𝐵(Σ))-countably compact subset of 𝑐𝑎_𝜆(Σ) (= the space of all 𝜆-absolutely continuous measures in 𝑐𝑎(Σ)). In view of the Radon-Nikodym theorem, for each 𝜇∈𝑐𝑎_𝜆(Σ) there exists uniquely 𝑓_𝜇∈𝐿¹(𝜆) such that 𝜇(𝐴)=∫_𝐴𝑓_𝜇𝑑𝜆 for each 𝐴∈Σ. It follows that for every 𝑔∈𝐿^∞(𝜆) we have ∫_Ω𝑓_𝜇𝑔𝑑𝜆=∫_Ω𝑔𝑑𝜇. Hence, the linear mapping 𝐹_𝜆∶𝑐𝑎_𝜆(Σ)∋𝜇↦𝑓_𝜇∈𝐿¹(𝜆) is (𝜎(𝑐𝑎_𝜆(Σ),𝐵(Σ)),𝜎(𝐿¹(𝜆),𝐿^∞(𝜆)))-continuous. Thus, the set 𝐹_𝜆(𝒞)={𝑓_𝜇𝑛∶𝑛∈ℕ} is relatively 𝜎(𝐿¹(𝜆),𝐿^∞(𝜆))-countably compact in 𝐿¹(𝜆). Hence, by the Eberlein-Smulian theorem {𝑓_𝜇𝑛∶𝑛∈ℕ} is relatively 𝜎(𝐿¹(𝜆),𝐿^∞(𝜆))-compact in 𝐿¹(𝜆). According to the Dunford-Pettis theorem (see [5, Chapter 3, Theorem  15]), the set {𝑓_𝜇𝑛∶𝑛∈ℕ} in 𝐿¹(𝜆) is uniformly integrable. Since 𝐴_𝑛↓∅, we have that 𝜆(𝐴_𝑛)→0, and hence 𝜇(𝐴_𝑛)=∫_𝐴𝑛𝑓_𝜇𝑛𝑑𝜆→0, contradicting |𝜇_𝑛(𝐴_𝑛)|≥𝛿 for 𝑛∈ℕ.
(iii)⇒(iv) Assume that sup_𝜇∈ℳ‖𝜇‖<∞ and ℳ is uniformly countably additive. Then, |ℳ| is uniformly countably additive (see [5, Chapter 1, Proposition  17]). Note that for 𝐴∈Σ, sup_{𝜈∈sol(ℳ)}|𝜈|(𝐴)=sup_𝜇∈ℳ|𝜇|(𝐴). It follows that sup_{𝜈∈sol(ℳ)}‖𝜈‖<∞ and sol(ℳ) is uniformly countably additive. Moreover, one can easily observe that for 𝐴∈Σ, sup_{𝜈∈conv(sol(ℳ))}|𝜈(𝐴)|=sup_{𝜈∈sol(ℳ)}|𝜈(𝐴)|and sup_{𝜈∈conv(sol(ℳ))}‖𝜈‖=sup_{𝜈∈sol(ℳ)}‖𝜈‖. It follows that conv(sol(ℳ)) is uniformly bounded and uniformly countably additive.
(iv)⇒(v) It is well known (see [11, Chapter 7, Theorem  13]).
(v)⇒(vi)⇒(i) It is obvious.

Recall that a locally convex space (𝐿,𝜂) is said to be strongly Mackey if every relatively 𝜎(𝐿^′_𝜂,𝐿)-countably compact set in 𝐿^′_𝜂 is 𝜂-equicontinuous.

Corollary 14. The space (𝐵(Σ),𝜏(𝐵(Σ),𝑐𝑎(Σ))) is strongly Mackey.

Proof. Let ℳ be a relatively 𝜎(𝑐𝑎(Σ),𝐵(Σ))-countably compact set in 𝑐𝑎(Σ). Then, by Proposition 13, the set cl_𝜎(absconvℳ) is absolutely convex and 𝜎(𝑐𝑎(Σ),𝐵(Σ))-compact. Hence, ⁰(cl_𝜎(absconvℳ)) is a 𝜏(𝐵(Σ),𝑐𝑎(Σ))-neighborhood of 0. It follows that ℳ is 𝜏(𝐵(Σ),𝑐𝑎(Σ))-equicontinuous.

