Kadec-Klee Properties of Calderón-Lozanovskiĭ Function Spaces

Abstract

We study Kadec-Klee properties with respect to global (local) convergence in measure. First, we present some results concerning Köthe spaces and Orlicz functions. Next, we shall give full criteria for Kadec-Klee properties with respect to global (local) convergence in measure in Calderón-Lozanovskiĭ function spaces. In particular, we obtain the full characterizations of Kadec-Klee properties in Orlicz-Lorentz function spaces, which have not been presented until now.

1. Introduction

The Kadec-Klee properties are fundamental notions in the theory of Banach function spaces (see [1, 2]). On the other hand, the Calderón-Lozanovskiĭ spaces are important class of Banach lattices, especially because of their role in the interpolation theory. Geometry of Calderón-Lozanovskiĭ spaces has been intensively developed during the last decades (see, e.g., [3–9]). The complete characterization of Kadec-Klee properties with respect to local 𝐻_𝑙 (global 𝐻_𝑔) convergence in measure for Orlicz function spaces 𝐿_𝜑 has been presented in [10]. The Orlicz spaces 𝐿_𝜑 are special case of Calderón-Lozanovskiĭ spaces 𝐸_𝜑. Thus, it is natural to investigate properties 𝐻_𝑙 and 𝐻_𝑔 in the spaces 𝐸_𝜑 generated by arbitrary Köthe spaces 𝐸. Since 𝐿¹∈𝐻_𝑙 and 𝐸_𝜑=𝐿_𝜑 when 𝐸=𝐿¹, so finding full criteria of 𝐻_𝑙 and 𝐻_𝑔 in Orlicz spaces is simpler than in general Calderón-Lozanovskiĭ spaces. The sufficient conditions of 𝐻_𝑙 in Calderón-Lozanovskiĭ function spaces 𝐸_𝜑 have been proved in [4, 6]. Note that the implication 𝐸_𝜑∈(𝐻_𝑙)⇒𝜑∈Δ^𝐸2 follows from the fact that property 𝐻_𝑙 implies order continuity in Köthe spaces and from results concerning copy of 𝑙^∞ in 𝐸_𝜑 (see [5, 10]). The implication 𝐸_𝜑∈(𝐻_𝑙)⇒𝐸∈(𝐻_𝑙) has not been discussed at all. Moreover, property 𝐻_𝑔 in 𝐸_𝜑 has not been considered early. We present full criteria of properties 𝐻_𝑙 and 𝐻_𝑔 in Calderón-Lozanovskiĭ function spaces 𝐸_𝜑. Furthermore, in the case of property 𝐻_𝑔, we prove some general results concerning Köthe spaces which will be essential in studying this property in spaces 𝐸_𝜑. The crucial difficulty in the proofs concerning property 𝐻_𝑔 in spaces 𝐸_𝜑 is the fact that property 𝐻_𝑔 needs not imply order continuity in general. Note that if 𝐸∈(𝐻_𝑙), then 𝐸∈(𝑂𝐶) (see [10]), so looking for complete characterizations in the spaces 𝐸_𝜑, the case of property 𝐻_𝑙 is easier than 𝐻_𝑔. As an application, we get the criteria of properties 𝐻_𝑙 and 𝐻_𝑔 in Orlicz-Lorentz function spaces (Λ_𝜔)_𝜑. A sufficient condition for property 𝐻_𝑙 of (Λ_𝜔)_𝜑 over a finite, atomless measure space has been presented in [6, Theorem  1]. Note also that the case of Kadec-Klee properties with respect to pointwise (uniform) convergence in Calderón-Lozanovskiĭ sequence spaces has been solved in [11].

2. Preliminaries

Let ℝ, ℝ₊, ℕ be the sets of real, nonnegative real, and positive integer numbers, respectively. As usual, 𝑆(𝑋) (resp., 𝐵(𝑋)) stands for the unit sphere (resp., the closed unit ball) of a real Banach space (𝑋,‖⋅‖_𝑋).

Let (𝑇,Σ,𝜇) be a 𝜎-finite and complete measure space. By 𝐿⁰=𝐿⁰(𝑇) we mean the set of all 𝜇-equivalence classes of real-valued measurable functions defined on 𝑇.

A Banach space 𝐸=(𝐸,‖⋅‖_𝐸) is said to be a Köthe space if 𝐸 is a linear subspace of 𝐿⁰ and(i)if 𝑥∈𝐸, 𝑦∈𝐿⁰, and |𝑦|≤|𝑥|  𝜇-a.e., then 𝑦∈𝐸 and ‖𝑦‖_𝐸≤‖𝑥‖_𝐸;(ii)there exists a function 𝑥 in 𝐸 that is positive on the whole 𝑇 (see [2, 12]).

Every Köthe space is a Banach lattice under the natural partial order (𝑥≥0 if 𝑥(𝑡)≥0 for 𝜇-a.e. 𝑡∈𝑇). In particular, if we consider the space 𝐸 over the nonatomic measure 𝜇, then we will say that 𝐸 is a Köthe function space. If we replace the measure space (𝑇,Σ,𝜇) by the counting measure space (ℕ,2^ℕ,𝑚), then we will say that 𝐸 is a Köthe sequence space.

The set 𝐸₊={𝑥∈𝐸∶𝑥≥0} is called the positive cone of 𝐸. For any subset 𝐴⊂𝐸, define 𝐴₊=𝐴∩𝐸₊.

By the symmetric Köthe function space 𝐸 (symmetric Banach function space) on 𝐼, where 𝐼=[0,1] or 𝐼=[0,∞) with the Lebesgue measure 𝑚, we mean a Köthe space 𝐸=(𝐸,‖⋅‖_𝐸) with the additional property that for any two equimeasurable functions 𝑥∼𝑦, 𝑥,𝑦∈𝐿⁰(𝐼) (i.e., they have the same distribution functions 𝑑_𝑥=𝑑_𝑦, where 𝑑_𝑥(𝜆)=𝑚({𝑡∈𝐼∶|𝑥(𝑡)|>𝜆), 𝜆≥0) and 𝑥∈𝐸 we have 𝑦∈𝐸 and ‖𝑥‖_𝐸=‖𝑦‖_𝐸. In particular, ‖𝑥‖_𝐸=‖𝑥^∗‖_𝐸, where𝑥^∗(𝑡)=inf𝜆>0∶𝑑_𝑥(𝜆)≤𝑡,𝑡>0.(2.1) For basic properties of symmetric spaces and rearrangements, we refer to [13, 14]. In the sequel, considering symmetric Köthe function space 𝐸 over measure space (𝑇,Σ,𝜇) we mean 𝑇=[0,1] or 𝑇=[0,∞) with the Lebesgue measure 𝜇.

A point 𝑥∈𝐸 is said to have order continuous norm if for any sequence (𝑥_𝑚) in 𝐸 such that 0≤𝑥_𝑚≤|𝑥| and 𝑥_𝑚→0  𝜇-a.e. we have ‖𝑥_𝑚‖_𝐸→0. A Köthe space 𝐸 is called order continuous (𝐸∈(𝑂𝐶)) if every element of 𝐸 has an order continuous norm (see [2, 12, 15]). As usual, 𝐸_𝑎 stands for the subspace of order continuous elements of 𝐸. It is known that 𝑥∈𝐸_𝑎 if and only if ‖𝑥𝜒_𝐴𝑛‖_𝐸↓0 for any sequence {𝐴_𝑛} satisfying 𝐴_𝑛↓∅ (i.e., 𝐴_𝑛⊃𝐴_𝑛+1 and 𝜇(⋂^∞𝑛=1𝐴_𝑛)=0). Clearly, 𝜇(⋂^∞𝑛=1𝐴_𝑛)=0 if and only if 𝜒_𝐴𝑛→0  𝜇-a.e. in 𝑇.

Recall that 𝐸 is said to have Kadec-Klee property (𝐸∈(KK) for short) whenever ‖𝑥_𝑛−𝑥‖→0 for any 𝑥 and {𝑥_𝑛} in 𝐸 satisfying 𝑥_𝑛→𝑥 in the weak topology 𝜎(𝐸,𝐸^∗) and ‖𝑥_𝑛‖→‖𝑥‖ (see [2]). This property is sometimes called the Radon-Riesz property or property 𝐻. It has been considered in many classes of Banach spaces (see [1, 4, 16–18]). If we replace the weak convergence 𝜎(𝐸,𝐸^∗) by the convergence in measure (𝑥→_𝑛𝜇𝑥), by the convergence in measure on every set of finite measure (𝑥→_𝑛𝜇𝑥 locally), or by the uniform convergence (𝑥_𝑛⇉𝑥), then we say that 𝐸 has the Kadec-Klee property with respect to convergence in measure, local convergence in measure, or uniform convergence, respectively (we will write 𝐸∈(𝐻_𝑔), 𝐸∈(𝐻_𝑙), and 𝐸∈(𝐻_𝑢)). Clearly, 𝐸∈(𝐻_𝑙)⇒𝐸∈(𝐻_𝑔)⇒𝐸∈(𝐻_𝑢). Moreover, the converse of any of these implications is not true in general (see Sections 3.1 and 3.2 below).

Given a property 𝑃, we will say that a Köthe space 𝐸 satisfies property 𝑃₊(𝐸∈(𝑃)₊) if 𝐸 satisfies property 𝑃 restricted only to nonnegative elements.

In the whole paper, 𝜑 denotes an Orlicz function, that is, 𝜑∶ℝ→[0,∞], it is convex, even, vanishing, and continuous at zero, left continuous on (0,∞) and not identically equal to zero. Denote𝑎_𝜑=sup{𝑢≥0∶𝜑(𝑢)=0},𝑏_𝜑=sup{𝑢≥0∶𝜑(𝑢)<∞}.(2.2) We write 𝜑>0 when 𝑎_𝜑=0 and 𝜑<∞ if 𝑏_𝜑=∞. Let 𝜑_𝑟=𝜑𝜒_𝐺𝜑, where𝐺_𝜑=𝑎^𝜑𝑎,𝑏_𝜑if𝜑𝑏_𝜑<∞,_𝜑,𝑏_𝜑otherwise.(2.3) Define on 𝐿⁰ a convex semimodular 𝐼_𝜑 by𝐼_𝜑(𝑥)=‖𝜑∘𝑥‖^𝐸∞iofth𝜑er∘w𝑥is∈e,𝐸,(2.4) where (𝜑∘𝑥)(𝑡)=𝜑(𝑥(𝑡)), 𝑡∈𝑇. By the Calderón-Lozanovskiĭ space 𝐸_𝜑 we mean𝐸_𝜑=𝑥∈𝐿⁰∶𝐼_𝜑(𝑐𝑥)<∞forsome𝑐>0,(2.5) equipped with so called Luxemburg norm defined by‖𝑥‖_𝜑=inf𝜆>0∶𝐼_𝜑𝑥𝜆≤1.(2.6) We generally assume that if 𝑏_𝜑<∞, then 𝑎_𝜑<𝑏_𝜑, because when 0<𝑎_𝜑=𝑏_𝜑, then 𝐸_𝜑=𝐿^∞ and ‖𝑥‖_𝜑=(1/𝑏_𝜑)‖𝑥‖_∞.

