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Lineability of nowhere monotone measures.

1 Introduction

Assume that M is a subset of a topological vector space X and [alpha] is a cardinal number. Then M is called lineable if M [union] {0} contains an infinite-dimensional linear subspace. More specifically, M is called [alpha]-lineable if M [union] {0} contains an a-dimensional linear subspace. If M [union] {0} contains a closed infinite-dimensional linear subspace it is called spaceable. If the set M [union] {0} contains an infinite-dimensional subspace that is dense in X, the set M is called dense-lineable. If M is dim(X)-(dense)lineable, we call it maximal (dense)-lineable. Finally, if M [union] {0} contains an infinitely generated algebra we call it algebrable.

The concept of lineability was coined by V. I. Gurariy in the early 2000's and it first appeared in print in [Gurariy and Quarta(2004)], [Aron et al.(2005)] and [Seoane-SepUlveda(2006)]. Note, however, that V. I. Gurariy's interest in linear structures in generally non-linear settings dates as far back as 1966 (see [Gurariy(1966)]). The study of large vector structures in sets of real and complex functions has attracted many mathematicians in the last decade. For example, in [Aron et al.(2005), Theorem 4.3], the authors prove that the set of everywhere surjective functions on R is [2.sup.[absolute value of R]]-lineable. This result has been further improved in [Gamez-Merino et al.(2010)], where the authors prove that the set of strongly everywhere surjective functions and the set of perfectly everywhere surjective functions are both [2.sup.[absolute value of R]]-lineable as well. In the same paper, the authors show that the set NMD of nowhere monotone everywhere differentiable functions on R is [absolute value of R]-lineable (this result has been a motivation for our study of lineability of the set of nowhere monotone measures).

As more and more examples of lineable sets were found, the questions of dense-lineability and spaceability gradually attracted more attention. In [Aron et al. (2009)] the authors show, among other results, that the set of nowhere differentiable functions, the set of non-analytic [C.sup.[infinity]] functions, the set of functions in [C.sup.m] \ [C.sup.n] (with m < n) and the set NM D (all considered on a non-empty closed interval [a, b]) are all dense-lineable in C([a, b]). In [Garcia et al. (2010)] the authors construct a Banach spaces of non-Riemann-integrable bounded functions that have an antiderivative at each point point of an interval, a Banach space of differentiable functions on [R.sup.d] failing the Denjoy-Clarkson property and a Banach space of infinitely differentiable functions that vanish at infinity and are not the Fourier transform of any Lebesgue-integrable function. In [Aron et al. (2006)], it is shown that given any Lebesgue-null subset J [subset] T the set of functions in C(T) whose Fourier series diverges in J contains a dense infinitely generated algebra. For some other recent results concerning algebrability see, for example, [Bayart and Quarta (2007)], [Aron et al. (2010)], or [Garcia-Pacheco et al. (2007)].

Even though most of the results on lineability focus on the study of sets of functions, papers concerning lineability of vector measures [Munoz-Fernandez et al.(2008)] and lineability of operators [Puglisi and Seoane-Sepulveda(2008)] have also been published. Some results concerning general properties of lineable sets have appeared, despite the fact that most literature deals with specific sets of functions or operators. For example, in [Aron et al. (2009), Theorem 2.2] the authors present a sufficient condition on a lineable subset of a separable Banach space to be dense-lineable. This technique allows for short, elegant proofs of several of the theorems mentioned above. These conditions have been further studied in a recent article [Bernal-Gonzalez and Cabrera(2014)] in which the authors prove an analogous result in the setting of general vector spaces. In another recent article (see [Ciesielski et al.(2014)]), the authors present the concept of maximal lineability cardinal number and use it to prove several results concerning lineability of specific sets of real functions. Lastly, we recommend the recently published survey [Bernal-Gonzalez et al.(2014)] to the interested reader. It provides an exceptionally well arranged list of the most important results in the field, far beyond the scope of this article.