Now we distinguish three natural classes of linear topologies on 𝐵(Σ).

Definition 15. A locally solid topology 𝜏 on 𝐵(Σ) is said to be(i)𝜎-Lebesgue if 𝑢_𝑛→0 for 𝜏whenever (𝑢_𝑛) is a sequence in 𝐵(Σ)⁺ such that 𝑢_𝑛↓0 in 𝐵_ℝ(Σ) (in the vector lattice sense).(ii)a 𝜎-smooth topology if 𝑓_𝑛→0 for 𝜏 whenever (𝑓_𝑛) is uniformly bounded sequence in 𝐵(Σ) such that 𝑓_𝑛(𝜔)→0 for all 𝜔∈Ω.(iii)a 𝜎-Dini topology if 𝑢_𝑛→0 for 𝜏 whenever (𝑢_𝑛) is a sequence in 𝐵(Σ)⁺ such that 𝑢_𝑛(𝜔)↓0 for all 𝜔∈Ω.

Note that every 𝜎-Lebesgue topology on 𝐵(Σ) is 𝜎-smooth, and every 𝜎-smooth topology on 𝐵(Σ) is 𝜎-Dini.

Now, we characterize the Mackey topology 𝜏(𝐵(Σ),𝑐𝑎(Σ)) on 𝐵(Σ).

Theorem 16. (i) 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is a locally convex-solid topology.
(ii) 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is the finest locally convex 𝜎-Lebesgue topology on 𝐵(Σ).
(iii) 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is the finest locally convex 𝜎-smooth topology on 𝐵(Σ).
(iv) 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is the finest locally convex 𝜎-Dini topology on 𝐵(Σ).

Proof. (i) Let 𝑉 be an absolutely convex and closed neighborhood of 0 for 𝜏(𝐵(Σ),𝑐𝑎(Σ)). Then, the polar 𝑉⁰ of 𝑉 (with respect to the solid dual system ⟨𝐵(Σ),𝑐𝑎(Σ)⟩) is 𝜎(𝑐𝑎(Σ),𝐵(Σ))-compact. In view of Proposition 13  conv(sol(𝑉⁰)) is relatively 𝜎(𝑐𝑎(Σ),𝐵(Σ))-compact and hence cl_𝜎(conv(sol(𝑉⁰))) (the closure is taken in 𝜎(𝑐𝑎(Σ),𝐵(Σ))) is absolutely convex and 𝜎(𝑐𝑎(Σ),𝐵(Σ))-compact. Hence, ⁰cl_𝜎con𝑣sol𝑉⁰⊂⁰sol𝑉⁰⊂⁰𝑉⁰=𝑉.(25) It follows that ⁰(sol(𝑉⁰)) is a convex and solid neighborhood of 0for 𝜏(𝐵(Σ),𝑐𝑎(Σ)) (see Proposition 8). This means that 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is a locally convex-solid topology on 𝐵(Σ).
(ii) Now we shall show that 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is a 𝜎-Lebesgue topology. Assume that (𝑢_𝑛) is a sequence in 𝐵(Σ)⁺ such that 𝑢_𝑛↓0 in the vector lattice 𝐵_ℝ(Σ). Let ℳ be a relatively 𝜎(𝑐𝑎(Σ),𝐵(Σ))-compact set in 𝑐𝑎(Σ). Then ℳ is uniformly bounded and uniformly countably additive (see Proposition 13), and it follows that |ℳ| is uniformly countably additive. Hence, by [10, Theorem 3] |ℳ| is 𝜏(𝐵_ℝ(Σ),𝑐𝑎_ℝ(Σ))-equicontinuous. Hence, the polar ⁰_ℝ|ℳ| of |ℳ| (with respect to the dual pair ⟨𝐵_ℝ(Σ),𝑐𝑎_ℝ(Σ)⟩) is a 𝜏(𝐵_ℝ(Σ),𝑐𝑎_ℝ(Σ)-neighborhood of 0 in 𝐵_ℝ(Σ). Let 𝜀>0 be given. Since 𝜏(𝐵_ℝ(Σ),𝑐𝑎_ℝ(Σ)) is a 𝜎-Lebesgue topology on 𝐵_ℝ(Σ) (see [6, Example 18, page 178], [6, Corollary 6.29 and Theorem 3.20]), there exists 𝑛_𝜀∈ℕ such that |𝑢_𝑛|∈𝜀(⁰_ℝ|ℳ|) for 𝑛≥𝑛_𝜀. Hence for 𝑛≥𝑛_𝜀 we get 𝑝_ℳ𝑢_𝑛=sup_Ω𝑢_𝑛𝑑𝜇∶𝜇∈ℳ≤sup_Ω𝑢_𝑛𝑑𝜇∶𝜇∈ℳ≤𝜀.(26) It follows that 𝑓_𝑛→0 for 𝜏(𝐵(Σ),𝑐𝑎(Σ)), and this means that 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is a 𝜎-Lebesgue topology on 𝐵(Σ).
Now assume that 𝜏 is a locally convex-solid 𝜎-Lebesgue topology on 𝐵(Σ). Then 𝐵(Σ)^′_𝜏⊂𝐵(Σ)^∗_𝑐, and hence, 𝜏⊂𝜏(𝐵(Σ),𝐵(Σ)^′_𝜏)⊂𝜏(𝐵(Σ),𝐵(Σ)^∗_𝑐)=𝜏(𝐵(Σ),𝑐𝑎(Σ)).
(iii) From (ii) it follows that 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is a 𝜎-smooth topology.
Now let 𝜏 be a locally convex 𝜎-smooth topology on 𝐵(Σ). Then by Proposition 12 we get 𝐵(Σ)^′_𝜏⊂𝐵(Σ)^∗_𝑐. Hence, 𝜏⊂𝜏(𝐵(Σ),𝐵(Σ)^′_𝜏)⊂𝜏(𝐵(Σ),𝐵(Σ)^∗_𝑐)=𝜏(𝐵(Σ),𝑐𝑎(Σ)).
(iv) Clearly 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is a 𝜎-Dini topology on 𝐵(Σ). Assume that 𝜏 is a locally convex 𝜎-Dini topology on 𝐵(Σ). In view of Proposition 12, 𝐵(Σ)^′_𝜏⊂𝐵(Σ)^∗_𝑐 and it follows that 𝜏⊂𝜏(𝐵(Σ),𝑐𝑎(Σ)).

