We estimate the measure of nonconvex total boundedness in terms of simpler quantitative characteristics in the space of measurable functionsā€‰ā€‰šæ0. A FrĆ©chet-Smulian type compactness criterion for convexly totally bounded subsets of šæ0 is established.

1. Introduction

In 1988, Idzik [1] proved that the answer to the well-known Schauder's problem [2, Problem 54]: does every continuous map š‘“āˆ¶š¾ā†’š¾ā€‰ā€‰defined on a convex compact subset of a Hausdorff topological linear space have a fixed point? is affirmative if š¾ is convexly totally bounded.

This notion was introduced by Idzik [1]: a subset š¾ of a topological linear space š‘‹ is said to be convexly totally bounded (ctb for short), if for every 0-neighborhood š‘ˆ there are š‘„1,ā€¦,š‘„š‘›āˆˆš¾ and convex subsets š¶1,ā€¦,š¶š‘› of š‘ˆ such that ā‹ƒš¾āŠ†š‘›š‘–=1(š‘„š‘–+š¶š‘–). If š‘‹ is locally convex, every convex compact subset of š‘‹ is ctb. This is not true, in general, if š‘‹ is nonlocally convex (see [3ā€“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., [6ā€“8]).

Let (šø,ā€–ā‹…ā€–šø) be a normed space, Ī© a nonempty set, and š’«(Ī©) the power set of Ī©. The space šæ0 is an š¹-normed linear space of šø-valued functions defined on Ī© which depends on an algebra š’œ in š’«(Ī©) and a submeasure šœ‚āˆ¶š’«(Ī©)ā†’[0,+āˆž].

We observe that the space šæ0 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 š‘€āŠ†šæ0, the Hausdorff measure of noncompactness š›¾(š‘€) of š‘€ is defined by īƒÆš›¾(š‘€)āˆ¶=infšœ€>0āˆ¶thereexistfunctionsš‘“1,ā€¦,š‘“š‘›āˆˆšæ0suchthatš‘€āŠ†š‘›īšš‘–=1ī€·š‘“š‘–+šµšœ€ī€·šæ0īƒ°,ī€øī€ø(1.1) where šµšœ€(šæ0)āˆ¶={š‘“āˆˆšæ0āˆ¶ā€–š‘“ā€–0ā‰¤šœ€}. In [10, 11], to estimate š›¾(š‘€) are used two quantitative characteristics šœ†(š‘€) and šœ”(š‘€) which measure, respectively, the degree of non equi-quasiboundedness and the degree of non equi-measurability of š‘€.

The main purpose of this note is to estimate the measure of nonconvex total boundedness in šæ0 and to characterize the convexly totally bounded subsets of šæ0. At this end we introduce two quantitative characteristics šœ†š‘(š‘€) and šœ”š‘(š‘€) involving convex sets, which measure the degree of nonconvex equi-quasiboundedness and the degree of nonconvex equi-measurability of š‘€, respectively. Then we establish some inequalities between š›¾š‘(š‘€), šœ†š‘(š‘€), šœ”š‘(š‘€), and šœ†(š‘€) that give, as a special case, a FrĆ©chet-Smulian type convex total boundedness criterion in the space šæ0. 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.