If 𝐸=𝐿¹(𝑒=𝑙¹), then 𝐸_𝜑(𝑒_𝜑) is the Orlicz function (sequence) space equipped with the Luxemburg norm. If 𝐸=Λ_𝜔—the Lorentz function space (𝑒=𝜆_𝜔—the Lorentz sequence space), then 𝐸_𝜑(𝑒_𝜑) is the corresponding Orlicz-Lorentz function (sequence) space denoted by (Λ_𝜔)_𝜑((𝜆_𝜔)_𝜑) and equipped with the Luxemburg norm (see [5, 8]).

We will assume in the whole paper that 𝐸 has the Fatou property, that is, if 0≤𝑥_𝑛↑𝑥∈𝐿⁰ with (𝑥_𝑛)^∞𝑛=1 in 𝐸 and sup_𝑛‖𝑥_𝑛‖_𝐸<∞, then 𝑥∈𝐸 and ‖𝑥‖_𝐸=lim_𝑛‖𝑥_𝑛‖_𝐸. Since 𝐸 has the Fatou property, 𝐸_𝜑 has also this property, whence 𝐸_𝜑 is a Banach space.

We say an Orlicz function 𝜑 satisfies condition Δ₂(0) (resp., Δ₂(∞)) if there exist 𝐾>0 and 𝑢₀>0 such that 𝜑(𝑢₀)>0  (resp., 𝜑(𝑢₀)<∞), and the inequality 𝜑(2𝑢)⩽𝐾𝜑(𝑢) holds for all 𝑢∈[0,𝑢₀] (resp., 𝑢∈[𝑢₀,∞)). If there exists 𝐾>0 such that 𝜑(2𝑢)⩽𝐾𝜑(𝑢) for all 𝑢⩾0, then we say that 𝜑 satisfies condition Δ₂(ℝ₊). We write for short 𝜑∈Δ₂(0), 𝜑∈Δ₂(∞), and 𝜑∈Δ₂(ℝ₊), respectively.

For a Köthe space 𝐸 and an Orlicz function 𝜑, we say that 𝜑 satisfies condition Δ^𝐸2 (𝜑∈Δ^𝐸2 for short) if(1)𝜑∈Δ₂(0) whenever 𝐸↪𝐿^∞,(2)𝜑∈Δ₂(∞) whenever 𝐿^∞↪𝐸,(3)𝜑∈Δ₂(ℝ₊) whenever neither 𝐿^∞↪𝐸 nor 𝐸↪𝐿^∞ (see [5]),

where the symbol 𝐸↪𝐹 stands for the continuous embedding of the space 𝐸 into the space 𝐹.

Relationships between the modular 𝐼_𝜑 and the norm ‖⋅‖_𝜑 are collected in [8].

3. Results

3.1. Köthe Spaces and Orlicz Functions

Given a Köthe function space 𝐸, we setm_𝐸=sup_{𝐵,𝜒𝐵∈𝐸}sup_{[(𝐴𝑛)⊂𝐵,𝜇(𝐴𝑛)→0]}inf_𝑛‖‖𝜒_𝐴𝑛‖‖_𝐸.(3.1)

Remark 3.1. It is easy to see that if 𝐸 is a symmetric Köthe function space, then m_𝐸=inf_{𝐴∈Σ,𝜇(𝐴)>0}‖‖𝜒_𝐴‖‖_𝐸.(3.2)

Recall that if 𝑥∈𝐿⁰, then supp𝑥={𝑡∈𝑇∶𝑥(𝑡)≠0}. For a Köthe space 𝐸⊂𝐿⁰(𝑇,Σ,𝜇), let 𝐹⊂𝐸 be a subset. By the support of 𝐹, we mean the set supp𝐹∈Σ satisfying(i)for each 𝑥∈𝐹, there is 𝐴∈Σ, with 𝜇(𝐴)=0, such that supp𝑥⊂𝐴∪supp𝐹,(ii)there is 𝑥∈𝐹 with 𝜇(supp𝐹⧵supp𝑥)=0.

Lemma 3.2. Suppose that 𝐸 is a Köthe function space. Consider the following statements:(i)𝐸∈(𝑂𝐶),(ii)𝐸∈(𝐻_𝑔)₊,(iii)For any 𝑢∈𝐸 and each sequence {𝐴_𝑛} in Σ the implication 𝜇𝐴_𝑛⟶0⟹‖‖𝑢𝜒_𝐴𝑛‖‖⟶0holds.(3.3)(iv)m_𝐸=0, (v)supp𝐸_𝑎=𝑇, (vi)𝐸_𝑎≠{0}. Then (i) ⇒ (iii) ⇒ (iv) ⇒ (v) ⇒ (vi) and (ii) ⇒ (iii). For a symmetric Köthe function space we have additionally (vi) ⇒ (iv). Furthermore, for a nonsymmetric Köthe function space, the implication (vi) ⇒ (iv) is not true in general. Finally, the implications (i) ⇒ (iii) ⇒ (iv) cannot be reversed in general.

Proof. The implications (i) ⇒ (iii) ⇒ (iv) and (v) ⇒ (vi) are obvious.
(iv) ⇒ (v). If supp𝐸_𝑎≠𝑇, then for each 𝑥∈𝐸 with supp𝑥⊂𝑇⧵𝑠upp𝐸_𝑎 we have 𝑥∉𝐸_𝑎. Let 𝑇₀⊂𝑇⧵supp𝐸_𝑎 be such that 0<𝜇(𝑇₀)<∞ and 𝜒_𝑇0∈𝐸. Then 𝑥₁=𝜒_𝑇0∉𝐸_𝑎 and consequently there is a sequence (𝐴_𝑛) of measurable pairwise disjoint sets with ⋃^∞𝑛=1𝐴_𝑛=𝑇₀ and a number 𝛿>0 satisfying ‖𝜒_𝐴𝑛‖≥𝛿 for each 𝑛 (see [19]). Since 𝜇(𝐴_𝑛)→0, we get m_𝐸≥𝛿>0.
(ii) ⇒ (iii). Assume that (iii) does not hold, then we find an element 𝑢∈𝐸, a number 𝛿>0, and a sequence (𝐴_𝑖) with 𝜇(𝐴_𝑖)→0 such that inf_𝑖‖𝑢𝜒_𝐴𝑖‖_𝐸≥𝛿. Without loss of generality, we may assume that ∑^∞𝑖=1𝜇(𝐴_𝑖)<∞. Set 𝑥_𝑘=sup𝑢𝜒_𝐴𝑖∶𝑖≥𝑘∈𝐿⁰.(3.4) Let 𝐴=⋃_𝑖𝐴_𝑖. Clearly, 𝑥_𝑘≤𝑢𝜒_𝐴∈𝐸. Put 𝑦=𝑢𝜒_𝐴 and 𝑦_𝑘=𝑦−𝑥_𝑘. Since {𝑥_𝑘} is decreasing, so {𝑦_𝑘} is increasing. Next, 𝑥_𝑘→0 in measure, because, for each 𝜀>0, 𝜇𝑡∈𝑇∶𝑥_𝑘(𝑡)>𝜀≤𝜇^∞_𝑖=𝑘𝐴_𝑖≤^∞_𝑖=𝑘𝜇𝐴_𝑖⟶0as𝑘⟶∞.(3.5) In particular, by 𝐸∈(𝐹𝑃), ‖𝑦_𝑘‖→‖𝑦‖. Finally, ‖𝑦_𝑘−𝑦‖=‖𝑥_𝑘‖≥‖𝑢𝜒_𝐴𝑘‖≥𝛿. Thus, 𝐸∉(𝐻_𝑔)₊.
Assume that 𝐸 is symmetric. We prove that (vi) ⇒ (iv). If m_𝐸>0, then we find a set 𝐵 with 𝜒_𝐵∈𝐸 and a sequence {𝐴_𝑛}⊂𝐵, 𝜇(𝐴_𝑛)→0 such that inf_𝑛‖𝜒_𝐴𝑛‖_𝐸≥m_𝐸/2. Let 𝑥∈𝐸⧵{0}. We prove that 𝑥∈𝐸⧵𝐸_𝑎. There is a number 𝛿>0 such that the set 𝐷={𝑡∈𝑇∶|𝑥(𝑡)|≥𝛿} has positive measure. We find a sequence {𝐶_𝑘} in 𝐷 of pairwise disjoint sets and a subsequence {𝐴_𝑛𝑘} of {𝐴_𝑛} satisfying 𝜇(𝐶_𝑘)≤(1/2^𝑘)𝜇(𝐷) and 𝜇(𝐶_𝑘)≥𝜇(𝐴_𝑛𝑘). Then, by the symmetry of 𝐸, ‖‖𝑥𝜒_𝐶𝑘‖‖_𝐸≥‖‖𝛿𝜒_𝐶𝑘‖‖_𝐸≥‖‖𝛿𝜒_𝐴𝑛𝑘‖‖_𝐸≥𝛿m^𝐸2,(3.6) for any 𝑘∈ℕ, whence 𝑥∉𝐸_𝑎.
Now we will prove that the implication (vi) ⇒ (iv) is not true in general in nonsymmetric case. Take 𝐸=𝐿¹[0,1]⊕𝐿^∞[1,2]₁=𝑥∈𝐿⁰[0,2]∶𝑥=𝑢+𝑤,𝑢∈𝐿¹[0,1],𝑤∈𝐿^∞[1,2],‖𝑥‖=‖𝑢‖₁+‖𝑤‖_∞.(3.7) Obviously, supp𝐸_𝑎=[0,1] and m_𝐸=1.
Now we will see that implications (i) ⇒ (iii) ⇒ (iv) cannot be reversed in general. If 𝐸=Λ_𝜔—the Lorentz space over (0,∞) with Lebesgue measure and ∫^∞0𝜔<∞, then 𝐸∈(𝐻_𝑔) and 𝐸∉(𝑂𝐶)(see [1, 8]), whence (iii) ⇏ (i) in general.
Finally, let 𝐸=𝐿_𝜑[0,1]—the Orlicz space with 𝜑<∞ and 𝜑∉Δ₂(∞). By the proof of Theorem  1 from [5], we find an element 𝑢∈𝐸 and a sequence {𝐴_𝑛} with 𝜇(𝐴_𝑛)→0 and ‖𝑢𝜒_𝐴𝑛‖_𝜑=1. On the other hand, for each 𝐵∈Σ with 𝑢=𝜒_𝐵∈𝐸, we claim that 𝜒_𝐵∈𝐸_𝑎, whence m_𝐸=0. Indeed, for any sequence {𝐴_𝑛} satisfying 𝐴_𝑛↓∅ and for each 𝜆>0, we have 𝐼_𝜑𝜆𝑢𝜒_𝐴𝑛=𝜑(𝜆)¹⁰𝜒_𝐴𝑛𝑑𝜇=𝜑(𝜆)𝜇𝐴_𝑛⟶0,(3.8) whence ‖𝑢𝜒_𝐴𝑛‖_𝜑→0. Thus, the implication (iii) ⇒ (iv) cannot be reversed in general.

