We estimate the measure of nonconvex total boundedness in terms of
simpler quantitative characteristics in the space of measurable functionsL0. A Fréchet-Smulian type compactness criterion for convexly totally
bounded subsets of L0 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 f:K→Kdefined 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 x1,…,xn∈K and convex subsets C1,…,Cn of U such that K⊆⋃i=1n(xi+Ci). 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., [6–8]).

Let (E,∥·∥E) be a normed space, Ω a nonempty set, and 𝒫(Ω) the power set of Ω. The space L0 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 L0 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 M⊆L0, the Hausdorff measure of noncompactness γ(M) of M is defined by
γ(M):=inf{⋃i=1nɛ>0:thereexistfunctionsf1,…,fn∈L0suchthatM⊆⋃i=1n(fi+Bɛ(L0))},
where Bɛ(L0)≔{f∈L0:∥f∥0≤ɛ}. 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 L0 and to characterize the convexly totally bounded subsets of L0. At this end we introduce two quantitative characteristics λc(M) and ωc(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 γc(M), λc(M), ωc(M), and λ(M) that give, as a special case, a Fréchet-Smulian type convex total boundedness criterion in the space L0. 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 inf∅:=+∞. Moreover, for two sets A and B, we denote by BA the set of all maps from A to B. We recall the definition of the space L0 (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‖f‖0:=inf{a>0:η({x∈Ω:‖f(x)‖E≥a})≤a}
defines a group pseudonorm on EΩ, that is, ∥0∥0=0, ∥-f∥0=∥f∥0, ∥f+g∥0≤∥f∥0+∥g∥0, for all f,g∈EΩ. Let L0:=L0(Ω,𝒜,E,η) be the closure of the linear space S:=span{yχA:y∈EandA∈𝒜} of E-valued 𝒜-simple functions in (EΩ,∥·∥0), where χA is the characteristics function of A. Identification of functions f,g∈EΩ for which ∥f-g∥0=0 turns (L0,∥·∥0) (the space of measurable functions) into an F-normed linear space in the sense of [19, page 38], that is, ∥f+g∥0≤∥f∥0+∥g∥0, limn→∞∥(1/n)f∥0=0, ∥λf∥0≤∥f∥0 for f,g∈L0 and |λ|≤1.

Let M⊆L0. The measure of nonconvex total boundedness γc(M) of M is defined by γc(M):=inf{⋃i=1n(fi+Ci)ɛ>0:thereexistfunctionsf1,…,fn∈L0andconvexsubsetsC1,…,CnofBɛ(L0)suchthatM⊆⋃i=1n(fi+Ci)}.
Clearly M is ctb if and only if γc(M)=0.

Let A1,…,Am∈𝒜 be a partition of Ω. We set
S(A1,…,Am):={s∈S:s=∑i=1myiχAi,whereyi∈Efori=1,…,m}.
In [11] the following two quantitative characteristic λ and ω are used to estimate γ in L0: λ(M):=inf{ɛ>0:thereexistsafinitesubsetGofEsuchthatM⊆(GΩ∩S)+Bɛ(L0)},ω(M):=inf{Amɛ>0:thereexistsapartitionA1,…,Am∈AofΩsuchthatM⊆S(A1,…,Am)+Bɛ(L0)Am}.

The set M is called equi-quasibounded if λ(M)=0 and equi-measurable if ω(M)=0.

Theorem 2.1 (see [<xref ref-type="bibr" rid="B17">11</xref>, Theorem 2.2.2]).

Let M⊆L0. Then
max{λ(M),ω(M)}≤γ(M)≤λ(M)+2ω(M).
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):σ(M):=inf{ɛ>0:thereexistsafinitesubsetGofEsuchthatforallf∈MthereisDf⊆Ωwithη(Df)≤ɛandf(Ω∖Df)⊆G+Bɛ(E)},
where Bɛ(E):={y∈E:∥y∥E≤ɛ}.

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

Proposition 2.2.

Let M⊆L0. Then λ(M)=σ(M).

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

Proposition 2.3.

Let M⊆L0. Then
γc(M)=inf{⋃i=1nɛ>0:thereexistfunctionss1,…,sn∈SandconvexsubsetsC1,…,CnofBɛ(L0)suchthatM⊆⋃i=1n(si+Ci)}.

3. Inequalities in the Space <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M110"><mml:mrow><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

In order to estimate the measure of nonconvex total boundedness in L0, we introduce the following two quantitative characteristics.

Definition 3.1.

Let M⊆L0. We define the following:
λc(M):=inf{⋃i=1nε>0:thereexistafinitesubsetGofEandconvexsubsetsC1,…,CnofBɛ(L0)suchthatM⊆(GΩ∩S)+⋃i=1nCi},ωc(M):=inf{⋃i=1nε>0:thereexistapartitionA1,…,Am∈AofΩandconvexsubsetsC1,…,CnofBɛ(L0)suchthatM⊆S(A1,…,Am)+⋃i=1nCi}.