2 Notation

For d [member of] N we denote by Ad the d-dimensional Lebesgue measure (we write [lambda] instead of [[lambda].sub.1]). For a function g : [R.sup.d] [right arrow] R we denote by {g > 0} the set of all x [member of] [R.sup.d] such that g(x) > 0. By c or [absolute value of R] we denote the cardinality of the set of real numbers. Furthermore, for P either R or a non-empty closed interval of R we denote by C(P) the set of all real continuous functions on P endowed with the uniform norm ||f||. For a locally compact Hausdorff space X we denote by [C.sub.c] (X) the set of all real continuous functions on X with compact support.

For a signed measure [mu] we denote by [[mu].sup.+] and [[mu].sup.-] the positive and negative variation of [mu], respectively. We denote by M(X) the set of all signed Radon measures on a locally compact Hausdorff space X endowed with the norm ||[mu]|| = [[mu].sup.+] (X) + [[mu].sup.-] (X). For [mu] [member of] M(X) we denote the support of [mu] by supp [mu]. Moreover, if [mu] and v are two measures on X such that for any open set G [subset] X, v(G) = 0 implies [mu](G) = 0, we write [mu] [much less than] v and say that [mu] is absolutely continuous with respect to v. For any given measure v [member of] M(X) we denote the closed subspace {[mu] [member of] M(X) : [mu] [much less than] v} by A[C.sub.v](X).

We recall the definitions of density topology and of approximately continuous functions here for the reader's convenience. A measurable set B [subset] [R.sup.d] is called density open if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for every x [member of] B. It is clear that every euclidean open set is also density open, that is, the density topology is finer than the euclidean topology.

A function F : [R.sup.d] [right arrow] R is called approximately continuous if the sets {F > [beta]} and {F < [beta]} are density open for all [beta] [member of] R. There are several mutually equivalent ways how to define approximately continuous functions. The reader interested in the density topology and in the properties of approximately continuous functions should consult [Bruckner(1994)], [Lukes and Maly(2005)] or [Lukes et al.(1986)] where these topics are presented in detail.

3 Nowhere Monotone Measures, Spaces with Humps

Let us start with a definition.

Definition 3.0.1. Let X be a locally compact Hausdorff space and y be a Radon measure on X. We say that y is nowhere monotone if [[mu].sup.+] (G) > 0 and [[mu].sup.-] (G) > 0 for every non-empty open set G [subset] X.

The concept of nowhere monotone measures has its origins in works concerning Choquet theory (see [McDonald(1971)], [McDonald(1973)]). However, the concept closely relates to that of nowhere monotone functions. Indeed, suppose that f is a nowhere monotone function of bounded variation on [0,1]. Then f is a distribution function of some nowhere monotone Lebesgue-Stieltjes measure. This example can be, in fact, taken as a direct motivation for defining these measures.

Once the relation between nowhere monotone functions and nowhere monotone measures has been established, various questions arise. We have already, albeit informally, established that such measures exist. In the following section we will prove that these measures form a residual set in the space of signed Radon measures M(X) on specific Hausdorff spaces. Note that this is analogous to the density of the set of nowhere monotone functions in C([a,b]) (see [Aron et al.(2009), Theorem 3.3]). To prove this result we will introduce the notion of spaces with humps.

Definition 3.0.2. Let A be a closed subspace of M (X). We say that A has humps, if there exists q [member of] (0,1) such that for every non-empty open set G [subset] X there exists [mu] [member of] A, such that ||[mu]|| = 1 and [member of] (G) [greater than or equal to] q.

The idea of humps first appeared in a slightly less general form in a Bachelor thesis of M. Kolar (see [Kolar(2009)]).

The following lemma establishes that for certain [mu] [member of] M (X) the spaces A[C.sub.[mu]] (X) have humps. This will prove useful later, when we study lineability of nowhere monotone measures.