Recall that if 𝐸∈(𝐻_𝑙)₊, then 𝐸∈(𝑂𝐶) (see [10]) and, as we have seen above, property 𝐻_𝑔 needs not imply 𝑂𝐶 in general. However, from the above result, it follows that if 𝐸∈(𝐻_𝑔)₊, then m_𝐸=0 and consequently supp𝐸_𝑎=𝑇. Note also that the implication (vi) ⇒ (v) for a symmetric Köthe function space has been proved in [8, Proposition  2.2].

Remark 3.3. It may happen that 𝜇(𝐴)<∞ and 𝜒_𝐴∉𝐸 only in a nonsymmetric Köthe space. However, if 𝜇(𝐴)<∞ and 𝜒_𝐴∉𝐸, then there is an increasing sequence (𝐴_𝑛) of subsets of 𝐴 satisfying ⋃_𝑛𝐴_𝑛=𝐴 and 𝜒_𝐴𝑛∈𝐸 for each 𝑛.

Proof. Let 𝑥₀ be a weak unit of 𝐸, that is, 𝑥₀(𝑡)>0 for each 𝑡∈𝑇. It is enough to take 𝐴_𝑛={𝑡∈𝐴∶𝑥₀(𝑡)>1/𝑛} for each 𝑛.

Remark 3.4. Let 𝐸 be a Köthe space. If 𝐸∈(𝐻_𝑢)₊, then ‖𝑥𝜒_𝐵𝑘‖→0 for each 𝑥∈𝐸₊, where 𝐵_𝑘={𝑡∈𝑇∶𝑥(𝑡)<1/𝑘}.

Proof. Let 𝑥∈𝐸₊. Clearly, 𝑥𝜒_{𝑇⧵𝐵𝑘}↑𝑥 and sup_𝑘‖𝑥𝜒_{𝑇⧵𝐵𝑘}‖<∞. By 𝐸∈(𝐹𝑃), we get ‖𝑥𝜒_{𝑇⧵𝐵𝑘}‖→‖𝑥‖. Moreover, 𝑥−𝑥𝜒_{𝑇⧵𝐵𝑘}→0 uniformly. Then ‖𝑥𝜒_{𝑇⧵𝐵𝑘}−𝑥‖→0, because 𝐸∈(𝐻_𝑢)₊.
It is easy to see that the above is not true if 𝐸∉(𝐻_𝑢)₊. It is enough to take 𝐸=𝐿^∞𝜔(0,∞)=𝑥∈𝐿⁰∶‖𝑥‖=supess_𝑡≥0𝜔(𝑡)||𝑥(𝑡)||<∞,(3.9) where 𝜔(𝑡)=𝑡 and 𝑥(𝑡)=1/𝑡. Then ‖𝑥𝜒_𝐵𝑘‖=1 for each 𝑘. On the other hand, taking 𝑥_𝑛=𝑥𝜒_[0,𝑛], we get ‖𝑥‖=‖𝑥_𝑛‖=1, 𝑥_𝑛→𝑥 uniformly and ‖𝑥_𝑛−𝑥‖=1, whence 𝐸∉(𝐻_𝑢)₊.

It is known that 𝐸∈(𝐻_𝑙) if and only if 𝐸∈(𝐻_𝑙)₊ whenever 𝐸∈(𝑂𝐶) (see [6]). Since (𝐻_𝑙)₊⇒(𝑂𝐶) (see [10]), so 𝐸∈(𝐻_𝑙) if and only if 𝐸∈(𝐻_𝑙)₊. The same proof as in [6] shows that if 𝐸∈(𝑂𝐶), then 𝐸∈(𝐻_𝑔) if and only if 𝐸∈(𝐻_𝑔)₊. However, the assumption 𝐸∈(𝑂𝐶) is now too strong in this case because there are Köthe spaces satisfying 𝐸∈(𝐻_𝑔) and 𝐸∉(𝑂𝐶). Hence, the following result is more natural and precise.

Lemma 3.5. Suppose 𝐸 is a Köthe space. Then 𝐸∈(𝐻_𝑔) if and only if 𝐸∈(𝐻_𝑔)₊.

Proof. The necessity is obvious. We prove the sufficiency. Let 𝐸∈(𝐻_𝑔)₊. Assume that 𝐸 is a Köthe function space. Let ‖𝑥_𝑛‖→‖𝑥‖ and 𝑥_𝑛→𝑥 in measure. For each 𝜂>0, we get 𝜇𝑡∈𝑇∶||||𝑥_𝑛(𝑡)||−||𝑥(𝑡)||||≥𝜂≤𝜇𝑡∈𝑇∶||𝑥_𝑛(𝑡)−𝑥(𝑡)||≥𝜂⟶0,(3.10) whence |𝑥_𝑛|→|𝑥| in measure. By the assumption 𝐸∈(𝐻_𝑔)₊, we conclude that ‖‖||𝑥_𝑛||−|𝑥|‖‖⟶0.(3.11) Then we find a subsequence (|𝑥_𝑛𝑚|)_𝑚 of (|𝑥_𝑛|)_𝑛, an element 𝑦∈𝐸₊, and a sequence 0≤𝛽_𝑚→0 such that ||𝑥_𝑛𝑚|−|𝑥||≤𝛽_𝑚𝑦 for each 𝑚 (see [12]). Denote (|𝑥_𝑛𝑚|)_𝑚 still by (|𝑥_𝑛|)_𝑛 and (𝛽_𝑚) by (𝛽_𝑛). Thus, we may assume that ||𝑥_𝑛|−|𝑥||≤𝑦 for all 𝑛. Set 𝑢=|𝑥|+𝑦. Let 𝜀>0. Applying Remark 3.4, with respective sets 𝐵_𝑘=𝑡∶𝑢(𝑡)<1𝑘,(3.12) take 𝑘 big enough to satisfy ‖𝑢𝜒_𝐵𝑘‖<𝜀/8. Moreover, ‖|𝑥_𝑛|−|𝑥|‖<𝜀/8 for sufficiently large 𝑛. Setting 𝐶_𝑛=𝑡∈𝑇∶0≤sgn𝑥(𝑡)𝑥_𝑛(𝑡)≤1,𝐷_𝑛=𝑡∈𝑇∶sgn𝑥(𝑡)𝑥_𝑛(𝑡)=−1,(3.13) we get ‖‖𝑥−𝑥_𝑛‖‖≤≤5‖‖𝜀|𝑥|−||𝑥_𝑛||𝜒_𝐶𝑛‖‖+‖‖𝑥−𝑥_𝑛𝜒_{𝐷𝑛∩𝐵𝑘}‖‖+‖‖𝑥−𝑥_𝑛𝜒_{𝐷𝑛⧵𝐵𝑘}‖‖8+‖‖𝑥−𝑥^𝑛𝜒_{𝐷𝑛⧵𝐵𝑘}‖‖.(3.14) Take 0<𝛿<𝜀16𝑘min‖𝑦1‖,‖𝑥1‖.(3.15) Denote 𝐹𝐺^𝑘𝑛=𝑡∈𝐷_𝑛⧵𝐵_𝑘∶||𝑥_𝑛(𝑡)−𝑥(𝑡)||>𝛿_𝑘𝑛,=𝑡∈𝐷_𝑛⧵𝐵_𝑘∶||𝑥_𝑛(𝑡)−𝑥(𝑡)||≤𝛿.(3.16) Note that 𝑥→_𝑛𝜇𝑥 and consequently 𝜇(𝐹^𝑘𝑛)→0 as 𝑛→∞. Then, by Lemma 3.2 (iii), ‖‖𝑥−𝑥_𝑛𝜒_{𝐷𝑛⧵𝐵𝑘}‖‖≤≤‖‖‖‖2𝑢𝑥𝜒−_𝐹𝑘𝑛𝑥_𝑛𝜒_𝐹^𝑘𝑛‖‖+‖‖𝑥−𝑥_𝑛𝜒_𝐺^𝑘𝑛‖‖‖‖+‖‖𝑥−𝑥_𝑛𝜒_𝐺^𝑘𝑛‖‖<𝜀8+‖‖𝑥−𝑥^𝑛𝜒_𝐺^𝑘𝑛‖‖,(3.17) for sufficiently large 𝑛. Finally, divide set 𝐺^𝑘𝑛 into two subsets 𝐻^𝑘𝑛=𝑡∈𝐺^𝑘𝑛∶||𝑥(𝑡)||≥12𝑘,𝐼^𝑘𝑛=𝑡∈𝐺^𝑘𝑛∶𝑦(𝑡)≥12𝑘.(3.18) Clearly, 𝐻^𝑘𝑛∪𝐼^𝑘𝑛=𝐺^𝑘𝑛 because 𝐺^𝑘𝑛⊂𝑇⧵𝐵_𝑘=𝑡∶||𝑥(𝑡)||+𝑦(𝑡)≥1𝑘.(3.19) Then, by (3.15), ‖‖𝑥−𝑥_𝑛𝜒_𝐺^𝑘𝑛‖‖≤≤2𝛿𝑘𝛿‖‖𝜒‖‖|_𝐻𝑘𝑛𝑥‖‖|𝜒+‖‖_𝐻𝑘𝑛𝜒_𝐼𝑘𝑛‖‖≤𝛿‖‖2𝑘|𝑥|𝜒_𝐻𝑘𝑛‖‖+‖‖2𝑘𝑦𝜒_𝐼𝑘𝑛‖‖‖‖+‖‖𝑦𝜒_𝐼𝑘𝑛‖‖<𝜀4.(3.20) Combining (3.14), (3.17), and (3.20), we get ‖𝑥−𝑥_𝑛‖≤𝜀.
If 𝐸 is a Köthe sequence space, the proof is the same, because for a counting measure space (ℕ,2^ℕ,𝑚) for each 𝑢∈𝐸 and any sequence (𝐴_𝑛) in 2^ℕ with 𝑚(𝐴_𝑛)→0, we have ‖𝑢𝜒_𝐴𝑛‖→0.

The straightforward calculation shows that we may equivalently take 𝑥 and {𝑥_𝑛} in 𝑆(𝐸) in the definition of property 𝐻_𝑙 (see also the proof below). However, the analogous problem for property 𝐻_𝑔 is a little more complicated. The crucial point is described in the following lemma.

Lemma 3.6. Suppose that 𝐸 is a symmetric Köthe function space. If 𝑥_𝑛→𝑥 in measure and ‖𝑥_𝑛‖→‖𝑥‖≠0, then 𝑥_𝑛/‖𝑥_𝑛‖→𝑥/‖𝑥‖ in measure. Moreover, the above implication is not true in general if 𝐸 is a nonsymmetric Köthe function space.