We call M convexly equi-quasibounded if λc(M)=0 and convexly equi-measurable if ωc(M)=0.

We observe that if E=ℝ, then the quantitative characteristics λc and ωc coincide with those introduced in [12].

We point out that the request of convexity plays a crucial role in the definition of the parameters λc and ωc, whereas it was not involved in the definition of λ and ω. We illustrate this with the following example.

Example 3.2.

Let L0:=L0([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 I1:=[0,1[, In:=[∑k=1n-1(1/k),∑k=1n(1/k)[ for n≥2, and Mn:={yχIn:y∈E} for n≥1. If M:=⋃n=1∞Mn, then λ(M)=1, ω(M)=0, λc(M)=ωc(M)=+∞.

Proof.

It is easy to check that λ(M)=1 and ω(M)=0. We are going to prove that λc(M)=+∞. On the contrary, suppose that λc(M)<α<+∞ then there exist a finite set G={z1,…,zp}⊆E and convex sets C1,…,Cm⊆Bα(L0) such that
M⊆(GΩ∩S)+⋃j=1mCj.
Set c:=max{∥zl∥E:l=1,…,p}. We have ∥s∥0≤c for all s∈GΩ∩S. Fix α¯>c+α, a natural number n¯ such that ∑n=1n¯(1/n)>mα¯, and y¯∈E with ∥y¯∥E>n¯·α¯.

Put
Sy¯:={f=y¯χIn:n=1,2,…,n¯},Sy¯j:={f∈Sy¯:thereissf∈GΩ∩Ssuchthatf-sf∈Cj}
for j=1,…,m.

Then it is easy to see that there exist j¯∈{1,…,m} and a subfamily {In1,…,Ink¯} of {I1,…,In¯} such that fnk∈Sy¯j¯ for k=1,…,k¯ and μ(⋃k=1k¯Ink)≥α¯. Moreover, a straightforward computation shows that
‖∑k=1k¯1k¯(fnk-sfnk)‖0>α,
which contradicts the convexity of the set Cj¯. Since λc(M)=+∞, the equality ωc(M)=+∞ is a consequence of Theorem 3.4.

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

Lemma 3.3.

Let A1,…,An∈𝒜 be a partition of Ω and H⊆S(A1,…,An). Then λ(H)=γc(H).

Proof.

Obviously, γ(H)≤γc(H). Since ω(H)=0, it follows from Theorem 2.1 and Proposition 2.2 that λ(H)=γ(H)=σ(H). Then it is sufficient to prove the inequality γc(H)≤σ(H) which is trivial if σ(H)=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 D0:=∅, D1,…,Dr⊆Ω, with η(Dj)≤α for j=1,…,r, such that

each Dj for j=1,…,r is the union of the members of a proper subfamily depending on j of the partition {A1,…,An};

for each s∈H, there is j∈{0,…,r} such that s(Ω∖Dj)⊆G+Bα(E).

For j=0,…,r we consider the following convex subsets of Bα(L0):
Cαj:={s∈S(A1,…,An):s(Ω∖Dj)⊆Bα(E)},
and we set
Hj:={s∈H:s(Ω∖Dj)⊆G+Bα(E)}.
We will prove that
H=⋃j=0rHj⊆[GΩ∩S(A1,…,An)]+⋃j=0rCαj.

It follows that γc(H)≤α, and therefore γc(H)≤σ(H).

Let s=∑i=1nyiχAi∈H. Suppose s∈Hj, j≥1 and Dj=⋃l=1kAil, where {Ai1,…,Aik} is a proper subfamily of the partition {A1,…,An}.

Put {Aik+1,…,Ain}:={A1,…,An}∖{Ai1,…,Aik}.

We have
s(Ω∖Dj)={yik+1,…,yin}⊆G+Bα(E).
Hence, for all l∈{k+1,…,n}, there is zil∈G such that yil-zil∈Bα(E). Then s=φ+h, where
φ:=∑l=k+1nzilχAil∈GΩ∩S(A1,…,An),h:=∑l=1kyilχAil+∑l=k+1n(yil-zil)χAil∈Cαj.
Therefore,
s∈[GΩ∩S(A1,…,An)]+Cαj.
Similarly, if s∈H0, we can prove that
s∈[GΩ∩S(A1,…,An)]+Cα0.
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 M⊆L0. Then,
max{λc(M),ωc(M)}≤γc(M)≤λ(M)+2ωc(M).

Proof.

We first prove the left inequality which is trivial if γc(M)=+∞. Assume that γc(M)<α<+∞. By Proposition 2.3, there are functions s1,…,sn∈S and convex sets C1,…,Cn in Bα(L0) such that
M⊆⋃i=1n(si+Ci).

Put F:=⋃i=1nsi(Ω) and let A1,…,Am∈𝒜 be a partition of Ω such that si|Aj is constant for i=1,…,n and j=i,…,m. Then,
M⊆(FΩ∩S(A1,…,Am))+⋃i=1nCi,
hence λc(M)<α, and ωc(M)<α.