Lemma 3.0.3. Suppose that p0 is a locally finite measure on X such that supp [[mu].sub.0] = X. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (X) has humps.

Proof. Let G [subset] X be a non-empty open set such that [absolute value of [[mu].sub.0] (G)] is finite and [absolute value of [[mu].sub.0]](G) =: [alpha] > 0. We may assume that [[mu].sup.+.sub.0] (G) [greater than or equal to] [alpha]/2 (otherwise consider -[[mu].sub.0] instead of [[mu].sub.0]). By inner regularity of [[mu].sup.+.sub.0] and [[mu].sup.-.sub.0] there exist disjoint compact sets [K.sup.+] [subset] G and [K.sup.-] [subset] G such that [[mu].sub.0] = [[mu].sup.+.sub.0] on [K.sup.+], [[mu].sub.0] = [[mu].sup.-.sub.0] on [K.sup.-] and [[mu].sup.+.sub.0] ([K.sup.+]) + [[mu].sup.-.sub.0] ([K.sup.-]) > 3[alpha]/4. Since X is Hausdorff, there exist disjoint open sets V and W such that [K.sup.+] [subset] V [subset] G and [K.sup.-] [subset] W [subset] G. By Urysohn's lemma we can construct functions f, g [member of] [C.sub.c] (X), 0 [less than or equal to] f [less than or equal to] 1, 0 [less than or equal to] g [less than or equal to] 1, f = 1 on [K.sup.+] and f = 0 on X \ V, g = 1 on [K.sup.~] and g = 0 on X \ W. Let h = f - g and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then ||[mu]|| [less than or equal to] [alpha] and

[mu](G) [greater than or equal to] [[mu].sup.+.sub.0] ([K.sup.+]) + [[mu].sup.-.sub.0] ([K.sup.-]) - [absolute value of [[mu].sub.0]] (G([K.sup.+] [union] [K.sup.-])) [greater than or equal to] 3[alpha]/4 - [alpha]/4 = [alpha]/2

It easily follows that the measure v := [mu]/||[mu]|| has all the desired properties.

Theorem 3.0.4. Let X be a locally compact second-countable Hausdorff space with no isolated points and let A [subset or equal to] M (X) be a closed subspace that has humps. Then the set of all nowhere monotone Radon signed measures on X is residual in A.

Proof. Let [{[G.sub.n]}.sub.n[member of]N] be a countable basis of open sets in X and denote by N(X) the set of all nowhere monotone measures in X. Define

[E.sup.+.sub.n] := {[mu] [member of] A : [[mu].sup.+] ([G.sub.n]) = 0} and [E.sup.-.sub.n] := {[mu] [member of] A : [[mu].sup.-] ([G.sub.n]) = 0}.

Clearly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Fix n [member of] N. To prove that [E.sup.+.sub.n] is closed, consider a sequence [{[[mu].sub.k]}.sub.k[member of]N] in [E.sup.+.sub.n] such that ||[[mu].sub.k] - [mu]|| [right arrow] 0 for some [mu] [member of] A. Then also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore (see [Lukes and Maly(2005), Theorem 17.4])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To prove that the interior of [E.sup.+.sub.n] is empty fix [mu] [member of] [E.sup.+.sub.n], [epsilon] > 0 and q [member of] (0,1) from Definition 3.0.2. Since X is Hausdorff and has no isolated points, [G.sub.n] is infinite. Therefore there exists z [member of] [G.sub.n] such that [[mu].sup.-] ({z}) < q[epsilon]/2. By regularity of [mu] there exists an open set G [subset] [G.sub.n] such that z [member of] G and [[mu].sup.-] {G) < q[epsilon]/2. Since A has humps, we can find v [epsilon] A such that ||v|| = [epsilon] and v(G) [greater than or equal to] q[epsilon]. Set [gamma] := [mu] + v. Then ||[mu] - [gamma]|| = [epsilon] and [[gamma].sup.+] ([G.sub.n]) [greater than or equal to] [[gamma].sup.+](G) [greater than or equal to] q[epsilon] - q[epsilon]/2 > 0, which means that [gamma] [not member of] [E.sup.+.sub.n].