Proof. Having our assumptions, for each 𝜀,𝜂>0, we get 𝜇𝑡∶||||𝑥^𝑛(𝑡)‖‖𝑥_𝑛‖‖−‖𝑥𝑥(‖𝑡)||||>𝜀=𝜇𝑡∶||||𝑥^𝑛(𝑡)≤𝜇𝑡∶‖‖𝑥1_𝑛‖‖−‖‖𝑥𝑥(_𝑛𝑡‖‖)+‖‖𝑥𝑥(_𝑛𝑡‖‖)−‖𝑥𝑥(‖𝑡)||||>𝜀‖‖𝑥_𝑛‖‖||𝑥^𝑛(𝑡)−𝑥(𝑡)||+||||1≤𝜇𝑡∶1‖‖𝑥_𝑛‖‖−‖𝑥1‖||||||𝑥(𝑡)||>𝜀‖‖𝑥_𝑛‖‖||𝑥^𝑛(𝑡)−𝑥(𝑡)||>𝜀2+𝜇𝑡∶||||‖‖𝑥1_𝑛‖‖−‖𝑥1‖||||||𝑥(𝑡)||>2𝜀(3.2.1) Since ‖𝑥_𝑛‖→‖𝑥‖, so ‖𝑥_𝑛‖>(1/2)‖𝑥‖ for 𝑛≥𝑁₀, whence 𝜇𝑡∶1‖‖𝑥_𝑛‖‖||𝑥^𝑛(𝑡)−𝑥(𝑡)||>𝜀2≤𝜇𝑡∶||𝑥^𝑛(𝑡)−𝑥(𝑡)||>𝜀‖𝑥‖4<𝜂2,(3.22) for 𝑛≥𝑁₁>𝑁₀. Moreover, applying the symmetry of 𝐸, we get 𝜇𝑡∶||||1‖‖𝑥_𝑛‖‖−‖𝑥1‖||||||𝑥(𝑡)||>2𝜀<2𝜂for𝑛≥𝑁².(3.23) Thus, 𝜇𝑡∶||||𝑥^𝑛(𝑡)‖‖𝑥_𝑛‖‖−‖𝑥𝑥(‖𝑡)||||>𝜀<𝜂,(3.24) for 𝑛≥max{𝑁₁,𝑁₂}.
Considering the second assertion, take 𝐸=𝐿^11/𝑡₃(1,∞)=𝑥∈𝐿⁰((1,∞),Σ,𝑚)∶‖𝑥‖=^∞1||𝑥(𝑡)||1𝑡³𝑑𝑡<∞.(3.25) Set 𝑥(𝑡)=𝑡 and 𝑥_𝑛(𝑡)=𝑡+(1/𝑛), then ‖𝑥‖=1, ‖𝑥_𝑛‖=1+(1/2𝑛), and 𝑥_𝑛→𝑥 in measure. However, 𝜇𝑡∶||||𝑥^𝑛(𝑡)‖‖𝑥_𝑛‖‖−‖𝑥𝑥(‖𝑡)||||>1=𝜇𝑡∶|||2𝑡𝑛−+21|||>1=∞,(3.26) for each 𝑛, whence (𝑥_𝑛/‖𝑥_𝑛‖)↛(𝑥/‖𝑥‖) in measure.

Corollary 3.7. Assume that 𝐸 is a symmetric Köthe function space. Then 𝐸∈(𝐻_𝑔) if and only if ‖𝑥_𝑛−𝑥‖→0 for any 𝑥∈𝑆(𝐸) and (𝑥_𝑛) in 𝑆(𝐸) satisfying 𝑥_𝑛→𝑥 in measure.

Proof. The necessity is obvious. We prove the sufficiency. Let 𝑥_𝑛→𝑥 in measure and ‖𝑥_𝑛‖→‖𝑥‖≠0. Taking 𝑦_𝑛=𝑥_𝑛/‖𝑥_𝑛‖ and 𝑦=𝑥/‖𝑥‖, we get 𝑦_𝑛→𝑦 in measure, by Lemma 3.6. By the assumption, we conclude that ‖𝑦_𝑛−𝑦‖→0. Notice that ||||1‖‖𝑥_𝑛‖‖‖‖𝑥^𝑛−𝑥‖‖−‖𝑥‖||||1≤‖‖‖𝑥‖‖𝑥_𝑛‖‖−‖𝑥1‖_𝑛||||||||‖‖𝑥_𝑛‖‖−‖‖𝑥𝑥_𝑛‖‖−−‖‖𝑥𝑥_𝑛‖‖+‖𝑥𝑥‖‖‖‖=‖‖‖‖‖𝑥𝑥^𝑛_𝑛‖‖−‖𝑥𝑥‖‖‖‖⟶0.(3.27) Since ‖𝑥_𝑛‖→‖𝑥‖≠0, so ‖𝑥_𝑛−𝑥‖→0 as desired.

We will need in the sequel the following.

Lemma 3.8. If 𝜑∈Δ₂(∞), then for any 𝑎>0, there are 𝜎∈(0,1] and 𝑢₁>𝑎_𝜑+𝑎 such that inf_{𝑢0∈(0,𝑎)}inf_𝑢≥𝑢1𝜑𝑢−𝑢₀𝜑(𝑢)−𝜑𝑢₀≥𝜎.(3.28)

Proof. Suppose conversely that there is 𝑎>0 such that for any 𝜎∈(0,1] and each 𝑢₁>𝑎_𝜑+𝑎, there holds inf_{𝑢0∈(0,𝑎)}inf_𝑢≥𝑢1𝜑𝑢−𝑢₀𝜑(𝑢)−𝜑𝑢₀<𝜎.(3.29) Taking 𝜎_𝑛=1/2^𝑛 and 𝑢_𝑛→∞, we get inf_{𝑢0∈(0,𝑎)}inf_{𝑢≥𝑢𝑛}𝜑(𝑢−𝑢₀)/(𝜑(𝑢)−𝜑(𝑢₀))<1/2^𝑛. We may assume that 𝑢_𝑛>2𝑎 for each 𝑛. Hence, for each 𝑛, one may choose 𝑣_𝑛≥𝑢_𝑛 and 𝑢₀∈(0,𝑎) with 𝜑(𝑣_𝑛−𝑢₀)/(𝜑(𝑣_𝑛)−𝜑(𝑢₀))≤1/2^𝑛. Consequently, we get 2^𝑛𝜑12𝑣^𝑛<2^𝑛𝜑𝑣_𝑛−𝑢₀<𝜑𝑣_𝑛−𝜑𝑢₀≤𝜑𝑣_𝑛,(3.30) for each 𝑛. Taking 𝑏_𝑛=(1/2)𝑣_𝑛, one has 2^𝑛𝜑(𝑏_𝑛)<𝜑(2𝑏_𝑛), whence 𝜑∉Δ₂(∞).

Lemma 3.9. (i) Suppose that 𝐸 is a symmetric Köthe space. If 𝑥_𝑛→𝑥 in measure and 𝜑∘𝑥_𝑛, 𝜑∘𝑥∈𝐸 are finitely valued a.e., then 𝜑∘𝑥_𝑛→𝜑∘𝑥 in measure. Furthermore, for a nonsymmetric Köthe space, the above implication is not true in general.
(ii) If 𝑥_𝑛→𝑥 in measure, 𝑥_𝑛,𝑥∈𝐸₊ and 𝜑^𝑟−1∘𝑥_𝑛, 𝜑^𝑟−1∘𝑥 are well-defined functions, then 𝜑^𝑟−1∘𝑥_𝑛→𝜑^𝑟−1∘𝑥 in measure.