Therefore,
max{λc(M),ωc(M)}≤γc(M).
We now prove the right inequality. Clearly, it is true if λ(M)=+∞ or ωc(M)=+∞. Assume that ωc(M)<β<+∞. By the definition of ωc, we can find a partition A1,…,Am∈𝒜 and convex sets K1,…,Km of Bβ(L0) such that
M⊆S(A1,…,Am)+⋃j=1mKj.
Set
H:=(M-⋃j=1mKj)∩S(A1,…,Am),
then we have λ(H)≤λ(M)+β. It easy to see that
M⊆H+⋃j=1mKj.

Therefore, by Lemma 3.3, we have that
γc(M)≤γc(H)+β=λ(H)+β≤λ(M)+β+β,
and so
γc(M)≤λ(M)+2ωc(M).
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 M of L0 is ctb if and only if λ(M)=ωc(M)=0.

Remark 3.6.

In [12], Trombetta proved that
max{λc(M),ωc(M)}≤γc(M)≤λc(M)+2ωc(M)
for a subset M⊆L0(Ω,𝒜,ℝ,η). Since λ(M)≤λc(M), Theorem 3.4 improves and generalizes to E-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(E)=+∞, it might exist some i∈{1,…,n} such that the set H(Ai):={s(Ai):s∈H} is bounded but not necessarily totally bounded.

The following example shows how the value of the parameters λ and λc changes when passing from the scalar case to the E-valued case.

Example 3.7.

Let M:={yχΩ:y∈B(E)}, where B(E)={y∈E:∥y∥E≤1}. Then λ(M)=λc(M)=0 if E is finite dimensional, λ(M)=λc(M)=1, otherwise. Moreover, since ωc(M)=0, we have λ(M)=λc(M)=γc(M).

Corollary 3.8.

Let M⊆L0. Then,
max{λ(M),ωc(M)}≤max{λ(M̅),ωc(M̅)}≤max{λc(M̅),ωc(M̅)}≤λ(M)+2ωc(M).

Proof.

It is sufficient to observe that γc(M)=γc(M̅).

In particular, M ctb implies λ(M)=λ(M̅)=0, λc(M)=λc(M̅)=0, and ωc(M)=ωc(M̅)=0.

The next two corollaries of Theorem 3.4 are useful in order to compute or to estimate γc in particular classes of subsets of L0. Moreover, the second one generalizes [12, Proposition 3.10].

Corollary 3.9.

Let M be a convexly equi-measurable subset of L0. Then, λ(M)=γc(M).

Corollary 3.10.

Let M be an equi-quasibounded subset of L0. Then,
max{λc(M),ωc(M)}≤γc(M)≤2ωc(M).

We observe that if K is a totally bounded subset of E and M:={f∈L0:f(Ω)⊆K}, then, since λ(M)=0, the inequalities (3.21) are true for M.

Example 3.11.

Let L0 be the space of Example 3.2 and assume that E is the Banach space l∞ of all sequences y=(ξ1,ξ2,…) with finite norm ∥y∥∞:=sup{|ξn|:n=1,2,…}. If K:={y∈l∞:|ξn|≤1/nforn=1,2,…} and M:={f∈L0:f(Ω)⊆K}, it is easy to prove that λc(M)=0 and ωc(M)=γc(M)=1. If B(l∞) is the closed unit ball of l∞ and M:={f∈L0:f(Ω)⊆B(l∞)}, the set M satisfies λ(M)=λc(M)=γc(M)=ωc(M)=1.

IdzikA.On γ-almost fixed point theorems. The single-valued caseGranasA.MauldinR. D.KKK-maps and their applications to nonlinear problemsDe PascaleE.TrombettaG.WeberH.Convexly totally bounded and strongly totally bounded sets. Solution of a problem of IdzikTrombettaG.A compact convex set not convexly totally boundedWeberH.Compact convex sets in non-locally convex linear spacesKimW. K.DingX.On generalized weight Nash equilibria for generalized multiobjective gamesParkS.Almost fixed points of multimaps having totally bounded rangesParkS.KangB. G.Generalized variational inequalities and fixed point theoremsDunfordN.SchwartzJ. T.AppellJ.De PascaleE.Some parameters associated with the Hausdorff measure of noncompactness in spaces of measurable functionsTrombettaG.WeberH.The Hausdorff measure of noncompactness for balls of F-normed linear spaces and for subsets of L0TrombettaG.The measures of nonconvex total boundedness and of nonstrongly convex total boundedness for subsets of L0CautyR.Solution du problème de point fixe de SchauderCautyR.Un théorème de point fixe pour les fonctions multivoques acycliquesIsacG.Erratum: on Rothe's fixed point theorem in a general topological vector spaceAskouraY.Godet-ThobieC.Fixed points in contractible spaces and convex subsets of topological vector spacesParkS.Remarks on recent results in analytical fixed point theoryAvalloneA.TrombettaG.Measures of noncompactness in the space L0 and a generalization of the Arzelà-Ascoli theoremJarchowH.