The sets [E.sup.+.sub.n] are closed and their interiors are empty, therefore they are nowhere dense. Using a similar argument we can prove that the sets [E.sup.-.sub.n] have the same properties.

Remark 3.0.5. Letting A = M(X) in the previous theorem answers the question of existence of nowhere monotone measures in M(X) for suitable choices of X. We will, however, use Theorem 3.0.4 to prove the existence and lineability of a more specific type of nowhere monotone measures.

A question one might ask at this point is under what assumptions can we expect A to have humps. It follows immediately from definition that no finite-dimensional subspace of M (X) can have this property. It is also not difficult to come up with an example of a subspace generated by a countably infinite sequence of measures that does (for example, for X = R consider the subspace A = [bar.span] {[[epsilon].sub.q], q [member of] Q}). However, not every infinitely generated closed subspace of M (X) has to have humps. Take for example A = [bar.span] {[[mu].sub.1], [[mu].sub.2], ...} such that [[union].sub.n] supp [[mu].sub.n] [subset] X \ G, where G is a non-empty open subset of X. But even when the the sets supp [[mu].sub.n] form a covering of X the space A need not have humps. We will formulate this as a separate result.

Theorem 3.0.6. There exists an infinite-dimensional space A [subset] M([0,1]), A = [bar.span] {[[mu].sub.1], [[mu].sub.2], ...} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

A does not have humps and the set of all nowhere monotone measures on [0,1] is residual in A.

Proof. Define the sequence of generating measures as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [I.sub.n] := [1 - [2.sup.n-1], 1 - [2.sup.n]]. It is obvious that (1) holds. To prove that A does not have humps, pick 0 < [epsilon] < 1/2 and find an open interval J [subset] [I.sub.1] such that [lambda](J) < [epsilon]. For any v [member of] span {[[mu].sub.1], [[mu].sub.2], ...}, ||v|| = 1 we have

[absolute value of v(J)] < [[mu].sub.1] (J) < 2[epsilon].

For a general [bar.v] [member of] A, ||[bar.v]|| = 1 find a sequence

[{[v.sub.k]}.sup.[infinity].sub.k=1] [subset] span{[[mu].sub.1], [[mu].sub.1], ...}, ||[v.sub.k]|| = 1, k [member of] N,

such that ||[bar.v] - [v.sub.k]|| [right arrow] 0. Then also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Using [Lukes and Maly(2005), Theorem 17.4] and the previous paragraph we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the same argument for -[bar.v] instead of [bar.v] yields the same inequality for [[bar.v].sup.-] (J). Hence we have

[absolute value of [bar.v](J)] < 2[epsilon].

This shows that A does not have humps.

Since the proof of residuality of nowhere monotone measures in A is very similar to the proof of Theorem 3.0.4 we will make free use of its notation. In fact, the proof of closedness of the sets [E.sup.+.sub.n] is exactly the same. To prove that these sets have empty interior, pick [mu] [member of] [E.sup.+.sub.n] and [epsilon] > 0. Since the support sets of generating measures form a covering of [0,1], there exists an interval [I.sub.m] such that [G.sub.m] [intersection] [I.sub.m] [not equal to] [empty set]. Set [kappa] = [mu] + [epsilon][[mu].sub.m]. Then ||[mu] - [kappa]|| = [epsilon] and [[kappa].sup.+] ([G.sub.m]) = [epsilon][[mu].sub.n] ([G.sub.m]) > 0, which means that [kappa] [not member of] [E.sup.+.sub.n]. This finishes the proof.