Proof. (i) Let 𝑥−→_𝑛𝜇𝑥.
(A) Suppose that 𝑏_𝜑=∞. Set 𝐴_𝑘=𝑡∈𝑇∶||𝑥(𝑡)||>𝑘.(3.31) Then, by symmetry of 𝐸,  𝜇(𝐴_𝑘)→0. Let 𝜂>0 and 𝜀>0. Take 𝑘₀ such that 𝜇(𝐴_𝑘0)<𝜂/3. We have 𝜇≤𝑡∈𝜇𝑇𝑡∶∈||𝜑𝑇⧵𝑥𝐴_𝑛(𝑡_𝑘0)−𝜑(𝑥(𝑡))||≥𝜀∶||𝜑𝑥_𝑛(𝑡)−𝜑(𝑥(𝑡))||≥𝜀+𝜂3.(3.32) Since 𝜑 is uniformly continuous on the interval [−𝑘₀,𝑘₀], so for each 𝛾>0 there is 𝛿=𝛿(𝛾,𝑘₀)>0 such that the implication |𝑢−𝑠|<𝛿⇒|𝜑(𝑢)−𝜑(𝑠)|<𝛾 is true for all 𝑢,𝑠∈[−𝑘₀,𝑘₀]. Set 𝛿=𝛿(𝜀,2𝑘₀) and 𝐴^1𝑛=𝑡∈𝑇⧵𝐴_𝑘0∶||𝑥_𝑛(𝑡)||>2𝑘₀,𝐴^2𝑛=𝑡∈𝑇⧵𝐴_𝑘0∶||𝑥_𝑛(𝑡)||≤2𝑘₀.(3.33) Consequently 𝜇≤𝑡∈𝜇𝑇𝐴∶_1𝑛||𝜑𝑥_𝑛(𝑡)−𝜑(𝑥(𝑡))||≥𝜀≤𝜇𝑡∈𝐴+_2𝑛𝜇𝑡∈𝐴^2𝑛∶||𝜑𝑥_𝑛(𝑡)−𝜑(𝑥(𝑡))||≥𝜀+3𝜂∶||𝑥_𝑛(𝑡)−𝑥(𝑡)||≥𝛿+2𝜂3≤𝜂(3.34) for sufficiently large 𝑛, because 𝐴^1𝑛={𝑡∈𝑇∶|𝑥_𝑛(𝑡)−𝑥(𝑡)|≥𝑘₀}. Thus 𝜑∘𝑥→_𝑛𝜇𝜑∘𝑥.
(B) Assume 𝜑(𝑏_𝜑)<∞. The proof is simpler than above because 𝑥_𝑛(𝑡)≤𝑏_𝜑 and 𝑥(𝑡)≤𝑏_𝜑 for 𝜇-a.e. 𝑡∈𝑇 and 𝜑 is uniformly continuous on the interval [−𝑏_𝜑,𝑏_𝜑].
(C) Suppose that 𝑏_𝜑<∞, 𝜑(𝑏_𝜑)=∞. Then 𝑥_𝑛(𝑡)<𝑏_𝜑 and 𝑥(𝑡)<𝑏_𝜑 for 𝜇-a.e. 𝑡∈𝑇. Set 𝐴_𝑘=𝑡∈𝑇∶||𝑥(𝑡)||>𝑏_𝜑−1𝑘.(3.35) Then 𝜇(𝐴_𝑘)→0, because 𝐸 is symmetric. Let 𝜂>0 and 𝜀>0. Take 𝑘₀ such that 𝜇(𝐴_𝑘0)<𝜂/3. We have 𝜇≤𝑡∈𝜇𝑇𝑡∶∈||𝜑𝑇⧵𝑥𝐴_𝑛(𝑡_𝑘0)−𝜑(𝑥(𝑡))||≥𝜀∶||𝜑𝑥_𝑛(𝑡)−𝜑(𝑥(𝑡))||≥𝜀+𝜇𝐴_𝑘0.(3.36) Since 𝜑 is uniformly continuous on the interval [−𝑢₀,𝑢₀] for each 𝑢₀<𝑏_𝜑, so for each 𝑢₀<𝑏_𝜑 and 𝛾>0 there is 𝛿=𝛿(𝛾,𝑢₀)>0 such that the implication |𝑢−𝑠|<𝛿⇒|𝜑(𝑢)−𝜑(𝑠)|<𝛾 is true for all 𝑢,𝑠∈[−𝑢₀,𝑢₀]. Put 𝛿=𝛿(𝜀,𝑏_𝜑−1/2𝑘₀) and 𝐴^1𝑛=𝑡∈𝑇⧵𝐴_𝑘0∶||𝑥_𝑛(𝑡)||>𝑏_𝜑−1𝐴_2𝑛2𝑘₀,=𝑡∈𝑇⧵𝐴_𝑘0∶||𝑥_𝑛(𝑡)||≤𝑏_𝜑−12𝑘₀.(3.37) Consequently 𝜇≤𝑡∈𝜇𝑇𝐴∶_1𝑛||𝜑𝑥_𝑛(𝑡)−𝜑(𝑥(𝑡))||≥𝜀≤𝜇𝑡∈𝐴+_2𝑛𝜇𝑡∈𝐴^2𝑛∶||𝜑𝑥_𝑛(𝑡)−𝜑(𝑥(𝑡))||≥𝜀+3𝜂∶||𝑥_𝑛(𝑡)−𝑥(𝑡)||≥𝛿+2𝜂3≤𝜂(3.38) for sufficiently large 𝑛, because 𝐴^1𝑛={𝑡∈𝑇∶|𝑥_𝑛(𝑡)−𝑥(𝑡)|≥1/2𝑘₀}. Thus 𝜑∘𝑥→_𝑛𝜇𝜑∘𝑥.
Now we will show that in nonsymmetric case the above implication is not true in general. Consider the Lebesgue measure space ((0,∞),Σ,𝑚). Take 𝜑(𝑢)=𝑢²,𝑥=_𝑛𝑛𝜒_{[𝑛,𝑛+1]},𝑥_𝑘=_𝑛𝑛+1𝑘𝜒^{[𝑛,𝑛+1]}.(3.39) Then 𝑥_𝑘→𝑥 in measure. On the other hand, |𝜑∘𝑥_𝑘−𝜑∘𝑥|=∑_𝑛((1/𝑘²)+(2𝑛/𝑘))𝜒_{[𝑛,𝑛+1]}, whence 𝜑∘𝑥_𝑘↛𝜑∘𝑥 in measure. Clearly, such element 𝜑∘𝑥 can be well defined only in some nonsymmetric Köthe space 𝐸.
(ii) First, we claim that for each 𝜀>0, there is 𝛿>0 such that the implication ||𝜑^𝑟−1(𝑢)−𝜑^𝑟−1(𝑣)||>𝜀⟹|𝑢−𝑣|>𝛿(3.40) is true for all 𝑢,𝑣≥0. First, notice that for each 𝜀>0, there is 𝛿>0 such that the implication ||𝜑^𝑟−1(𝑢)−𝜑^𝑟−1(0)||>𝜀⟹|𝑢|>𝛿(3.41) is true for all 𝑢≥0, because 𝜑^𝑟−1 is continuous at zero. Next, defining 𝑔(𝑢)=𝜑^𝑟−1(𝑢)−𝜑^𝑟−1(0), we see that 𝑔 is subadditive on ℝ₊, because 𝑔=𝜓⁻¹, where 𝜓 is an Orlicz function with 𝜓>0. Consequently, 𝜑^𝑟−1(𝑢−𝑣)−𝜑^𝑟−1(0)=𝑔(𝑢−𝑣)≥𝑔(𝑢)−𝑔(𝑣)=𝜑^𝑟−1(𝑢)−𝜑^𝑟−1(𝑣).(3.42) Hence, if |𝜑^𝑟−1(𝑢)−𝜑^𝑟−1(𝑣)|>𝜀, then |𝜑^𝑟−1(𝑢−𝑣)−𝜑^𝑟−1(0)|>𝜀, whence |𝑢−𝑣|>𝛿, by (3.41). Thus, the claim is proved. Assume that 𝑥_𝑛, 𝑥∈𝐸₊ and 𝜑^𝑟−1∘𝑥_𝑛, 𝜑^𝑟−1∘𝑥 are well-defined functions and 𝑥_𝑛→𝑥 in measure, then, applying (3.40), for each 𝜀>0, we get 𝜇𝑡∈𝑇∶||𝜑^𝑟−1∘𝑥_𝑛−𝜑^𝑟−1∘𝑥(𝑡)||>𝜀≤𝜇𝑡∈𝑇∶||𝑥_𝑛−𝑥(𝑡)||>𝛿⟶0.(3.43)

3.2. Calderón-Lozanovskiĭ Function Spaces

We still assume below that 𝐸 is a Köthe function space.

It is known that 𝐸_𝜑∈(𝑂𝐶) if and only if 𝐸∈(𝑂𝐶) and 𝜑∈Δ^𝐸2 (see Corollary  12 from [9]). Considering some weaker property than 𝑂𝐶, we get the following.

Lemma 3.10. Suppose that 𝑏_𝜑=∞, then 𝐸_𝑎≠{0} if and only if (𝐸_𝜑)_𝑎≠{0}.

Proof. Necessity
Let 𝑧∈𝐸_𝑎⧵{0}. We find a number 𝑚>0 such that 𝜇(𝐴)=𝜇{𝑡∈𝑇∶|𝑧(𝑡)|>1/𝑚}>0. Taking 𝑤=𝜑⁻¹(1)𝜒_𝐴, we get 𝜑∘𝑤∈𝐸_𝑎. Moreover, since 𝑏_𝜑=∞, 𝜑 satisfies the local Δ^𝐸2 condition with respect to the element 𝑤 (see [9]). Consequently, 𝑤∈(𝐸_𝜑)_𝑎, by Theorem  11 from [9].
Sufficiency
We apply Theorem  11 from [9]. Let 𝑥∈(𝐸_𝜑)_𝑎⧵{0}, then 𝜑∘𝑥∈𝐸_𝑎. If 𝜇{𝑡∶|𝑥(𝑡)|>𝑎_𝜑}>0, then 𝜇(𝐴_𝑚)>0 for some 𝑚∈ℕ, where 𝐴_𝑚={𝑡∈𝑇∶|𝑥(𝑡)|≥(1+(1/𝑚))𝑎_𝜑}. Thus, 𝜒_𝐴𝑚∈𝐸_𝑎, and we are done, if 𝜇{𝑡∶|𝑥(𝑡)|≤𝑎_𝜑}=𝜇(𝐴)>0 then 𝜒_𝐴∈𝐸_𝑎⧵{0}, by Theorem  11 from [9].

Theorem 3.11. 𝐸_𝜑∈(𝐻_𝑙) if and only if(a)𝐸∈(𝐻_𝑙), (b)𝜑∈Δ₂(∞) whenever 𝐿^∞↪𝐸 and 𝜑∈Δ₂(ℝ₊) provided 𝐿↪^∞╱𝐸.