A characterization of relative weak compactness in the Banach space 𝑐𝑎(Σ) in terms of uniform boundedness and uniform countable additivity of measures belongs to the classical results in measure theory (see [11, Chapter 7, Theorem 13], [8, Theorem 11.6], [12, Theorem 1.1]). Now, using the properties of the Mackey topology 𝜏(𝐵(Σ),𝑐𝑎(Σ)) we are ready to add Grothendieck type conditions to this characterization.

Corollary 17. For a subset ℳ of 𝑐𝑎(Σ) the following statements are equivalent:(i)ℳ is a relatively weakly compact. (ii)sup_𝜇∈ℳ‖𝜇‖<∞ and ∫_Ω𝑓_𝑛𝑑𝜇→0 uniformly for 𝜇∈ℳ whenever (𝑓_𝑛) is a uniformly bounded sequence in 𝐵(Σ) such that 𝑓_𝑛(𝜔)→0 for all 𝜔∈Ω.(iii)sup_𝜇∈ℳ‖𝜇‖<∞ and ∫_Ω𝑢_𝑛𝑑𝜇→0 uniformly for 𝜇∈ℳ whenever (𝑢_𝑛) is a sequence in 𝐵(Σ)⁺ such that 𝑢_𝑛(𝜔)↓0 for all 𝜔∈Ω.