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 infāˆ…āˆ¶=+āˆž. Moreover, for two sets š“ and šµ, we denote by šµš“ the set of all maps from š“ to šµ. We recall the definition of the space šæ0 (see [11, 18]). Let (šø,ā€–ā‹…ā€–šø) be a normed space, Ī© a nonempty set, š’œ an algebra in the power set ā„™(Ī©) of Ī©, and šœ‚āˆ¶ā„™(Ī©)ā†’[0,āˆž] a submeasure. Thenā€–š‘“ā€–0ī€½ā€–āˆ¶=infš‘Ž>0āˆ¶šœ‚ī€·ī€½š‘„āˆˆĪ©āˆ¶ā€–š‘“(š‘„)šøī€¾ā‰„š‘Žī€¾ī€øā‰¤š‘Ž(2.1) defines a group pseudonorm on šøĪ©, that is, ā€–0ā€–0=0, ā€–āˆ’š‘“ā€–0=ā€–š‘“ā€–0, ā€–š‘“+š‘”ā€–0ā‰¤ā€–š‘“ā€–0+ā€–š‘”ā€–0, for all š‘“,š‘”āˆˆšøĪ©. Let šæ0āˆ¶=šæ0(Ī©,š’œ,šø,šœ‚) be the closure of the linear space š‘†āˆ¶=span{š‘¦šœ’š“āˆ¶š‘¦āˆˆšøandš“āˆˆš’œ} of šø-valued š’œ-simple functions in (šøĪ©,ā€–ā‹…ā€–0), where šœ’š“ is the characteristics function of š“. Identification of functions š‘“,š‘”āˆˆšøĪ© for which ā€–š‘“āˆ’š‘”ā€–0=0 turns (šæ0,ā€–ā‹…ā€–0) (the space of measurable functions) into an š¹-normed linear space in the sense of [19, page 38], that is, ā€–š‘“+š‘”ā€–0ā‰¤ā€–š‘“ā€–0+ā€–š‘”ā€–0, limš‘›ā†’āˆžā€–(1/š‘›)š‘“ā€–0=0, ā€–šœ†š‘“ā€–0ā‰¤ā€–š‘“ā€–0 for š‘“,š‘”āˆˆšæ0 and |šœ†|ā‰¤1.

Let š‘€āŠ†šæ0. The measure of nonconvex total boundedness š›¾š‘(š‘€) of š‘€ is defined by š›¾š‘īƒÆ(š‘€)āˆ¶=infšœ€>0āˆ¶thereexistfunctionsš‘“1,ā€¦,š‘“š‘›āˆˆšæ0š¶andconvexsubsets1,ā€¦,š¶š‘›ofšµšœ€ī€·šæ0ī€øsuchthatš‘€āŠ†š‘›īšš‘–=1ī€·š‘“š‘–+š¶š‘–ī€øīƒ°.(2.2) Clearly š‘€ is ctb if and only if š›¾š‘(š‘€)=0.

Let š“1,ā€¦,š“š‘šāˆˆš’œ be a partition of Ī©. We set š‘†ī€·š“1,ā€¦,š“š‘šī€øīƒÆāˆ¶=š‘ āˆˆš‘†āˆ¶š‘ =š‘šī“š‘–=1š‘¦š‘–šœ’š“š‘–,whereš‘¦š‘–īƒ°.āˆˆšøforš‘–=1,ā€¦,š‘š(2.3) In [11] the following two quantitative characteristic šœ† and šœ” are used to estimate š›¾ in šæ0: šœ†ī€½ī€·šŗ(š‘€)āˆ¶=infšœ€>0āˆ¶thereexistsaļ¬nitesubsetšŗofšøsuchthatš‘€āŠ†Ī©ī€øāˆ©š‘†+šµšœ€ī€·šæ0,ī€½ī€øī€¾šœ”(š‘€)āˆ¶=infšœ€>0āˆ¶thereexistsapartitionš“1,ā€¦,š“š‘šī€·š“āˆˆš’œofĪ©suchthatš‘€āŠ†š‘†1,ā€¦,š“š‘šī€ø+šµšœ€ī€·šæ0.ī€øī€¾(2.4)

The set š‘€ is called equi-quasibounded if šœ†(š‘€)=0 and equi-measurable if šœ”(š‘€)=0.

Theorem 2.1 (see [11, Theorem 2.2.2]). Let š‘€āŠ†šæ0. Then max{šœ†(š‘€),šœ”(š‘€)}ā‰¤š›¾(š‘€)ā‰¤šœ†(š‘€)+2šœ”(š‘€).(2.5) In particular, š‘€ is totally bounded if and only if šœ†(š‘€)=šœ”(š‘€)=0.