Proof. Since the sufficiency has been proved in [4, Theorem  11] (we need also to apply [7, Lemma  2]), we need only to prove the necessity. Since 𝐸_𝜑∈(𝐻_𝑙), so 𝐸_𝜑∈(𝑂𝐶) (see [10, Proposition  2.1]), and consequently, 𝐸∈(𝑂𝐶), 𝜑<∞, and 𝜑∈Δ^𝐸2 (see [9, Corollary  12]). Notice that 𝐸∈(𝑂𝐶) implies that 𝐸↪^╱𝐿^∞. Indeed, if 𝐸↪𝐿^∞, then there is a decreasing sequence (𝐴_𝑛) in Σ with 0<𝜇(𝐴_𝑛)→0 and 𝜒_𝐴𝑛∈𝐸, whence 1=‖𝜒_𝐴𝑛‖_𝐿∞≤𝑀‖𝜒_𝐴𝑛‖_𝐸, so 𝐸∉(𝑂𝐶). Thus, (b) is proved. Suppose that 𝐸∉(𝐻_𝑙). Recall that property 𝐻_𝑙 can be equivalently considered only on the positive cone 𝐸₊ [6, Proposition  1]. Consequently, we find 𝑥 and {𝑥_𝑛} in (𝑆(𝐸))₊ with 𝑥→_𝑛𝜇𝑥 locally in measure and ‖𝑥_𝑛−𝑥‖_𝐸≥𝜀. Set 𝑦_𝑛=𝜑^𝑟−1∘𝑥_𝑛 and 𝑦=𝜑^𝑟−1∘𝑥, then 𝐼_𝜑(𝑦)=𝐼_𝜑(𝑦_𝑛)=1, whence 𝑦,𝑦_𝑛∈𝑆(𝐸_𝜑). Moreover, 𝑦→_𝑛𝜇𝑦 locally in measure, because, by 𝜎 finiteness of the measure space (𝑇,Σ,𝜇), the local convergence in measure can be equivalently replaced by convergence 𝜇-a.e. (also Lemma 3.9(ii) can be applied). It is enough to prove that there is 𝜂>0 with ‖‖𝑦_𝑛−𝑦‖‖_𝜑≥𝜂,(3.44) for infinitely many 𝑛. The similar condition has been proved in [8, Theorem  2.11], but under additional assumption that 𝜑>0. Here, the lack of this assumption requires new techniques in comparison with the respective proof in [8, Theorem  2.11]. We have ‖‖𝜑∘𝑦_𝑛−𝜑∘𝑦‖‖_𝐸≥𝜀.(3.45) Denote 𝐴_𝑛=𝑡∈𝑇∶𝜑𝑦_𝑛(𝑡)≥𝜑(𝑦(𝑡)),𝐵_𝑛=𝑇⧵𝐴_𝑛.(3.46) By (3.45), we get max‖‖𝜑∘𝑦_𝑛−𝜑∘𝑦𝜒_𝐴𝑛‖‖_𝐸,‖‖𝜑∘𝑦_𝑛−𝜑∘𝑦𝜒_𝐵𝑛‖‖_𝐸≥𝜀2,(3.47) for each 𝑛. We assume that ‖(𝜑∘𝑦_𝑛−𝜑∘𝑦)𝜒_𝐴𝑛‖_𝐸≥𝜀/2 for infinitely many 𝑛∈ℕ, because otherwise the proof is analogous. We divide the proof into two cases.(I)Suppose that 𝐿^∞↪𝐸 and 𝜑∈Δ₂(∞). Let 𝑐=𝜑^𝑟−1(𝜀/32‖𝜒_𝑇‖_𝐸)>𝑎_𝜑. Denoting𝐴^1𝑛=𝑡∈𝐴_𝑛∶𝑦_𝑛(𝑡)<𝑐,𝐴^2𝑛=𝑡∈𝐴_𝑛∶𝑦_𝑛(𝑡)≥𝑐,(3.48) we get 𝜀2≤‖‖𝜑∘𝑦^𝑛−𝜑∘𝑦𝜒_𝐴𝑛‖‖_𝐸≤‖‖𝜑∘𝑦_𝑛−𝜑∘𝑦𝜒_𝐴^2𝑛‖‖_𝐸+𝜀4.(3.49) Thus, ‖‖𝜑∘𝑦_𝑛−𝜑∘𝑦𝜒_𝐴^2𝑛‖‖_𝐸≥𝜀4.(3.50) Set 𝑐₁=𝑐+𝑎^𝜑2,𝐵^1𝑛=𝑡∈𝐴^2𝑛∶𝑦(𝑡)<𝑐₁,𝐵^2𝑛=𝑡∈𝐴^2𝑛∶𝑦(𝑡)≥𝑐₁.(3.51) The remaining proof of the case (I) we divide into two parts.(1)Suppose that ‖(𝜑∘𝑦_𝑛−𝜑∘𝑦)𝜒_𝐵^1𝑛‖_𝐸≥𝜀/8. By Lemma 3.8, for 𝑎=𝑐₁, there are 𝜎>0 and 𝑢₁>0 such that inf_{𝑢0∈0,𝑐1}inf_𝑢≥𝑢1𝜑𝑢−𝑢₀𝜑(𝑢)−𝜑𝑢₀≥𝜎.(3.52) Set 𝐶^1𝑛=𝑡∈𝐵^1𝑛∶𝑦_𝑛(𝑡)<𝑢₁,𝐶^2𝑛=𝑡∈𝐵^1𝑛∶𝑦_𝑛(𝑡)≥𝑢₁.(3.53)(a)Assume that ‖(𝜑∘𝑦_𝑛−𝜑∘𝑦)𝜒_𝐶^1𝑛‖_𝐸≥𝜀/16. Taking 0<𝛼<(𝑐−𝑐₁)/𝑢₁, we get ‖‖‖𝜑∘𝑦^𝑛−𝑦𝛼𝜒^𝐶1𝑛‖‖‖_𝐸≥‖‖‖𝜑𝑐−𝑐¹𝛼𝜒^𝐶1𝑛‖‖‖_𝐸≥‖‖𝜑𝑢₁𝜒_𝐶^1𝑛‖‖_𝐸≥‖‖𝜑∘𝑦_𝑛−𝜑∘𝑦𝜒_𝐶^1𝑛‖‖_𝐸≥𝜀16.(3.54) Thus, ‖𝑦_𝑛−𝑦‖_𝜑≥𝜂₁=min{1,𝛼𝜀/16}.(b)Let ‖(𝜑∘𝑦_𝑛−𝜑∘𝑦)𝜒_𝐶^2𝑛‖_𝐸≥𝜀/16. Thus, by Lemma 3.8, ‖‖𝜑∘𝑦_𝑛−𝑦𝜒_𝐶^2𝑛‖‖_𝐸≥𝜎2‖‖𝜑∘𝑦^𝑛−𝜑∘𝑦𝜒_𝐶^2𝑛‖‖_𝐸≥𝜎𝜀32,(3.55) and consequently ‖𝑦_𝑛−𝑦‖_𝜑≥𝜂₂=min{1,𝜎𝜀/32} (cf. [8, Lemma  1.1]).(2) Assume that ‖(𝜑∘𝑦_𝑛−𝜑∘𝑦)𝜒_𝐵^1𝑛‖_𝐸<𝜀/8, then, by (3.50), ‖‖𝜑∘𝑦_𝑛−𝜑∘𝑦𝜒_𝐵^2𝑛‖‖_𝐸≥𝜀8.(3.56) Since 𝜑∈Δ₂(∞), for 𝑙=1+𝜀/32 and 𝑢₁=𝑐₁, there is 𝑎=𝑎(𝑙,𝑢₁)∈(0,1) such that 𝜑((1+𝑎)𝑢)≤𝑙𝜑(𝑢),(3.57) for every 𝑢≥𝑐₁ (see [20, Theorem  1.13(4)]). Moreover, we can choose 𝑎>0 satisfying 𝑎1+𝑎𝜑𝑐¹‖‖𝜒_𝑇‖‖_𝐸<𝜀32.(3.58) Let 𝐵^𝑛21=𝑡∈𝐵^2𝑛∶𝑦_𝑛−𝑦(𝑡)<𝑎𝑐¹𝐵1+𝑎_𝑛22,=𝑡∈𝐵^2𝑛∶𝑦_𝑛−𝑦(𝑡)≥𝑎𝑐¹1+𝑎,(3.59) then, by (3.57) and (3.58), using convexity of 𝜑, we get ||𝜑∘𝑦_𝑛−𝜑∘𝑦||𝜒_𝐵^𝑛21=𝜑∘𝑦_𝑛−𝑦+𝑦−𝜑∘𝑦𝜒_𝐵^𝑛21≤𝑎1+𝑎𝜑∘1+𝑎𝑎𝑦^𝑛−𝑦+11+𝑎𝜑∘((1+𝑎)𝑦)−𝜑∘𝑦𝜒^𝐵𝑛21≤𝑎1+𝑎𝜑∘1+𝑎𝑎𝑦^𝑛−𝑦+11+𝑎𝑙𝜑∘𝑦−𝜑∘𝑦𝜒^𝐵𝑛21=𝑎1+𝑎𝜑∘1+𝑎𝑎𝑦^𝑛−𝑦+𝜀/32−𝑎1+𝑎𝜑∘𝑦𝜒^𝐵𝑛21.(3.60) Note that 𝑓(𝑎)=(𝜀/32−𝑎)/(1+𝑎) is a decreasing function of 𝑎>0. Moreover, 𝑎≤𝜀/32 by convexity of 𝜑 and inequality (3.57). Hence, by (3.58), ‖(𝜑∘𝑦_𝑛−𝜑∘𝑦)𝜒_𝐵^𝑛21‖_𝐸<𝜀/16. Then, by (3.56), ‖‖𝜑∘𝑦_𝑛−𝜑∘𝑦𝜒_𝐵^𝑛22‖‖_𝐸≥𝜀16.(3.61) Since 𝜑∈Δ₂(∞), for 𝑢₃=𝑐₁ and 𝛽=(1+𝑎)/𝑎, there is 𝑘₂>0 such that 𝜑(𝛽𝑢)≤𝑘₂𝜑(𝑢) for each 𝑢≥𝑢₃ (see [20]). Taking 0<𝛾<𝑎/(1+𝑎) and applying (3.57), we get ||𝜑∘𝑦_𝑛−𝜑∘𝑦||𝜒_𝐵^𝑛22=𝜑∘𝑦_𝑛−𝑦+𝑦−𝜑∘𝑦𝜒_𝐵^𝑛22≤𝜑∘𝑦^𝑛−𝑦𝛾+𝑦−𝜑∘𝑦𝜒^𝐵𝑛22≤𝑎1+𝑎𝜑∘1+𝑎𝑎𝑦^𝑛−𝑦𝛾+1+1𝑎𝜑∘((1+𝑎)𝑦)−𝜑∘𝑦𝜒^𝐵𝑛22≤𝑎1+𝑎𝑘²𝜑∘𝑦^𝑛−𝑦𝛾+1+1𝑎𝑙𝜑∘𝑦−𝜑∘𝑦𝜒^𝐵𝑛22=𝑎1+𝑎𝑘²𝜑∘𝑦^𝑛−𝑦𝛾+𝜀/132+−𝑎𝑎𝜑∘𝑦𝜒^𝐵𝑛22.(3.62) Then ‖𝜑∘((𝑦_𝑛−𝑦)/𝛾)‖_𝐸≥𝜀(1+𝑎)/32𝑎𝑘₂, and consequently ‖𝑦_𝑛−𝑦‖_𝜑≥𝜂₃=min{1,𝛾𝜀(1+𝑎)/32𝑎𝑘₂} (see [8, Lemma  1.1]).
Combining cases (1) and (2), we get (3.44) with 𝜂=min{𝜂₁,𝜂₂,𝜂₃}.
(II) Suppose that 𝐿↪^∞╱𝐸. Since 𝜑∈Δ₂(ℝ₊), so for every 𝑙>1, there is 𝑎=𝑎(𝑙)∈(0,1) such that 𝜑((1+𝑎)𝑢)≤𝑙𝜑(𝑢) for every 𝑢≥0 (see [20, Theorem  1.13(4)]). Moreover, for every 𝛽>0, there is 𝑘>0 such that 𝜑(𝛽𝑢)≤𝑘𝜑(𝑢) for each 𝑢≥0. Then the proof is analogous as in case (I) (it is simpler and shorter).

Now we will consider the Kadec-Klee property with respect to global convergence in measure. Theorem 3.19 below presents the full characterization of property 𝐻_𝑔. We divide the whole proof into several lemmas for readers convenience.

Lemma 3.12. If 𝐸_𝜑∈(𝐻_𝑔)₊, then m_𝐸=0.

