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 . A Fréchet-Smulian type compactness criterion for convexly totally bounded subsets of is established.
In 1988, Idzik  proved that the answer to the well-known Schauder's problem [2, Problem 54]: does every continuous map defined on a convex compact subset of a Hausdorff topological linear space have a fixed point? is affirmative if is convexly totally bounded.
This notion was introduced by Idzik : a subset of a topological linear space is said to be convexly totally bounded (ctb for short), if for every 0-neighborhood there are and convex subsets of such that . If is locally convex, every convex compact subset of is ctb. This is not true, in general, if is nonlocally convex (see [3–5]).
In 1993, De Pascale et al.  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., [6–8]).
Let be a normed space, a nonempty set, and the power set of . The space is an -normed linear space of -valued functions defined on which depends on an algebra in and a submeasure .
We observe that the space defined above is a generalization of the space of measurable functions introduced in [9, Chapter III], in order to develop the integration theory with respect to finitely additive measures. Recall that given , the Hausdorff measure of noncompactness of is defined by where . In [10, 11], to estimate are used two quantitative characteristics and which measure, respectively, the degree of non equi-quasiboundedness and the degree of non equi-measurability of .
The main purpose of this note is to estimate the measure of nonconvex total boundedness in and to characterize the convexly totally bounded subsets of . At this end we introduce two quantitative characteristics and involving convex sets, which measure the degree of nonconvex equi-quasiboundedness and the degree of nonconvex equi-measurability of , respectively. Then we establish some inequalities between , , , and that give, as a special case, a Fréchet-Smulian type convex total boundedness criterion in the space . This generalizes previous results of Trombetta . 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  contains some unsolved gaps (see [14, 15], Mathscinet review of , and Zentralblatt Math review of ). However, the results of this paper are meant to be independent from the Schauder's problem.
2. 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 . Moreover, for two sets and , we denote by the set of all maps from to . We recall the definition of the space (see [11, 18]). Let be a normed space, a nonempty set, an algebra in the power set of , and a submeasure. Then defines a group pseudonorm on , that is, , , , for all . Let be the closure of the linear space of -valued -simple functions in , where is the characteristics function of . Identification of functions for which turns (the space of measurable functions) into an -normed linear space in the sense of [19, page 38], that is, , , for and .
Let . The measure of nonconvex total boundedness of is defined by Clearly is ctb if and only if .
Let be a partition of . We set In  the following two quantitative characteristic and are used to estimate in :
The set is called equi-quasibounded if and equi-measurable if .
Theorem 2.1 (see [11, Theorem 2.2.2]). Let . Then In particular, is totally bounded if and only if .
Moreover, in , it is defined the following quantitative characteristic which is useful for the calculation of : where .
The following result was established in [18, Proposition 2.1].
Proposition 2.2. Let . Then .
We omit the proof of the following proposition which is similar to the proof of Proposition 2.6 of .
Proposition 2.3. Let . Then
3. Inequalities in the Space
In order to estimate the measure of nonconvex total boundedness in , we introduce the following two quantitative characteristics.
Definition 3.1. Let . We define the following:
We call convexly equi-quasibounded if and convexly equi-measurable if .
We observe that if , then the quantitative characteristics and coincide with those introduced in .
We point out that the request of convexity plays a crucial role in the definition of the parameters and , whereas it was not involved in the definition of and . We illustrate this with the following example.
Example 3.2. Let , where is the algebra of all Lebesgue-measurable subsets of the interval and the Lebesgue measure. Let , for , and for . If , then , , .
Proof. It is easy to check that and . We are going to prove that . On the contrary, suppose that then there exist a finite set and convex sets such that
Set . We have for all . Fix , a natural number such that , and with .
Put for .
Then it is easy to see that there exist and a subfamily of such that for and . Moreover, a straightforward computation shows that which contradicts the convexity of the set . Since , the equality is a consequence of Theorem 3.4.
The following lemma is crucial in the proof of Theorem 3.4.
Lemma 3.3. Let be a partition of and . Then .
Proof. Obviously, . Since , it follows from Theorem 2.1 and Proposition 2.2 that . Then it is sufficient to prove the inequality which is trivial if .
Assume that . By the definition of we can find a finite set , containing the origin and sets , , with for , such that(i)each for is the union of the members of a proper subfamily depending on of the partition ;(ii)for each , there is such that .
For we consider the following convex subsets of : and we set We will prove that
It follows that , and therefore .
Let . Suppose , and , where is a proper subfamily of the partition .
We have Hence, for all , there is such that . Then , where Therefore, Similarly, if , we can prove that Thus, (3.7) immediately follows from (3.10) and (3.11).
We are now in a position to prove the main result of this note.
Theorem 3.4. Let . Then,
Proof. We first prove the left inequality which is trivial if . Assume that . By Proposition 2.3, there are functions and convex sets in such that
Put and let be a partition of such that is constant for and . Then, hence , and .
Therefore, We now prove the right inequality. Clearly, it is true if or . Assume that . By the definition of , we can find a partition and convex sets of such that Set then we have . It easy to see that
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.
Corollary 3.5. A subset of is ctb if and only if .
We point out that the approach used in the scalar case  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 , it might exist some such that the set is bounded but not necessarily totally bounded.
The following example shows how the value of the parameters and changes when passing from the scalar case to the -valued case.
Example 3.7. Let , where . Then if is finite dimensional, , otherwise. Moreover, since , we have .
Corollary 3.8. Let . Then,
Proof. It is sufficient to observe that .
In particular, ctb implies , , and .
Corollary 3.9. Let be a convexly equi-measurable subset of . Then, .
Corollary 3.10. Let be an equi-quasibounded subset of . Then,
We observe that if is a totally bounded subset of and , then, since , the inequalities (3.21) are true for .
Example 3.11. Let be the space of Example 3.2 and assume that is the Banach space of all sequences with finite norm . If and , it is easy to prove that and . If is the closed unit ball of and , the set satisfies .
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