Moreover, in [18], it is defined the following quantitative characteristic šœŽ(š‘€) which is useful for the calculation of šœ†(š‘€):šœŽ(š‘€)āˆ¶=inf{šœ€>0āˆ¶thereexistsaļ¬nitesubsetšŗofšøsuchthatforallš‘“āˆˆš‘€thereisš·š‘“ī€·š·āŠ†Ī©withšœ‚š‘“ī€øī€·ā‰¤šœ€andš‘“Ī©ā§µš·š‘“ī€øāŠ†šŗ+šµšœ€ī€¾,(šø)(2.6) where šµšœ€(šø)āˆ¶={š‘¦āˆˆšøāˆ¶ā€–š‘¦ā€–šøā‰¤šœ€}.

The following result was established in [18, Proposition 2.1].

Proposition 2.2. Let š‘€āŠ†šæ0. Then šœ†(š‘€)=šœŽ(š‘€).

We omit the proof of the following proposition which is similar to the proof of Proposition 2.6 of [12].

Proposition 2.3. Let š‘€āŠ†šæ0. Then š›¾š‘īƒÆ(š‘€)=infšœ€>0āˆ¶thereexistfunctionsš‘ 1,ā€¦,š‘ š‘›š¶āˆˆš‘†andconvexsubsets1,ā€¦,š¶š‘›ofšµšœ€ī€·šæ0ī€øsuchthatš‘€āŠ†š‘›īšš‘–=1ī€·š‘ š‘–+š¶š‘–ī€øīƒ°.(2.7)

3. Inequalities in the Space šæ0

In order to estimate the measure of nonconvex total boundedness in šæ0, we introduce the following two quantitative characteristics.

Definition 3.1. Let š‘€āŠ†šæ0. We define the following: šœ†š‘īƒÆš¶(š‘€)āˆ¶=infšœ€>0āˆ¶thereexistaļ¬nitesubsetšŗofšøandconvexsubsets1,ā€¦,š¶š‘›ofšµšœ€ī€·šæ0ī€øī€·šŗsuchthatš‘€āŠ†Ī©ī€ø+āˆ©š‘†š‘›īšš‘–=1š¶š‘–īƒ°,šœ”š‘īƒÆ(š‘€)āˆ¶=infšœ€>0āˆ¶thereexistapartitionš“1,ā€¦,š“š‘šš¶āˆˆš’œofĪ©andconvexsubsets1,ā€¦,š¶š‘›ofšµšœ€ī€·šæ0ī€øī€·š“suchthatš‘€āŠ†š‘†1,ā€¦,š“š‘šī€ø+š‘›īšš‘–=1š¶š‘–īƒ°.(3.1)

We call š‘€ā€‚convexly equi-quasibounded if šœ†š‘(š‘€)=0 and convexly equi-measurable if šœ”š‘(š‘€)=0.

We observe that if šø=ā„, then the quantitative characteristics šœ†š‘ and šœ”š‘ coincide with those introduced in [12].

We point out that the request of convexity plays a crucial role in the definition of the parameters šœ†š‘ and šœ”š‘, whereas it was not involved in the definition of šœ† and šœ”. We illustrate this with the following example.

Example 3.2. Let šæ0āˆ¶=šæ0([0,+āˆž[,š’œ,šø,šœ‚), where š’œ is the algebra of all Lebesgue-measurable subsets of the interval [0,+āˆž[ and šœ‚|š’œ the Lebesgue measure. Let š¼1āˆ¶=[0,1[, š¼š‘›āˆ‘āˆ¶=[š‘›āˆ’1š‘˜=1āˆ‘(1/š‘˜),š‘›š‘˜=1(1/š‘˜)[ for š‘›ā‰„2, and š‘€š‘›āˆ¶={š‘¦šœ’š¼š‘›āˆ¶š‘¦āˆˆšø} for š‘›ā‰„1. If ā‹ƒš‘€āˆ¶=āˆžš‘›=1š‘€š‘›, then šœ†(š‘€)=1, šœ”(š‘€)=0, šœ†š‘(š‘€)=šœ”š‘(š‘€)=+āˆž.