Proof. If m_𝐸>0, then there are 𝛿>0, a set 𝐵,𝜇(𝐵)>0,𝜒_𝐵∈𝐸, and a sequence (𝐴_𝑛)⊂𝐵 with 𝜇(𝐴_𝑛)→0 and inf_𝑛‖𝜒_𝐴𝑛‖_𝐸≥𝛿. Without loss of generality, we may assume that ∑^∞𝑖=1𝜇(𝐴_𝑖)<∞. Set 𝑧_𝑛=sup𝜒_𝐴𝑖∶𝑖≥𝑛∈𝐿⁰.(3.63) Let 𝐴=⋃_𝑖𝐴_𝑖. Clearly, 𝑧_𝑛≤𝜒_𝐴≤𝜒_𝐵∈𝐸. Put 𝑦=𝜒_𝐴 and 𝑦_𝑛=𝑦−𝑧_𝑛. Choose 𝑎>0 such that 𝑎∈𝑎𝑎_𝜑=𝑏_𝜑if𝜑𝑏_𝜑‖‖𝜒_𝐴‖‖_𝐸<1,𝑎∈𝑎_𝜑,𝑏_𝜑if𝜑𝑏_𝜑<∞,𝜑𝑏_𝜑‖‖𝜒_𝐴‖‖_𝐸≥1,,𝑏_𝜑if𝜑𝑏_𝜑=∞,(3.64) and 𝜑(𝑎)‖𝜒_𝐴‖_𝐸=1 in the last two cases. Take 𝑥=𝑎𝑦 and 𝑥_𝑛=𝑎𝑦_𝑛, then ‖𝑥_𝑛‖_𝜑≤‖𝑥‖_𝜑=1. It is obvious that ‖𝑥_𝑛‖_𝜑=1 when 𝑎=𝑏_𝜑. We show that ‖𝑥_𝑛‖_𝜑→1 in the case 𝑎<𝑏_𝜑. Note that 𝑧_𝑛→0 in measure (see the proof of Lemma 3.2), 𝑧_𝑛↓, and consequently ‖𝑦_𝑛‖_𝐸→‖𝑦‖_𝐸, because 𝐸∈(𝐹𝑃). If ‖𝑥_𝑛‖_𝜑≤1−𝛼 for some 𝛼>0 and infinitely many 𝑛, then we get 1=𝜑(𝑎)‖‖𝜒_𝐴‖‖_𝐸<𝜑𝑎1−𝛼‖‖𝜒^𝐴‖‖_𝐸⟵𝜑𝑎1−𝛼‖‖𝑦^𝑛‖‖_𝐸=𝐼_𝜑𝑥^𝑛1−𝛼≤1,(3.65) which is a contradiction. Moreover, 𝑥_𝑛→𝑥 in measure. Since ‖‖𝑥−𝑥_𝑛‖‖_𝜑≥𝐼_𝜑𝑥−𝑥_𝑛≥𝜑(𝑎)‖‖𝜒_𝐴𝑛‖‖_𝐸≥𝜑(𝑎)𝛿,(3.66) we conclude that 𝐸_𝜑∉(𝐻_𝑔)₊.

Remark 3.13. If 𝐸_𝜑∈(𝐻_𝑔)₊, then 𝐸↪^╱𝐿^∞.

Proof. It is immediate, since if 𝐸↪𝐿^∞, then m_𝐸>0. By Lemma 3.12, we get 𝐸_𝜑∉(𝐻_𝑔)₊.

From Lemmas 3.12 and 3.2, we get immediately the following.

Corollary 3.14. If 𝐸_𝜑∈(𝐻_𝑔)₊, then supp𝐸_𝑎=𝑇.

Lemma 3.15. If 𝐸_𝜑∈(𝐻_𝑔), then 𝜑(𝑏_𝜑)=∞.

Proof. Suppose that 𝐸_𝜑∈(𝐻_𝑔) and 𝜑(𝑏_𝜑)<∞. By Lemma 3.12, we get m_𝐸=0. Let 𝐴∈Σ, 𝜇(𝐴)>0, and 𝜇(𝑇⧵𝐴)>0 be such that 𝜒_𝐴∈𝐸 and 𝜑(𝑏_𝜑)‖𝜒_𝐴‖_𝐸≤1/2. Let (𝐴_𝑛)⊂𝑇⧵𝐴 be such that 0<𝜇(𝐴_𝑛)→0 and 𝜑(𝑏_𝜑)‖𝜒_𝐴𝑛‖_𝐸≤1/2. Taking 𝑥=𝑏_𝜑𝜒_𝐴 and 𝑥_𝑛=𝑏_𝜑𝜒_{𝐴∪𝐴𝑛}, we get ‖𝑥‖_𝜑=‖𝑥_𝑛‖_𝜑=1, 𝑥_𝑛→𝑥 in measure and ‖𝑥_𝑛−𝑥‖_𝜑=1, whence 𝐸_𝜑∉(𝐻_𝑔).

Lemma 3.16. Suppose that (𝑏_𝜑<∞ and 𝜑(𝑏_𝜑)=∞) or (𝑏_𝜑=∞ and 𝜑∉Δ₂(∞)). If 𝐸_𝑎≠{0}, then 𝐸_𝜑∉(𝐻_𝑔).

Proof. It is enough to apply order isometry 𝑃∶𝑙^∞→𝐸_𝜑 constructed in the proof of [5, Theorem  1] and take 𝑧=𝜒_ℕ and 𝑧_𝑛=𝑧−𝑒_𝑛, where 𝑒_𝑛 is the 𝑛th unit vector in 𝑙^∞. If 𝑦=𝑃𝑧 and 𝑦_𝑛=𝑃𝑧_𝑛, then 𝑦_𝑛→𝑦 in measure, because, without loss of generality, one can take in that proof a sequence {𝐴_𝑛} satisfying min{𝜇(𝐴_𝑛),‖𝜒_𝐴𝑛‖_𝐸}≤2^−𝑛. Moreover, ‖𝑦_𝑛‖_𝜑=‖𝑦‖_𝜑=‖𝑦_𝑛−𝑦‖_𝜑=1.

Lemma 3.17. (i) If 𝐿↪^∞╱𝐸, 𝑎_𝜑>0, 𝜑(𝑏_𝜑)=∞, then 𝐸_𝜑∉(𝐻_𝑔).
(ii) If 𝐿↪^∞╱𝐸, supp𝐸_𝑎=𝑇, 𝑎_𝜑=0, 𝑏_𝜑=∞ and 𝜑∉Δ₂(0),
then 𝐸_𝜑∉(𝐻_𝑔).

Proof. (i) Suppose that 𝐿↪^∞╱𝐸, 𝜑(𝑏_𝜑)=∞, and 𝑎_𝜑>0. If 𝑇₁∪𝑇₂=𝑇, 𝑇₁∩𝑇₂=∅, 𝜇(𝑇₁)>0, and 𝜇(𝑇₂)>0, then 𝜒_𝑇1∉𝐸 or 𝜒_𝑇2∉𝐸. Assume that 𝜒_𝑇1∉𝐸. Setting 𝑥=𝑎_𝜑𝜒_𝑇1 and 𝑥_𝑛=𝑥+𝑎_𝑛𝜒_𝐴𝑛, where (𝐴_𝑛)⊂𝑇₂, 𝜒_𝐴𝑛∈𝐸, 𝜇(𝐴_𝑛)→0, and 𝜑(𝑎_𝑛)‖𝜒_𝐴𝑛‖_𝐸=1, we get ‖𝑥‖_𝜑=‖𝑥_𝑛‖_𝜑=1, 𝑥_𝑛→𝑥 in measure and ‖𝑥_𝑛−𝑥‖_𝜑=1, whence 𝐸_𝜑∉(𝐻_𝑔).
(ii) The proof is analogous to that of [10, Proposition  2.3]. In fact, it shows even more that in this case, 𝐸_𝜑∉(𝐻_𝑢). By Lemma 3.10, we get (𝐸_𝜑)_𝑎≠{0}. Let 𝑥∈(𝐸_𝜑)_𝑎, 𝑥≥0, and ‖𝑥‖_𝜑=1.
First, we prove that 𝜑∘(2𝑥)∈𝐸_𝑎. Since 𝑥∈𝑆((𝐸_𝜑)_𝑎), so 𝜑∘𝑥∈𝐸_𝑎 and 𝜑 satisfies the local Δ^𝐸2 condition with respect to the element 𝑥 by Theorem  11 from [9] (notice 𝑎_𝜑=0, whence Δ^𝐸2(𝑥𝜒_𝑇⧵𝐴)=Δ^𝐸2(𝑥) in Theorem  11 from [9]). Furthermore, by Lemma  9 from [9], we conclude that 𝜑∘(2𝑥)∈𝐸_𝑎, because 𝑏_𝜑=∞.
We claim that there is a sequence {𝐴_𝑛} in Σ satisfying 𝜒_𝐴𝑛∉𝐸 and 𝐼_𝜑(2𝑥𝜒_𝐴𝑛)≤2^−𝑛 for each 𝑛. Since the measure space (𝑇,Σ,𝜇) is 𝜎-finite, there is an increasing sequence {𝐶_𝑛} with 0<𝜇(𝐶_𝑛)<∞ for any 𝑛∈ℕ and ⋃_𝑛𝐶_𝑛=𝑇. Applying Remark 3.3, without loss of generality, we may assume that 𝜒_𝐶𝑛∈𝐸 for each 𝑛 (see also [12]). Consequently, ‖𝜑∘(2𝑥)𝜒_{𝑇⧵𝐶𝑛}‖_𝐸→0. Moreover, 𝜒_{𝑇⧵𝐶𝑛}∉𝐸, because 𝜒_𝑇∉𝐸 and 𝜒_𝐶𝑛∈𝐸. Thus, the claim is proved with {𝐴_𝑛} being the respective subsequence of {𝑇⧵𝐶_𝑛}. Since 𝜑∉Δ₂(0), there is a sequence {𝑢_𝑛} in ℝ₊ with 𝑢_𝑛→0 and 𝜑2𝑢_𝑛>2^𝑛𝜑𝑢_𝑛,(3.67) for any 𝑛∈ℕ. Notice that if 𝜒_𝐴∈𝐸_𝑎, then the submeasure 𝜈_𝐴(𝐵)=‖𝜒_𝐵‖_𝐸, 𝐵∈Σ∩𝐴 has the Darboux property in the interval [0,‖𝜒_𝐴‖_𝐸] (see [21]). We claim that for any 𝑛, there is a set 𝐵_𝑛⊂𝐴_𝑛 with 𝜑𝑢_𝑛‖‖𝜒_𝐵𝑛‖‖_𝐸=2^−𝑛.(3.68) Let 𝑛∈ℕ. First, by 𝐸∈(𝐹𝑃), there is a set 𝐷_𝑛⊂𝐴_𝑛 with 𝜇(𝐷_𝑛)<∞, 𝜒_𝐷𝑛∈𝐸, and 𝜑(𝑢_𝑛)‖𝜒_𝐷𝑛‖_𝐸>2⋅2^−𝑛. Next, since supp𝐸_𝑎=𝑇 and 𝐸∈(𝐹𝑃), there is a set 𝐸_𝑛⊂𝐷_𝑛 with 𝜒_𝐸𝑛∈𝐸_𝑎 and 𝜑(𝑢_𝑛)‖𝜒_𝐸𝑛‖_𝐸>(3/2)⋅2^−𝑛. Finally, there is a set 𝐵_𝑛⊂𝐸_𝑛 with 𝜑(𝑢_𝑛)‖𝜒_𝐵𝑛‖_𝐸=2^−𝑛. Taking 𝑥_𝑛=𝑥+𝑢^𝑛2𝜒^𝐵𝑛=𝑥𝜒_{𝑇⧵𝐵𝑛}+𝑥+𝑢^𝑛2𝜒^𝐵𝑛,(3.69) we can complete arguments as in the proof of [10,Proposition  2.3].

Lemma 3.18. Suppose that 𝐸 is a symmetric Köthe function space.(i)If 𝐿^∞↪𝐸, 𝐸∈(𝐻_𝑔) and 𝜑∈Δ₂(∞), then 𝐸_𝜑∈(𝐻_𝑔).(ii)If 𝐿↪^∞╱𝐸, 𝐸∈(𝐻_𝑔) and 𝜑∈Δ₂(ℝ₊), then 𝐸_𝜑∈(𝐻_𝑔).