4 Lineability of Nowhere Monotone [[lambda].sub.d]-differentiable Measures

In this section we will prove that not only there exists a nowhere monotone measure On [R.sup.d] that is a.e. differentiable with respect to the d-dimensional Lebesgue measure but also that the set of all such measures is [absolute value of R]-lineable.

Corollary 4.0.1. There exists a nowhere monotone Radon signed measure [mu] on [R.sup.d] that is absolutely continuous with respect to Lebesgue measure [[lambda].sub.d].

Proof. According to Lemma 3.0.3 the set {[mu] [member of] M([R.sup.d]) : [mu] [much less than] [[lambda].sub.d]} has humps. It remains to use Theorem 3.0.4.

To prove the existence of an everywhere differentiable nowhere monotone measure we need the following theorem (for details see [Lukes et al.(1986), Chapter 3]).

Theorem 4.0.2. Let E [subset] [R.sup.d] be a density open [F.sub.[sigma]] set. Then there exists an approximately continuous function [phi], 0 [less than or equal to] [phi] [less than or equal to] 1, such that {[phi] > 0} = E.

Theorem 4.0.3. Let f : [R.sup.d] [right arrow] R be a measurable function. There exists a Radon measure [mu] on [R.sup.d] that is absolutely continuous with respect to [[lambda].sub.d] and such that D[mu]/D[[lambda].sub.d] exists everywhere on [R.sup.d]. Moreover,

P:= {D{mu]/D[[lambda].sub.d] > 0} [subset] {f > 0}, N := {D{mu]/D[[lambda].sub.d] > 0} [subset] {f < 0}

and

[[lambda].sub.d]({f > 0}\P)= [[lambda].sub.d]({f < 0}\N)= 0.

Proof. Let [??] be the interior of the set {f > 0} in the density topology. By the Lebesgue density theorem, [[lambda].sub.d]({f > 0} \ [??]) = 0. By regularity of Lebesgue measure, there exists an [F.sub.[sigma]] set P [subset] [??] such that

[[lambda].sub.d]({f > 0}\ P)= [[lambda].sub.d]([??] \ P) = 0.

It is obvious that P is also open in the density topology. By Theorem 4.0.2 there exists an approximately continuous function [phi] such that 0 [less than or equal to] [phi] [less than or equal to] 1, {[phi] > 0} = P. We may assume that [phi] [member of] [L.sup.1] ([R.sup.d]) (otherwise consider the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] instead).

Similarly, there exists a density open [F.sub.[sigma]] set N such that N [subset] {f < 0},

[[lambda].sub.d]({f < 0} \ N) = 0

and an approximately continuous function [psi] [member of] [L.sup.1] ([R.sup.d]) such that 0 [less than or equal to] [psi] [less than or equal to] 1 and {[phi] > 0} = N. Let [mu] be a measure on [R.sup.d] such that = D[mu]/D[[lambda].sub.d] = [psi] - [phi]. It is easy to check that the sets P, N and the measure [mu] have all the desired properties.

Theorem 4.0.4. There exists a nowhere monotone Radon signed measure [mu] on [R.sup.d] that is everywhere differentiable with respect to [[lambda].sub.d].

Proof. Let v be a nowhere monotone measure that is absolutely continuous with respect to [[lambda].sub.d] (the existence of such a measure is guaranteed by Corollary 4.0.1) and let f : [R.sup.d][right arrow] R be a representative of the Radon-Nikodym derivative Dv/D[[lambda].sub.d]. It remains to use the previous theorem to finish the proof.

Remark 4.0.5. We obtain an even stronger result in the case d = 1. The distribution function of the measure constructed in the previous theorem is then even everywhere differentiable in the classical sense.

Theorem 4.0.6. The set of all nowhere monotone almost everywhere differentiable Radon signed measures on [R.sup.d] is [absolute value of R]-lineable.