Proof. It is easy to check that šœ†(š‘€)=1 and šœ”(š‘€)=0. We are going to prove that šœ†š‘(š‘€)=+āˆž. On the contrary, suppose that šœ†š‘(š‘€)<š›¼<+āˆž then there exist a finite set šŗ={š‘§1,ā€¦,š‘§š‘}āŠ†šø and convex sets š¶1,ā€¦,š¶š‘šāŠ†šµš›¼(šæ0) such that ī€·šŗš‘€āŠ†Ī©ī€ø+āˆ©š‘†š‘šīšš‘—=1š¶š‘—.(3.2) Set š‘āˆ¶=max{ā€–š‘§š‘™ā€–šøāˆ¶š‘™=1,ā€¦,š‘}. We have ā€–š‘ ā€–0ā‰¤š‘ for all š‘ āˆˆšŗĪ©āˆ©š‘†. Fix š›¼>š‘+š›¼, a natural number š‘› such that āˆ‘š‘›š‘›=1(1/š‘›)>š‘šš›¼, and š‘¦āˆˆšø with ā€–š‘¦ā€–šø>š‘›ā‹…š›¼.
Put š‘†š‘¦ī€½āˆ¶=š‘“=š‘¦šœ’š¼š‘›āˆ¶š‘›=1,2,ā€¦,š‘›ī€¾,š‘†š‘—š‘¦ī€½āˆ¶=š‘“āˆˆš‘†š‘¦āˆ¶thereisš‘ š‘“āˆˆšŗĪ©āˆ©š‘†suchthatš‘“āˆ’š‘ š‘“āˆˆš¶š‘—ī€¾(3.3) for š‘—=1,ā€¦,š‘š.
Then it is easy to see that there exist š‘—āˆˆ{1,ā€¦,š‘š} and a subfamily {š¼š‘›1,ā€¦,š¼š‘›š‘˜} of {š¼1,ā€¦,š¼š‘›} such that š‘“š‘›š‘˜āˆˆš‘†š‘—š‘¦ for š‘˜=1,ā€¦,š‘˜ and ā‹ƒšœ‡(š‘˜š‘˜=1š¼š‘›š‘˜)ā‰„š›¼. Moreover, a straightforward computation shows that ā€–ā€–ā€–ā€–š‘˜ī“š‘˜=11š‘˜(š‘“š‘›š‘˜āˆ’š‘ š‘“š‘›š‘˜)ā€–ā€–ā€–ā€–0>š›¼,(3.4) which contradicts the convexity of the set š¶š‘—. Since šœ†š‘(š‘€)=+āˆž, the equality šœ”š‘(š‘€)=+āˆž is a consequence of Theorem 3.4.

The following lemma is crucial in the proof of Theorem 3.4.

Lemma 3.3. Let š“1,ā€¦,š“š‘›āˆˆš’œ be a partition of Ī© and š»āŠ†š‘†(š“1,ā€¦,š“š‘›). Then šœ†(š»)=š›¾š‘(š»).

