On Convex Total Bounded Sets in the Space of Measurable Functions

We estimate the measure of nonconvex total boundedness in terms of 
simpler quantitative characteristics in the space of measurable functions   𝐿 0 . A Frechet-Smulian type compactness criterion for convexly totally 
bounded subsets of 𝐿 0 is established.


Introduction
In 1988, Idzik 1 proved that the answer to the well-known Schauder's problem 2, Problem 54 : does every continuous map f : K → K defined on a convex compact subset of a Hausdorff topological linear space have a fixed point? is affirmative if K is convexly totally bounded.
This notion was introduced by Idzik 1 : a subset K of a topological linear space X is said to be convexly totally bounded ctb for short , if for every 0-neighborhood U there are x 1 , . . ., x n ∈ K and convex subsets C 1 , . . ., C n of U such that K ⊆ n i 1 x i C i .If X is locally convex, every convex compact subset of X is ctb.This is not true, in general, if X is nonlocally convex see 3-5 .In 1993, De Pascale et al. 3 defined the measure of nonconvex total boundedness, modelled on Idzik's concept, that may be regarded as the analogue of the well-known notion of Hausdorff measure of noncompactness in nonlocally convex linear spaces.The above notions of ctb set and of nonconvex total boundedness are especially useful working in the setting of nonlocally convex topological linear spaces see, e.g., [6][7][8] .
Let E, • E be a normed space, Ω a nonempty set, and P Ω the power set of Ω.The space L 0 is an F-normed linear space of E-valued functions defined on Ω which depends on an algebra A in P Ω and a submeasure η : P Ω → 0, ∞ .
We observe that the space L 0 defined above is a generalization of the space of measurable functions introduced in 9, Chapter III , in order to develop the integration theory Journal of Function Spaces and Applications with respect to finitely additive measures.Recall that given M ⊆ L 0 , the Hausdorff measure of noncompactness γ M of M is defined by where B ε L 0 : {f ∈ L 0 : f 0 ≤ ε}.In 10, 11 , to estimate γ M are used two quantitative characteristics λ M and ω M which measure, respectively, the degree of non equi-quasiboundedness and the degree of non equi-measurability of M.
The main purpose of this note is to estimate the measure of nonconvex total boundedness in L 0 and to characterize the convexly totally bounded subsets of L 0 .At this end we introduce two quantitative characteristics λ c M and ω c M involving convex sets, which measure the degree of nonconvex equi-quasiboundedness and the degree of nonconvex equi-measurability of M, respectively.Then we establish some inequalities between γ c M , λ c M , ω c M , and λ M that give, as a special case, a Fréchet-Smulian type convex total boundedness criterion in the space L 0 .This generalizes previous results of Trombetta 12 .Finally, we point out that it is not so clear if the Schauder's problem has been solved in its generality.In particular, the proof given by Cauty in 13 contains some unsolved gaps see 14, 15 , Mathscinet review of 16 , and Zentralblatt Math review of 17 .However, the results of this paper are meant to be independent from the Schauder's problem.

Definitions and Preliminaries
For the remainder of this section we present some definitions and known results which will be needed throughout this paper.
We use the convention inf ∅ : ∞.Moreover, for two sets A and B, we denote by B A the set of all maps from A to B. We recall the definition of the space L 0 see 11,18 .Let E, • E be a normed space, Ω a nonempty set, A an algebra in the power set P Ω of Ω, and η : P Ω → 0, ∞ a submeasure.Then • 0 the space of measurable functions into an F-normed linear space in the sense of 19, page 38 , that is, In 11 the following two quantitative characteristic λ and ω are used to estimate γ in L 0 : The set M is called equi-quasibounded if λ M 0 and equi-measurable if ω M 0.
In particular, M is totally bounded if and only if λ M ω M 0.
Moreover, in 18 , it is defined the following quantitative characteristic σ M which is useful for the calculation of λ M : where B ε E : {y ∈ E : y E ≤ ε}.
The following result was established in 18, Proposition 2.1 .
We omit the proof of the following proposition which is similar to the proof of Proposition 2.6 of 12 .

Inequalities in the Space L 0
In order to estimate the measure of nonconvex total boundedness in L 0 , we introduce the following two quantitative characteristics.