Proof. (i)⇒(ii) Assume that ℳ is relatively weakly compact. Then, in view of Corollary 14  ℳ is bounded and 𝜏(𝐵(Σ),𝑐𝑎(Σ))-equicontinuous. Let (𝑓_𝑛) be a uniformly bounded sequence in 𝐵(Σ) such that 𝑓_𝑛(𝜔)→0 for all 𝜔∈Ω. Since 𝜏(𝐵(Σ),𝑐𝑎(Σ)) is a 𝜎-smooth topology on 𝐵(Σ) (see Theorem 16), we obtain that 𝑓_𝑛→0 for 𝜏(𝐵(Σ),𝑐𝑎(Σ)). Now let 𝜀>0 be given. Then, there exists a 𝜏(𝐵(Σ),𝑐𝑎(Σ))-neighborhood 𝑉 of 0 such that sup_𝜇∈ℳ|∫_Ω𝑓𝑑𝜇|≤𝜀 for all 𝑓∈𝑉. Hence, there exists 𝑛_𝜀∈ℕ such that 𝑓_𝑛∈𝑉 for 𝑛≥𝑛_𝜀, and it follows that sup_𝜇∈ℳ|∫_Ω𝑓_𝑛𝑑𝜇|≤𝜀 for 𝑛≥𝑛_𝜀, as desired.
(ii)⇒(iii) It is obvious.
(iii)⇒(i) Assume that (iv) holds and let 𝐴_𝑛↓∅, (𝐴_𝑛)⊂Σ. Then, 𝟙_𝐴𝑛(𝜔)↓0 for all 𝜔∈Ω. Hence sup_𝜇∈ℳ|𝜇(𝐴_𝑛)|=sup_𝜇∈ℳ|∫_Ω𝟙_𝐴𝑛𝑑𝜇|→0, that is, ℳ is uniformly countably additive. Hence, ℳ is relatively weakly compact (see [11, Chapter 7, Theorem 13]).

Remark 18. One can observe that Corollary 17 is closely related to a Grothendieck’s characterization of relative weak compactness in the space of bounded complex Radon measures on a locally compact space (see [13, Theorem 2]).

Now we define a class of linear operators on 𝐵(Σ).

Definition 19. Let (𝐸,𝜉) be a locally convex Hausdorff space (briefly, lcHs). A linear operator 𝑇∶𝐵(Σ)→𝐸 is said to be 𝜎-order continuous if for each sequence (𝑓_𝑛) in 𝐵(Σ),|𝑓_𝑛|−−→^(o)0 in 𝐵_ℝ(Σ) implies 𝑇(𝑓_𝑛)→0 for 𝜉.

For terminology and basic results concerning the integration with respect to vector measures, we refer the reader to [14–17].

Let (𝐸,𝜉) be a quasicomplete lcHs. Let 𝑚∶Σ→𝐸 be a 𝜉-bounded measure (i.e., the range of 𝑚 is 𝜉-bounded in 𝐸). Given 𝑓∈𝐵(Σ), let (𝑠_𝑛) be a sequence of Σ-simple scalar functions that converges uniformly to 𝑓 on Ω. Following [14, Definition 1] we say that 𝑓 is 𝑚-integrable and define _Ω𝑓𝑑𝑚∶=𝜉−lim_Ω𝑠_𝑛𝑑𝑚.(27) The ∫_Ω𝑓𝑑𝑚 is well defined (see [14, Lemma 5]) and the map 𝑇_𝑚∶𝐵(Σ)→𝐸 given by 𝑇_𝑚(𝑓)=∫_Ω𝑓𝑑𝑚 is (‖⋅‖,𝜉)-continuous and linear, and for each 𝑒^′∈𝐸^′_𝜉𝑒^′_Ω𝑓𝑑𝑚=_Ω𝑓𝑑𝑒^′∘𝑚for𝑓∈𝐵(Σ)(28) (see [14, Lemma 5]). Conversely, let 𝑇∶𝐵(Σ)→𝐸 be a (‖⋅‖,𝜉)-continuous linear operator, and let 𝑚(𝐴)=𝑇(𝟙_𝐴) for 𝐴∈Σ. Then, 𝑚∶Σ→𝐸 is a 𝜉-bounded vector measure, called the representing measure of 𝑇 and 𝑇_𝑚(𝑓)=𝑇(𝑓) for 𝑓∈𝐵(Σ) (see [14, Definition 2]).

Recall that a measure 𝑚∶Σ→𝐸 is said to be 𝜉-countably additive if 𝑚(𝐴_𝑛)→0 for 𝜉 whenever 𝐴_𝑛↓∅, (𝐴_𝑛)⊂Σ.