Proof. Obviously, š›¾(š»)ā‰¤š›¾š‘(š»). Since šœ”(š»)=0, it follows from Theorem 2.1 and Proposition 2.2 that šœ†(š»)=š›¾(š»)=šœŽ(š»). Then it is sufficient to prove the inequality š›¾š‘(š»)ā‰¤šœŽ(š») which is trivial if šœŽ(š»)=šœ‚(Ī©).
Assume that šœŽ(š»)<š›¼<šœ‚(Ī©). By the definition of šœŽ we can find a finite set šŗāŠ†šø, containing the origin 0āˆˆšø and sets š·0āˆ¶=āˆ…, š·1,ā€¦,š·š‘ŸāŠ†Ī©, with šœ‚(š·š‘—)ā‰¤š›¼ for š‘—=1,ā€¦,š‘Ÿ, such that(i)each š·š‘— for š‘—=1,ā€¦,š‘Ÿ is the union of the members of a proper subfamily depending on š‘— of the partition {š“1,ā€¦,š“š‘›};(ii)for each š‘ āˆˆš», there is š‘—āˆˆ{0,ā€¦,š‘Ÿ} such that š‘ (Ī©ā§µš·š‘—)āŠ†šŗ+šµš›¼(šø).
For š‘—=0,ā€¦,š‘Ÿ we consider the following convex subsets of šµš›¼(šæ0): š¶š‘—š›¼ī€½ī€·š“āˆ¶=š‘ āˆˆš‘†1,ā€¦,š“š‘›ī€øī€·āˆ¶š‘ Ī©ā§µš·š‘—ī€øāŠ†šµš›¼ī€¾,(šø)(3.5) and we set š»š‘—ī€½ī€·āˆ¶=š‘ āˆˆš»āˆ¶š‘ Ī©ā§µš·š‘—ī€øāŠ†šŗ+šµš›¼ī€¾.(šø)(3.6) We will prove that š»=š‘Ÿīšš‘—=0š»š‘—āŠ†ī€ŗšŗĪ©ī€·š“āˆ©š‘†1,ā€¦,š“š‘›+ī€øī€»š‘Ÿīšš‘—=0š¶š‘—š›¼.(3.7)
It follows that š›¾š‘(š»)ā‰¤š›¼, and therefore š›¾š‘(š»)ā‰¤šœŽ(š»).
Let āˆ‘š‘ =š‘›š‘–=1š‘¦š‘–šœ’š“š‘–āˆˆš». Suppose š‘ āˆˆš»š‘—, š‘—ā‰„1 and š·š‘—=ā‹ƒš‘˜š‘™=1š“š‘–š‘™, where {š“š‘–1,ā€¦,š“š‘–š‘˜} is a proper subfamily of the partition {š“1,ā€¦,š“š‘›}.
Put {š“š‘–š‘˜+1,ā€¦,š“š‘–š‘›}āˆ¶={š“1,ā€¦,š“š‘›}ā§µ{š“š‘–1,ā€¦,š“š‘–š‘˜}.
We have š‘ ī€·Ī©ā§µš·š‘—ī€ø=ī‚†š‘¦š‘–š‘˜+1,ā€¦,š‘¦š‘–š‘›ī‚‡āŠ†šŗ+šµš›¼(šø).(3.8) Hence, for all š‘™āˆˆ{š‘˜+1,ā€¦,š‘›}, there is š‘§š‘–š‘™āˆˆšŗ such that š‘¦š‘–š‘™āˆ’š‘§š‘–š‘™āˆˆšµš›¼(šø). Then š‘ =šœ‘+ā„Ž, where šœ‘āˆ¶=š‘›ī“š‘™=š‘˜+1š‘§š‘–š‘™šœ’š“š‘–š‘™āˆˆšŗĪ©ī€·š“āˆ©š‘†1,ā€¦,š“š‘›ī€ø,ā„Žāˆ¶=š‘˜ī“š‘™=1š‘¦š‘–š‘™šœ’š“š‘–š‘™+š‘›ī“š‘™=š‘˜+1ī€·š‘¦š‘–lāˆ’š‘§š‘–š‘™ī€øšœ’š“š‘–š‘™āˆˆš¶š‘—š›¼.(3.9) Therefore, ī€ŗšŗš‘ āˆˆĪ©ī€·š“āˆ©š‘†1,ā€¦,š“š‘›ī€øī€»+š¶š‘—š›¼.(3.10) Similarly, if š‘ āˆˆš»0, we can prove that ī€ŗšŗš‘ āˆˆĪ©ī€·š“āˆ©š‘†1,ā€¦,š“š‘›ī€øī€»+š¶0š›¼.(3.11) 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 š‘€āŠ†šæ0. Then, ī€½šœ†maxš‘(š‘€),šœ”š‘ī€¾(š‘€)ā‰¤š›¾š‘(š‘€)ā‰¤šœ†(š‘€)+2šœ”š‘(š‘€).(3.12)

