Measures of Noncompactness in Regular Spaces

Abstract Previous results by the author on the connection between three measures of noncompactness obtained for ${{\mathcal{L}}_{p}}$ are extended to regular spaces of measurable functions. An example is given of the advantages of some cases in comparison with others. Geometric characteristics of regular spaces are determined. New theorems for $\left( k,\,\beta \right)$ -boundedness of partially additive operators are proved.


Introduction
A condensing operator is a mapping under which the image of any set is, in a certain sense, more compact than the set itself. The degree of noncompactness of a set is measured by means of functions called measures of noncompactness (MNCs for brevity). Condensing operators have properties similar to compact ones. In particular, the theory of rotation of completely continuous vector fields, the Schauder-Tikhonov fixed point principle, and the Fredholm-Riesz-Schauder theory of linear equations with compact operators admit natural generalizations to condensing operators. Therefore, the theory of MNCs and condensing operators has applications in different areas of mathematics. For example, a technique connected with MNCs and condensing operators is used in the study of differential equations in infinite dimensional spaces, function-differential equations of neutral type, integral equations, as well as some types of partial differential equations (see, for example, [1,3]).
In this paper we investigate the relationships among three different MNCs, and we will illustrate with examples the advantages of some MNCs over the others.

Basic Notions
Let E be a Banach space. Given a bounded subset U of E, the Hausdorff measure of noncompactness χ E (U ) = χ(U ) is defined as the infimum of all ε > 0 such that there exists a finite ε-net for U in E.
The measure of noncompactness β E (U ) = β(U ) of U ⊂ E is defined as the supremum of all numbers r > 0 such that there exists an infinite sequence in U with u n − u m r for every n = m.

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We denote by B(u 0 , r) = {u ∈ E : u − u 0 r} the closed ball in E of radius r and with the center u 0 , and by B = B(θ, 1) the unit ball with the center θ where θ is zero element.
Let Ω be some subset of R n , and let µ(Ω) < ∞, µ be a continuous measure; i.e., each subset D ⊂ Ω, µ(D) > 0, can be split into two subsets of the same measure.
A Banach space E of real-valued measurable functions on Ω is an ideal space if it satisfies the following condition: if a function v belongs to E, u is a measurable function, and the inequality |u| |v| is fulfilled almost everywhere, then u also belongs to E, and u E v E . An ideal space E is a regular space (see [4,12,13]) if each function u ∈ E has an absolutely continuous norm: lim µ(D)→0 P D u E = 0. In particular, where u 0 ∈ E is any fixed function with positive values, conventionally called the unit of space E, D(u, T, u 0 ) = {s ∈ Ω : |u(s)| > Tu 0 (s)} for an arbitrary number T > 0, and the symbol P D u denotes the multiplication operator by characteristic function κ D of any subset D ⊂ Ω. Define L ∞ (u 0 ) to be a Banach space of all real-valued measurable functions on Ω, with the norm u L∞(u0) = inf{λ : |u| ≤ λu 0 a.e.} (L ∞ (1) = L ∞ ). It is a non-regular space.
We list the following examples of regular spaces for u 0 ≡ 1: • the Orlicz spaces.

N. A. Erzakova
As in [5][6][7][8][9][10], for any regular space E the symbol ν E (U ) denotes the measure of the non-uniform absolute equicontinuity of norms U ⊂ E: which is considered an MNC. In particular, The measure ν E (U ) has all properties of ϕ mentioned above, excluding the regularity, since the equality ν E (U ) = 0 is possible on noncompact sets.
Also it has been proved in [5] and [6] that if U is a bounded subset of a regular space E, then ν E (U ) ≤ χ E (U ); if U is, in addition, compact in measure, then ν E (U ) = χ E (U ) . Below we will prove similar properties for β.
Here compactness in measure [1, 4.9.1] means compactness in the normed space S of all measurable, almost everywhere finite functions u, equipped with the norm u = in f {s + µ{t : |u(t)| s}}.
The following two statements, which will be prove below, are general in nature, i.e., valid for an arbitrary Banach space E.