3.1
We call M convexly equi-quasibounded if λ c M 0 and convexly equi-measurable if ω c M 0. We observe that if E R, then the quantitative characteristics λ c and ω c coincide with those introduced in 12 .
We point out that the request of convexity plays a crucial role in the definition of the parameters λ c and ω c , whereas it was not involved in the definition of λ and ω.We illustrate this with the following example.
Example 3.2.Let L 0 : L 0 0, ∞ , A, E, η , where A is the algebra of all Lebesgue-measurable subsets of the interval 0, ∞ and η| A the Lebesgue measure.Let I 1 : 0, 1 , I n : Proof.It is easy to check that λ M 1 and ω M 0. We are going to prove that λ c M ∞.On the contrary, suppose that λ c M < α < ∞ then there exist a finite set G {z 1 , . . ., z p } ⊆ E and convex sets C 1 , . . ., C m ⊆ B α L 0 such that

3.2
Set c : max{ z l E : l 1, . . ., p}.We have s 0 ≤ c for all s ∈ G Ω ∩ S. Fix α > c α, a natural number n such that n n 1 1/n > mα, and y ∈ E with y E > n • α.Put S y : f yχ I n : n 1, 2, . . ., n , S j y : f ∈ S y : there is Then it is easy to see that there exist j ∈ {1, . . ., m} and a subfamily {I n 1 , . . ., which contradicts the convexity of the set The following lemma is crucial in the proof of Theorem 3.4.η Ω .Assume that σ H < α < η Ω .By the definition of σ we can find a finite set G ⊆ E, containing the origin 0 ∈ E and sets D 0 : ∅, D 1 , . . ., D r ⊆ Ω, with η D j ≤ α for j 1, . . ., r, such that i each D j for j 1, . . ., r is the union of the members of a proper subfamily depending on j of the partition {A 1 , . . ., A n };

Lemma 3.3. Let
ii for each s ∈ H, there is j ∈ {0, . . ., r} such that s Ω \ D j ⊆ G B α E .
For j 0, . . ., r we consider the following convex subsets of B α L 0 : and we set We will prove that

3.7
It follows that γ c H ≤ α, and therefore γ c H ≤ σ H .

Journal of Function Spaces and Applications
Let We have Hence, for all l ∈ {k 1, . . ., n}, there is z i l ∈ G such that y i l − z i l ∈ B α E .Then s ϕ h, where ϕ : 3.9 Therefore,
We are now in a position to prove the main result of this note.
Theorem 3.4.Let M ⊆ L 0 .Then, Proof.We first prove the left inequality which is trivial if γ c M ∞.Assume that γ c M < α < ∞.By Proposition 2.3, there are functions s 1 , . . ., s n ∈ S and convex sets C 1 , . . ., C n in B α L 0 such that

3.13
Put F : n i 1 s i Ω and let A 1 , . . ., A m ∈ A be a partition of Ω such that s i | A j is constant for i 1, . . ., n and j i, . . ., m.Then, 3.17 Therefore, by Lemma 3.3, we have that and so The proof is complete.
As a corollary of Theorem 3.4, we obtain the following Fréchet-Smulian type convex total boundedness criterion.We point out that the approach used in the scalar case 12 in order to prove Theorem 3.7 cannot be used in our framework.The crucial difference is in the proof of Lemma 3.3.In fact, if dim E ∞, it might exist some i ∈ {1, . . ., n} such that the set H A i : {s A i : s ∈ H} is bounded but not necessarily totally bounded.
The following example shows how the value of the parameters λ and λ c changes when passing from the scalar case to the E-valued case.

3.21
Proof.It is sufficient to observe that γ c M γ c M .In particular, M ctb implies λ M λ M 0, λ c M λ c M 0, and ω c M ω c M 0.
The next two corollaries of Theorem 3.4 are useful in order to compute or to estimate γ c in particular classes of subsets of L 0 .Moreover, the second one generalizes 12, Proposition 3.10 .

3 . 14 hence⎠
λ c M < α, and ω c M < α.Therefore, max{λ c M , ω c M } ≤ γ c M .* We now prove the right inequality.Clearly, it is true if λ M ∞ or ω c M ∞.Assume that ω c M < β < ∞.By the definition of ω c , we can find a partition A 1 , . . ., A m ∈ A and convex sets K 1 , . . ., K m of B β L 0 such that M ⊆ S A 1 , . . .∩ S A 1 , . . ., A m , 3.16 then we have λ H ≤ λ M β.It easy to see that

Corollary 3 . 5 .Remark 3 . 6 .
A subset M of L 0 is ctb if and only if λ M ω c M 0. In 12 , Trombetta proved that max{λ c M , ω c M } ≤ γ c M ≤ λ c M 2ω c M 3.20for a subset M ⊆ L 0 Ω, A, R, η .Since λ M ≤ λ c M , Theorem 3.4 improves and generalizes to E-valued case the above inequalities.