Proof. We first prove the left inequality which is trivial if š›¾š‘(š‘€)=+āˆž. Assume that š›¾š‘(š‘€)<š›¼<+āˆž. By Proposition 2.3, there are functions š‘ 1,ā€¦,š‘ š‘›āˆˆš‘† and convex sets š¶1,ā€¦,š¶š‘› in šµš›¼(šæ0) such that š‘€āŠ†š‘›īšš‘–=1ī€·š‘ š‘–+š¶š‘–ī€ø.(3.13)
Put ā‹ƒš¹āˆ¶=š‘›š‘–=1š‘ š‘–(Ī©) and let š“1,ā€¦,š“š‘šāˆˆš’œ be a partition of Ī© such that š‘ š‘–|š“š‘— is constant for š‘–=1,ā€¦,š‘› and š‘—=š‘–,ā€¦,š‘š. Then, ī€·š¹š‘€āŠ†Ī©ī€·š“āˆ©š‘†1,ā€¦,š“š‘š+ī€øī€øš‘›īšš‘–=1š¶š‘–,(3.14) hence šœ†š‘(š‘€)<š›¼, and šœ”š‘(š‘€)<š›¼.
Therefore, ī€½šœ†maxš‘(š‘€),šœ”š‘ī€¾(š‘€)ā‰¤š›¾š‘(š‘€).(āˆ—) We now prove the right inequality. Clearly, it is true if šœ†(š‘€)=+āˆž or šœ”š‘(š‘€)=+āˆž. Assume that šœ”š‘(š‘€)<š›½<+āˆž. By the definition of šœ”š‘, we can find a partition š“1,ā€¦,š“š‘šāˆˆš’œ and convex sets š¾1,ā€¦,š¾š‘š of šµš›½(šæ0) such that ī€·š“š‘€āŠ†š‘†1,ā€¦,š“š‘šī€ø+š‘šīšš‘—=1š¾š‘—.(3.15) Set īƒ©š»āˆ¶=š‘€āˆ’š‘šīšš‘—=1š¾š‘—īƒŖī€·š“āˆ©š‘†1,ā€¦,š“š‘šī€ø,(3.16) then we have šœ†(š»)ā‰¤šœ†(š‘€)+š›½. It easy to see that š‘€āŠ†š»+š‘šīšš‘—=1š¾š‘—.(3.17)
Therefore, by Lemma 3.3, we have that š›¾š‘(š‘€)ā‰¤š›¾š‘(š»)+š›½=šœ†(š»)+š›½ā‰¤šœ†(š‘€)+š›½+š›½,(3.18) and so š›¾š‘(š‘€)ā‰¤šœ†(š‘€)+2šœ”š‘(š‘€).(3.19) The proof is complete.

As a corollary of Theorem 3.4, we obtain the following FrƩchet-Smulian type convex total boundedness criterion.

Corollary 3.5. A subset š‘€ of šæ0 is ctb if and only if šœ†(š‘€)=šœ”š‘(š‘€)=0.

Remark 3.6. In [12], Trombetta proved that ī€½šœ†maxš‘(š‘€),šœ”š‘ī€¾(š‘€)ā‰¤š›¾š‘(š‘€)ā‰¤šœ†š‘(š‘€)+2šœ”š‘(š‘€)(3.20) for a subset š‘€āŠ†šæ0(Ī©,š’œ,ā„,šœ‚). Since šœ†(š‘€)ā‰¤šœ†š‘(š‘€), Theorem 3.4 improves and generalizes to šø-valued case the above inequalities.