Lemma 2.1 Let U be an arbitrary bounded infinite subset of a Banach space E. Then
for every ε > 0 there exists an element u ∈ U such that the ball B(u, β E (U )+ε) contains an infinite subset of U .
Proof Let u 1 ∈ U be an arbitrary element. Choose ε > 0. If the ball B(u 1 , β E (U ) + ε) contains an infinite subset of U , then the proof of the lemma is complete; otherwise, there exists an element u 2 / ∈ B(u 1 , β E (U ) + ε), u 2 ∈ U . Similarly, if the ball B(u 2 , β E (U ) + ε) does not contain an infinite subset of U , there exists an element u 3 ∈ U such that

etc.
By the definition of β E (U ) this process terminates on some step n, since by the construction, for any i = j (1 i, j n), Lemma 2.1 is proved.

Lemma 2.2 Let U be an arbitrary bounded infinite subset of a Banach space E. Then
for each ε > 0 a set U contains an infinite subset such that the distance between any two elements is less than or equal to β E (U ) + ε.
Proof By Lemma 2.1, for an arbitrary ε > 0 in U there exists an element u 1 such that the ball B(u 1 , β E (U ) + ε) contains an infinite subset U 1 ⊂ U . Now we apply Lemma 2.1 to the set U 1 \{u 1 }. Taking into account the inequality β E (U 1 ) β E (U ), we choose an element u 2 = u 1 , such that the ball B(u 2 , β E (U ) + ε) contains an infinite set U 2 ⊂ U 1 , etc.
Since on n-th step we obtain an infinite subset U n ⊂ U n−1 , this process does not stop and we build an infinite sequence {u n }, the distance between any two members of which is not greater than β E (U ) + ε.
Lemma 2.2 is proved.

Connection Between MNCs and Geometrical Characteristics of Regular Spaces
Let E be a regular space. (i) u n , n ∈ N, have pairwise disjoint supports; (ii) lim n→∞ u n E = 1; (iii) the measure of the support supp u n tends to zero as n → ∞; (iv) there exists a strictly increasing sequence of positive numbers {T n } with lim n→∞ T n = ∞ such that the inequality T n−1 u 0 (s) |u n (s)| < T n u 0 (s) holds for all n ∈ N and s ∈ supp u n . Let The upper bound follows from the triangle inequality and Condition (ii). The lower bound is a consequence of Conditions (i) and (ii), since E is an ideal space.
We suppose further that the norm in a regular space also satisfies the following condition: for any sequences of subsets {D n }, {D * n } in Ω such that D n ∩ D * n = ∅ for all n and lim n→∞ max{µ(D n ), µ(D * n )} = 0, there is no bounded sequence {u n } of functions in E such that and we get a contradiction to (3.3).

Lemma 3.2 Let U be an arbitrary bounded subset of a regular space E with
If U is compact in measure, we can choose {u n } to satisfy, in addition, Proof Let U be an arbitrary bounded subset of a regular space E with ν E (U ) > 0. By (2.2), there exists a strictly increasing sequence of numbers {T n }, lim n→∞ T n = ∞, and a sequence of functions {u n } ⊆ U , for which the equality Note that (2.1) implies lim n→∞ P D(um,Tn,u0) u E = 0 for each fixed m. Considering a subsequence (for our convenience, we do not change the notation), we may assume that ν E (U ) = lim n→∞ P Dn u n E , where D n = {s ∈ Ω : T n u 0 (s) |u n (s)| < T n+1 u 0 (s)}.
It follows from the boundedness of U [13, Theorem 1], that Therefore, lim n→∞ µ( D n ) = 0. Extracting subsequences, we may assume that µ ∞ k=n+1 D k are small enough and the difference between P Dn u n E and P Dn u n E is slight for D n = D n \ ∞ k=n+1 D k . Eventually, we get a sequence {u n } such that ν E (U ) = lim n→∞ P Dn u n E and the sets D n are pairwise disjoint. Note that ν E {u n } = ν E (U ) and by the remark before the lemma, ν E {v n } = 0 for v n = u n − P Dn u n .
As consequence, we obtain