Proof First, put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], x [member of] [R.sup.d], for [alpha] [member of] R. According to Theorem 4.0.4, there exists a nowhere monotone measure [mu] such that D[mu]/D[[lambda].sub.d] = f exists everywhere on [R.sup.d]. Define the measures [[mu].sub.[alpha]] for [alpha] [member of] R as follows

D[[mu].sub.[alpha]]/D[[lambda].sub.d] := f x [f.sub.[alpha]]. (2)

Since

{(f x [f.sub.[alpha]]) > 0} = {f > 0} [intersection] {[f.sub.[alpha]] [not equal to] 0}, {(f x [f.sub.[alpha]]) < 0} = {f < 0} [intersection] {[f.sub.[alpha]] [not equal to] 0},

the measures pa are nowhere monotone. Consider a measure

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some n [member of] N and some [b.sub.i] not all zero. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[[lambda].sub.d]-almost everywhere. Denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The function g is constant on every sphere centered in the origin. A simple observation yields that its restriction on any one-dimensional subspace of [R.sup.d] takes zero value in at most countably many points. Thus we get

[[lambda].sub.d] ({g = 0}) = 0.

Hence v is non-trivial. Furthermore, since

{Dv/D[[lambda].sub.d] > 0} = ({f > 0} [intersection] {g > 0}) [union] <{f < 0} [intersection] {g < 0}), {Dv/D[[lambda].sub.d] < 0} = ({f > 0} [intersection] {g < 0}) [union] <{f < 0} [intersection] {g > 0}),

the measure v is nowhere monotone. To finish the proof consider the set

span{[[mu].sub.[alpha]] : [alpha] [member of] R}.

Remark 4.0.7. The idea of using the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to produce the set {[[mu].sub.[alpha]] : [alpha] [member of] R} was first used in [Gamez-Merino et al.(2010)], where it was used to prove that the set of nowhere monotone everywhere differentiable functions on R is [absolute value of R]-lineable. The fact that this method could have been used in our proof is not too surprising taking in mind our previous commentary on the analogy between nowhere monotone functions and nowhere monotone measures.

5 Maximal Dense-lineability of Nowhere Monotone Measures in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d])

In this section we aim to prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]) is even maximal dense-lineable in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]). Let us first recall the notion of strong sets (see [Aron et al.(2009)]): if A and B are subsets of a vector space X, then A is said to be stronger than B if A + B [subset or equal to] A. The following recent result (see [Bernal-Gonzalez and Cabrera(2014), Theorem 2.3 (c)]) will play a crucial role in our proof.

Theorem 5.0.1. Assume that X is a topological vector space. Let A [subset] X. Suppose that there exists a subset B [subset] X such that A is stronger than B and B is dense-lineable. If the origin possesses a fundamental system U of neighborhoods with card(U) [less than or equal to] dim(X), A is maximal lineable and A [intersection] B = [empty set], then A is maximal dense-lineable. In particular, the same conclusion follows if X is metrizable, A is maximal lineable and A [intersection] B = [empty set].

Note that the above theorem strengthens the result in [Aron et al.(2009), Theorem 2.2]. Neither separability, nor metrizability of X are needed as a general assumption. Moreover, if the sets A and B are disjoint, the above theorem even provides an estimate of the dimension of the obtained subspaces.

Some additional notation is required in the following: We denote by

[C.sup.1.sub.k] [subset] [0,1], k [member of] N,

the fat Cantor set constructed by the well-known inductive process in which in the n-th step we remove [2.sup.n-1] intervals of length [4.sup.-(k+n)]. For any k, d [member of] N, d > 1 we denote by [C.sup.d.sub.k] [subset] [[0,1].sup.d] the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The sets [{[C.sup.d.sub.k]}.sub.d,k[member of]N] are closed and nowhere dense in the respective spaces. They also satisfy the following property: For any d [member of] N and [epsilon] > 0 there exists k [member of] N such that

[[lanbda].sub.d]([C.sup.d.sub.k]) [greater than or equal to] 1 - [epsilon].