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(šø)=+āˆž, it might exist some š‘–āˆˆ{1,ā€¦,š‘›} such that the set š»(š“š‘–)āˆ¶={š‘ (š“š‘–)āˆ¶š‘ āˆˆš»} is bounded but not necessarily totally bounded.

The following example shows how the value of the parameters šœ† and šœ†š‘ changes when passing from the scalar case to the šø-valued case.

Example 3.7. Let š‘€āˆ¶={š‘¦šœ’Ī©āˆ¶š‘¦āˆˆšµ(šø)}, where šµ(šø)={š‘¦āˆˆšøāˆ¶ā€–š‘¦ā€–šøā‰¤1}. Then šœ†(š‘€)=šœ†š‘(š‘€)=0 if šø is finite dimensional, šœ†(š‘€)=šœ†š‘(š‘€)=1, otherwise. Moreover, since šœ”š‘(š‘€)=0, we have šœ†(š‘€)=šœ†š‘(š‘€)=š›¾š‘(š‘€).

Corollary 3.8. Let š‘€āŠ†šæ0. Then, ī€½maxšœ†(š‘€),šœ”š‘ī€¾ī‚†šœ†ī‚€(š‘€)ā‰¤maxš‘€ī‚,šœ”š‘ī‚€š‘€ī‚†šœ†ī‚ī‚‡ā‰¤maxš‘ī‚€š‘€ī‚,šœ”š‘ī‚€š‘€ī‚ī‚‡ā‰¤šœ†(š‘€)+2šœ”š‘(š‘€).(3.21)

Proof. It is sufficient to observe that š›¾š‘(š‘€)=š›¾š‘(š‘€).
In particular, š‘€ ctb implies šœ†(š‘€)=šœ†(š‘€)=0, šœ†š‘(š‘€)=šœ†š‘(š‘€)=0, and šœ”š‘(š‘€)=šœ”š‘(š‘€)=0.

The next two corollaries of Theorem 3.4 are useful in order to compute or to estimate š›¾š‘ in particular classes of subsets of šæ0. Moreover, the second one generalizes [12, Proposition 3.10].

Corollary 3.9. Let š‘€ be a convexly equi-measurable subset of šæ0. Then, šœ†(š‘€)=š›¾š‘(š‘€).

Corollary 3.10. Let š‘€ be an equi-quasibounded subset of šæ0. Then, ī€½šœ†maxš‘(š‘€),šœ”š‘ī€¾(š‘€)ā‰¤š›¾š‘(š‘€)ā‰¤2šœ”š‘(š‘€).(āˆ—āˆ—)

We observe that if š¾ is a totally bounded subset of šø and š‘€āˆ¶={š‘“āˆˆšæ0āˆ¶š‘“(Ī©)āŠ†š¾}, then, since šœ†(š‘€)=0, the inequalities (3.21) are true for š‘€.

Example 3.11. Let šæ0 be the space of Example 3.2 and assume that šø is the Banach space š‘™āˆž of all sequences š‘¦=(šœ‰1,šœ‰2,ā€¦) with finite norm ā€–š‘¦ā€–āˆžāˆ¶=sup{|šœ‰š‘›|āˆ¶š‘›=1,2,ā€¦}. If š¾āˆ¶={š‘¦āˆˆš‘™āˆžāˆ¶|šœ‰š‘›|ā‰¤1/š‘›forš‘›=1,2,ā€¦} and š‘€āˆ¶={š‘“āˆˆšæ0āˆ¶š‘“(Ī©)āŠ†š¾}, it is easy to prove that šœ†š‘(š‘€)=0 and šœ”š‘(š‘€)=š›¾š‘(š‘€)=1. If šµ(š‘™āˆž) is the closed unit ball of š‘™āˆž and š‘€āˆ¶={š‘“āˆˆšæ0āˆ¶š‘“(Ī©)āŠ†šµ(š‘™āˆž)}, the set š‘€ satisfies šœ†(š‘€)=šœ†š‘(š‘€)=š›¾š‘(š‘€)=šœ”š‘(š‘€)=1.