The following characterization of 𝜎-order continuous operators from 𝐵(Σ) into a quasicomplete lcHs (𝐸,𝜉) displays the close connection between the Mackey topology 𝜏(𝐵(Σ),𝑐𝑎(Σ)) on 𝐵(Σ) and 𝐸-valued 𝜉-countably additive measures.

Proposition 20. Assume that (𝐸,𝜉) is a quasicomplete lcHs. Then for a 𝜉-bounded measure 𝑚∶Σ→𝐸 the following statements are equivalent:(i)𝑇_𝑚 is (𝜏(𝐵(Σ),𝑐𝑎(Σ)),𝜏(𝐸,𝐸^′_𝜉))-continuous. (ii)𝑇_𝑚 is (𝜏(𝐵(Σ),𝑐𝑎(Σ)),𝜉)-continuous. (iii)𝑇_𝑚 is sequentially (𝜏(𝐵(Σ),𝑐𝑎(Σ)),𝜉)-continuous. (iv)𝑇_𝑚 is 𝜎-order continuous. (v)𝑇_𝑚(𝑓_𝑛)→0 for 𝜉 whenever (𝑓_𝑛) is a uniformly bounded sequence in 𝐵(Σ) such that 𝑓_𝑛(𝜔)→0 for all 𝜔∈Ω.(vi)𝑇_𝑚(𝑢_𝑛)→0 for 𝜉 for each sequence (𝑢_𝑛) in 𝐵(Σ)⁺ such that 𝑢_𝑛(𝜔)→0 for all 𝜔∈Ω.(vii)𝑚 is 𝜉-countably additive.

Proof. (i)⇒(ii)⇒(iii) It is obvious.
(iii)⇒(iv) Assume that 𝑇_𝑚 is sequentially (𝜏(𝐵(Σ),𝑐𝑎(Σ)),𝜉)-continuous, and let (𝑓_𝑛) be a sequence in 𝐵(Σ) such that |𝑓_𝑛|−−→^(o)0 in 𝐵_ℝ(Σ). Then, by Theorem 16, 𝑓_𝑛→0 for 𝜏(𝐵(Σ),𝑐𝑎(Σ)). Hence, 𝑇_𝑚(𝑓_𝑛)→0 for 𝜉, that is, 𝑇 is 𝜎-order continuous.
(iv)⇒(v) Assume that 𝑇_𝑚 is 𝜎-order continuous, and assume that (𝑓_𝑛) is a uniformly bounded sequence in 𝐵(Σ) such that 𝑓_𝑛(𝜔)→0 for all 𝜔∈Ω. Then |𝑓_𝑛|−−→^(o)0 in 𝐵_ℝ(Σ) (see the proof of Proposition 12). Hence, 𝑇_𝑚(𝑓_𝑛)→0 for 𝜉, as desired.
(v)⇒(vi) It is obvious.
(vi)⇒(vii) Assume that (vi) holds, and let 𝐴_𝑛↓∅, (𝐴_𝑛)⊂Σ. Then, 𝟙_𝐴𝑛(𝜔)↓0 for all 𝜔∈Ω, and it follows that 𝜇(𝐴_𝑛)=𝑇_𝑚(𝟙_𝐴𝑛)→0 for 𝜉. This means that 𝑚 is 𝜉-countably additive.
(vii)⇒(i) Assume that 𝑚 is 𝜉-countably additive. Then 𝑒^′∘𝑚∈𝑐𝑎(Σ) for each 𝑒^′∈𝐸^′_𝜉, and in view of (28) and Proposition 12 we get 𝑒^′∘𝑇_𝑚∈𝐵(Σ)^∗_𝑐 for each 𝑒^′∈𝐸^′_𝜉. Hence, 𝑇_𝑚 is (𝜎(𝐵(Σ),𝐵(Σ)^∗_𝑐),𝜎(𝐸,𝐸^′_𝜉))-continuous (see [7, Theorem  9.26]). It follows that 𝑇_𝑚 is (𝜏(𝐵(Σ),𝐵(Σ)^∗_𝑐),𝜏(𝐸,𝐸^′_𝜉))-continuous (see [7, Example 11, page 149]).

Acknowledgments

The author wishes to thank the referee for useful suggestions that have improved the paper.