Proof. (i) Applying Lemma 3.5 and Corollary 3.7, it is enough to take 𝑥,𝑥_𝑛∈(𝑆(𝐸_𝜑))₊, 𝑛∈ℕ, and 𝑥_𝑛→𝑥 in measure. Then ‖𝜑∘𝑥‖_𝐸=‖𝜑∘𝑥_𝑛‖_𝐸=1, by 𝜑∈Δ₂(∞)=Δ^𝐸2. Moreover, 𝜑∘𝑥→_𝑛𝜇𝜑∘𝑥 by Lemma 3.9(i). Consequently, ‖𝜑∘𝑥_𝑛−𝜑∘𝑥‖_𝐸→0, because 𝐸∈(𝐻_𝑔). Applying superadditivity 𝜑 on ℝ₊, we get ‖‖𝜑∘𝑥_𝑛−𝑥‖‖_𝐸≤‖‖𝜑∘𝑥_𝑛−𝜑∘𝑥‖‖_𝐸⟶0.(3.70) Take 𝜆>1. We need to show that ‖‖𝜑∘𝜆𝑥_𝑛−𝑥‖‖_𝐸⟶0.(3.71) Let 𝜀>0. The remaining part of the proof we divide into two parts.
(A) Suppose that 𝑎_𝜑>0. Take 𝛿>0 such that 𝜑((1+𝛿)𝑎_𝜑)‖𝜒_𝑇‖_𝐸<𝜀/3. Put 𝐴𝐴_𝑛1𝑛=𝑡∈𝑇∶𝜆||𝑥_𝑛(𝑡)−𝑥(𝑡)||>(1+𝛿)𝑎_𝜑,𝐴_2𝑛=𝑡∈𝐴_𝑛∶||𝑥_𝑛(𝑡)−𝑥(𝑡)||>(1+𝛿)𝑎_𝜑,=𝑡∈𝐴_𝑛∶||𝑥_𝑛(𝑡)−𝑥(𝑡)||≤(1+𝛿)𝑎_𝜑.(3.72) By 𝜑∈Δ₂(∞), there is 𝐾(𝜆,(1+𝛿)𝑎_𝜑)=𝐾>0 such that 𝜑(𝜆𝑢)≤𝐾𝜑(𝑢) for each 𝑢≥(1+𝛿)𝑎_𝜑. Consequently, by (3.70), ‖‖𝜑∘𝜆𝑥_𝑛−𝑥‖‖_𝐸≤‖‖𝜑∘𝜆𝑥_𝑛−𝑥𝜒_𝐴𝑛‖‖_𝐸+‖‖𝜑∘𝜆𝑥_𝑛−𝑥𝜒_{𝑇⧵𝐴𝑛}‖‖_𝐸≤‖‖𝜑∘𝜆𝑥_𝑛−𝑥𝜒_𝐴^1𝑛‖‖_𝐸+‖‖𝜑∘𝜆𝑥_𝑛−𝑥𝜒_𝐴^2𝑛‖‖_𝐸+𝜑(1+𝛿)𝑎_𝜑‖‖𝜒_{𝑇⧵𝐴𝑛}‖‖_𝐸≤𝐾‖‖𝜑∘𝑥_𝑛−𝑥𝜒_𝐴^1𝑛‖‖_𝐸+𝜑𝜆(1+𝛿)𝑎_𝜑‖‖𝜒_𝐴^2𝑛‖‖_𝐸+𝜀3<23𝜀+𝜑𝜆(1+𝛿)𝑎^𝜑‖‖𝜒_𝐴^2𝑛‖‖_𝐸,(3.73) for sufficiently large 𝑛. Recall that m_𝐸=0, because 𝐸∈(𝐻_𝑔) (see Lemma 3.2). Moreover, 𝐴^2𝑛⊂𝐴_𝑛, whence 𝜇(𝐴^2𝑛)≤𝜇(𝐴_𝑛)→0, because 𝑥_𝑛→𝑥 in measure. Consequently, ‖𝜒_𝐴^2𝑛‖_𝐸→0. Thus, ‖𝜑∘(𝜆(𝑥_𝑛−𝑥))‖_𝐸≤𝜀 for sufficiently large 𝑛.
(B) If 𝑎_𝜑=0, then we follow as in (A) but with the number 𝑢₀>0 in place of (1+𝛿)𝑎_𝜑 satisfying 𝜑(𝜆𝑢₀)‖𝜒_𝑇‖_𝐸<𝜀/2. We also set 𝐴_𝑛={𝑡∈𝑇∶|𝑥_𝑛(𝑡)−𝑥(𝑡)|>𝑢₀}.
(ii) The proof is analogous and simpler than in case (i).

Theorem 3.19. Suppose that 𝐸 is a symmetric Köthe function space. The Calderón-Lozanovskiĭ function space 𝐸_𝜑 has the Kadec-Klee property with respect to convergence in measure if and only if:(a)𝐸∈(𝐻_𝑔),(b)𝜑∈Δ₂(∞) whenever 𝐿^∞↪𝐸 and 𝜑∈Δ₂(ℝ₊) provided 𝐿↪^∞╱𝐸.

Proof. The necessity By Lemma 3.12, we conclude that m_𝐸=0, whence supp𝐸_𝑎=𝑇. We also have 𝐸↪^╱𝐿^∞ (see Remark 3.13). By Lemmas 3.15 and 3.16, we get 𝑏_𝜑=∞ and 𝜑∈Δ₂(∞). We consider two cases.(A)Assume that 𝐿^∞↪𝐸. To prove 𝐸∈(𝐻_𝑔), we follow analogously as in the proof of condition 𝐸∈(𝐻_𝑙) in Theorem 3.11 (case I). We apply also Lemma 3.5, Corollary 3.7, and Lemma 3.9(ii).(B)Assume that 𝐿↪^∞╱𝐸, then 𝜑∈Δ₂(0), by Lemma 3.17. Consequently, 𝜑∈Δ₂(ℝ₊), and to prove that 𝐸∈(𝐻_𝑔), see the proof of Theorem 3.11 (case II).
The sufficiency is done by Lemma 3.18.

Remark 3.20. Property 𝐻_𝑢 does not imply 𝐻_𝑔 in general. If 𝐿^∞↪𝐸, then 𝜒_𝑇∈𝐸_𝜑, because there is 𝜆>0 with 𝜑(𝜆)<∞, whence 𝜑∘(𝜆𝜒_𝑇)∈𝐸. Consequently, taking 𝑥_𝑛→𝑥 uniformly, we get ‖𝑥_𝑛−𝑥‖_𝜑≤𝜀‖𝜒_𝑇‖_𝜑 for sufficiently large 𝑛. Hence, 𝐸_𝜑∈(𝐻_𝑢). However, if 𝜑∉Δ₂(∞), then 𝐸_𝜑∉(𝐻_𝑔).

Denote by (𝐼,Σ,𝑚) the Lebesgue measure space with 𝐼=[0,1] or 𝐼=[0,∞). Recall that the function 𝜔∶[0,𝛾)→𝑅₊ with 𝛾=𝑚(𝐼) is said to be the weight function, if it is nonnegative, nonincreasing and locally integrable function with the respect to the Lebesgue measure 𝑚. Then the Lorentz function space Λ_𝜔 consists of all functions 𝑥∈𝐿⁰(𝐼,Σ,𝑚) such that ‖𝑥‖=∫^𝛾0𝑥^∗(𝑡)𝜔(𝑡)𝑑𝑡<∞.

Applying Theorems 3.11 and 3.19 with 𝐸=Λ_𝜔, we get the criteria for Orlicz-Lorentz function spaces. Note that the implication (iii) ⇒ (i) in the case (a) has been proved directly in [6, Theorem  1]. The sufficient condition for case (b)(ii) has been proved in [22, Theorem  6] but under additional assumption.

Corollary 3.21. Let (Λ_𝜔)_𝜑 be the Orlicz-Lorentz function space.(a)If 𝐼=[0,1], then the following assertions are equivalent:(i)(Λ_𝜔)_𝜑∈(𝐻_𝑙), (ii)(Λ_𝜔)_𝜑∈(𝐻_𝑔), (iii)𝜑∈Δ₂(∞). (b)Suppose that 𝐼=[0,∞).(i)(Λ_𝜔)_𝜑∈(𝐻_𝑙) if and only if ∫^∞0𝜔=∞ and 𝜑∈Δ₂(ℝ₊),(ii)(Λ_𝜔)_𝜑∈(𝐻_𝑔) if and only if 𝜑∈Δ₂(ℝ₊).

Proof. Recall that Λ_𝜔∈(𝐻_𝑔) for any weight 𝜔 [1, Corollary  1.3]. Thus, (a) and (b) (ii) follow immediately. Clearly, each symmetric Köthe space is separable if and only if it is order continuous (see [13]). Consequently, from [1,Theorem  3.3], [8,Lemma  3.2], it follows that Λ_𝜔∈(𝐻_𝑙) if and only if ∫^∞0𝜔=∞. Thus, (b) (i) follows.

Obviously, since 𝐿¹∈(𝐻_𝑙), from Theorems 3.11 and 3.19 with 𝐸=𝐿¹, we obtain the characterization of 𝐻_𝑙 and 𝐻_𝑔 for Orlicz spaces proved directly in [10, Theorem  2.1].

4. Open Questions

(1) We have seen that Lemma 3.9(i) is crucial in the proof of sufficiency in the main Theorem 3.19. Moreover, Lemma 3.9(i) is not true for a nonsymmetric Köthe space. Consequently, natural questions are as follows(a)are conditions presented in Theorem 3.19 sufficient also in a nonsymmetric case (different proof)? or, if the answer is no,(b)how the necessary and sufficient conditions for 𝐻_𝑔 are in general?

Looking for full criteria in nonsymmetric case, we are not allowed to apply Corollary 3.7 in the proof of sufficiency of Theorem 3.19.

Note also that the symmetry of 𝐸 is applied in the proof of necessity of Theorem 3.19, by using Corollary 3.7. However, it seems that these parts go also if we drop the symmetry of 𝐸 (after minor changes).

(2) We say that 𝐸 has property 𝐻^𝑠𝑔 provided for any 𝑥∈𝑆(𝐸) and (𝑥_𝑛) in 𝑆(𝐸) with 𝑥_𝑛→𝑥 in measure we have ‖𝑥_𝑛−𝑥‖→0. Clearly, by Corollary 3.7, 𝐸∈(𝐻^𝑠𝑔) if and only if 𝐸∈(𝐻_𝑔) whenever 𝐸 is a symmetric Köthe function space. Are properties 𝐻^𝑠𝑔 and 𝐻_𝑔 different in nonsymmetric case?

Acknowledgment

This work was supported partially by the State Committee for Scientific Research, Poland, Grant no. N N201 362236.