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The constructed sequence of u n = P Dn u n satisfies Conditions (i),(iii), and (iv) from Definition 3.1. Condition (ii) is replaced by the condition lim n→∞ u n E = ν E (U ). Therefore Since E is an ideal space, we have for any m = n, and by (3.4), The first part of Lemma 3.2 is proved. Note that by (iii) the sequence { u n } tends by measure to zero. Let U be compact in measure. Then {u n } is compact in measure too. Therefore, the sequence {u n − u n } is compact in measure too.
As it was proved in [5], [6], in this case χ E {u n − u n } = ν E {u n − u n }. By the remark before the lemma, ν E {u n − u n } = 0. Hence χ E {u n − u n } = 0. By the definition of the Hausdorff MNC, for every ε > 0 there exists a finite ε-net C = {c 1 , c 2 , . . . , c N } ⊂ E such that {u n − u n } ⊂ C +εB. Since C is finite, we can choose an infinite subsequence (with the same notation as before) that satisfies {u n − u n } ⊂ c * +εB for some c * ∈ C. As a result, we have | u n − u m E − u n − u m E | 2ε. Now we decrease ε and extract a subsequence (which we denote again by {u n }) such that The second part of Lemma 3.2 is proved, since lim m→∞ lim n→∞ u n − u m E c E ν E (U ).

Theorem 3.3 In a regular space E the MNCs ν and β are related by the inequality
If U is compact in measure and ν E (U ) = 0, then by the compactness criterion in regular spaces ( [11][12][13]) U is relatively compact and β E (U ) = 0. Thus the assertion for the case ν E (U ) = 0 holds. Therefore, we assume that ν E (U ) > 0.
Let a sequence {u n } be as in Lemma 3.2. Choose ε > 0. By Lemma 2.2 we can assume without loss of generality that u m − u n E β E {u n } + ε for all n and m.
Thus by virtue of the monotonicity of β, we obtain for all n and m.
Since we can take ε arbitrarily small and c E ν E (U ) lim m→∞ lim n→∞ u n − u m E by Lemma 3.2, we get the assertion of the first part of Theorem 3.3.

N. A. Erzakova
Let U be compact in measure. By the definition of β, for given ε > 0, there exists a sequence {w n } such that w n − w m β E (U ) − ε for all n = m. Hence by Lemma 3.2, we can extract a subsequence {u n } of {w n } such that which finishes the proof of Theorem 3.3, since ε can be arbitrarily small.

Example 3.5 c Λ
Proof Since by [11,15.1] the set of all finite-valued functions is dense in Λ 1/p , 1 p < ∞, without loss of generality, we may assume that S consists of sequences of finite-valued functions. By Definition 3.1, {u n } is a sequence of functions with disjoint supports such that there exists strictly increasing sequence of positive numbers {T n }, such that T n−1 |u n (s)| < T n for all s ∈ supp u n .

MNC β of Bounded Subsets in L ∞
The aim of this section is to show that β L1 (V ) (2 − r/(aµ(Ω))r follows from the inclusion V ⊂ B L∞ (θ, a) ∩ B L1 (θ, r). We start with some particular cases.
Throughout this section U denotes the set of all measurable functions on Ω with values in the set {−1, 0, 1}.
Below we use the proportionality β and χ in the separable Hilbert space:
Next we prove that for the given ε, there exists an infinite subset U 2 ⊂ U 1 such that for any two elements u, v ∈ U 2 we have Indeed, by (4.1), Therefore, by Lemma 2.2, the set U 1 includes an infinite subset U 2 such that which completes the proof of (4.3). Note that (4.3) implies ω uv (ω − ξ uv /2)/2 + ε. Thus for every u, v ∈ U 2 , u = v, whence we obtain the assertion of Lemma 4.1, since ε can be arbitrarily small. Proof If n = 1, the assertion follows from Lemma 4.1, the semi-homogeneity of β, and the inequality α 1 > 0. Therefore, we assume the validity of the assertion for some n > 1 and prove that it remains true when we replace n with n+1. Without loss of generality, we may assume that α 1 < α 2 < · · · < α n < α n+1 . Then the algebraic additivity of β and the equality Using the inductive assumption, we get Now we are ready to prove the main result of the section. Proof By the definition of β, for every ε > 0, the set U contains an infinite sequence {u k }, satisfying the inequality u k − u m L1 β L1 (U ) − ε for all k = m.
Note that {u k } is bounded in L ∞ . Therefore, considering a subsequence, we may assume that there exists lim k→∞ u k = r 1 r. Now we consider approximations of {u k } by functions satisfying the assumptions of Lemma 4.2. Considering limit points of sets of values of every ω k for a fixed k, taking subsequences once again, and using the continuity of the measure µ, we may assume that there exists a sequence { u k } of elements satisfying the assumptions of Lemma 4.2 with the same α i and ω j , such that u k − u k L1 < ε for all k ∈ N. Thus, β L1 (U )−ε u k − u m L1 +2ε for any k = m. By Lemma 2.2, without loss of generality we may assume that u k − u m L1 β L1 { u k }+ε.

(k, β)-boundedness of Partially Additive Operators
Let E and E 1 be Banach spaces. We recall from [1, 1.5.1] that a continuous operator A : G ⊆ E → E 1 (not necessarily linear) is said to be condensing with respect to MNC ϕ, if for any bounded subset U ⊂ G with noncompact closure, the inequality ϕ E1 (AU ) < ϕ E (U ) holds.
A continuous operator A : G ⊆ E → E 1 is called (k, ϕ)-bounded with respect to MNC ϕ, if there exists a constant k > 0 such that ϕ E1 (AU ) kϕ E (U ) for any bounded subset U ⊂ G.
If k < 1 then (k, ϕ)-bounded operator A is condensing with respect to MNC ϕ. The converse, in general, is not true.
Let E be a regular space. We consider partially additive operators A : E → E 1 [11, 17.4]. In particular, partially additive operators satisfy the condition Let U be any bounded subset from E. We denote the following as in [7][8][9]: Evidently, in the case of a linear bounded operator the constant k(U, A, E, E 1 ) does not exceed the norm of the operator. For a nonlinear operator, even if it is partially additive and bounded, this constant is either finite or infinite.
Proof By (5.1), the assumption of partially additivity of A, and the algebraic additivity of β, we obtain for any V ⊆ U , for any T > 0. From here, taking into account the monotonicity of β and the inequality k(V, A, E, E 1 ) k(U, A, E, E 1 ) for every V ⊆ U , we obtain the assertion of Lemma 5.1.

Theorem 5.2 Suppose that A satisfies the conditions of Lemma 5.1. Then the operator
Proof Applying Lemma 5.1, Theorem 3.3, and the semi-homogeneity of β, we obtain Theorem 5.2 is proved.
(ii) Let A be a linear operator, acting from L p in L ∞ (1 p < ∞). Then A as an operator from L p in L q is a (2 (p−q)/pq A , β)-bounded operator for 1 q 2, and a (2 1−1/p−1/q A , β)-bounded operator for 2 < q < ∞ [10, Theorem 2]. Note that [11,Lemma 5.3] implies that any linear integral operator is compact as an operator L ∞ → L 1 .

K(t, s)u(s)ds
is measurable for every u ∈ L 1 , and its norm L 1 is less than or equal to u L1 . Therefore, the operator K satisfies all conditions of Theorem 5.3 (see also the remark before the example) and, in addition, K L1→L∞ = K L1→L1 = 1. Thus by Theorem 5.3, K is (1/2, β)-bounded and, therefore, β-condensing.
Remark MNCs χ and β were cosidered in the works of L. S. Gol'denshtein, I. Gohberg, A. S. Markus, V. Istrǎtescu, J. Daneš, and others. Detailed description of bibliographic information is given in [1]. In particular, the author has proved the algebraic semi-additivity, the invariance under passage to the convex hull of β and proportionality formula (4.1) (see the references in [1, 1.8.3, 4.9.9]). The formula (4.1) and the algebraic semi-additivity of β were also obtained independently by the authors of [2].