We also denote

[I.sup.d.sub.n] := [[-n, n].sup.d], n, d [member of] N.

Each [I.sup.d.sub.n] can be written as a union of [(2n).sup.d] cubes of [[lambda].sub.d]-measure 1. We denote the set of these cubes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 5.0.2. Let us denote for d [member of] N

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then [B.sub.d] has the following properties:

(i) [B.sub.d] is nonempty,

(ii) [B.sub.d] [union] {0} is closed under linear combinations,

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]) [intersection] [B.sub.d] = [empty set],

(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]) is stronger than [B.sub.d].

Proof. To prove (i) consider v [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]) defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Clearly v [member] [B.sub.d]. Properties (ii) and (iii) follow trivially from the definition of [B.sub.d]. To prove (iv), let a [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]), b [member of] [B.sub.d], and I [subset] [R.sup.d] open. Since [bar.spt]{Db/D[lambda]} is nowhere dense, there exists a nonempty open set I' [subset] I such that [absolute value of b](I') = 0. Thus

[(a + b).sup.+] (I) [greater than or equal to] [(a + b).sup.+] (I') = [a.sup.+] (I') > 0,

[(a + b).sup.-] (I) [greater than or equal to] [(a + b).sup.-] (I') = [a.sup.-] (I') > 0,

and a + b [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] {[R.sup.d]).

Lemma 5.0.3. The set [B.sub.d] defined in Lemma 5.0.2 is dense in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]) for any d [member of] N.

Proof. Let [mu] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]) \ [B.sub.d], ||[mu]|| = 1 and [epsilon] > 0 be given. Find n [member of] N such that [absolute value of [mu]]([R.sup.d]\[I.sup.d.sub.n]) [less than or equal to] [epsilon]/2. Denote

c := [[lambda].sub.d] ([bar.spt] {D[mu]/D[[lambda].sub.d]} [intersection] [I.sup.d.sub.n]

and let [epsilon]' := min {[epsilon]/2, c/2}. Find k [member of] N such that

[[lambda].sub.d]([C.sup.d.sub.k]) [greater than or equal to] 1 - [epsilon]'/[(2n).sup.d].

For every j [member of] {1, ..., [(2n).sup.d]} denote by [C.sup.d.sub.k,j] the copy of [C.sup.d.sub.k] in [I.sup.d.sub.n,j] and let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally, set

Dv/D[[lambda].sub.d] := D[mu]/D[[lambda].sub.d] [chi][C.sub.[epsilon]'].

We claim that v [member of] [B.sub.d]. Indeed, since

[bar.spt]{Dv/D[[lambda].sub.d]} [subset or equal to] [C.sub.[epsilon]'],

we have v [member of] [B.sub.d] [union] {0}. But v = 0 would imply

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a contradiction. Thus, v [member of] [B.sub.d]. Furthermore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This finishes the proof.

Theorem 5.0.4. The set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]) is maximal dense-lineable in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]) for any d [member of] N.

Proof. As mentioned at the beginning of this section, we aim to use Theorem 5.0.1. If follows from Riesz' theorem (see [Rudin(1987), Theorem 6.19]) that the cardinality of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]) is c. Thus, by Theorem 4.0.6, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.d]) is maximal-lineable in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]([R.sup.d]). Furthermore, it follows from Lemma 5.0.2(ii) and Lemma 5.0.3 that [B.sup.d] is dense-lineable in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]([R.sup.d]). Disjointness of [B.sub.d] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]([R.sup.d]) follows from Lemma 5.0.2(iii) and by Lemma 5.0.2(iv) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (R) is stronger than [B/sib/d]. The result thus follows from Theorem 5.0.1.

6 Final Remarks, Open Problems

Further inspection of spaces with humps could provide some useful results. The author would be interested in finding out whether some useful characterizations of these spaces can be found. One could also ask whether restricting the values of the coefficient q to some strictly smaller interval would yield some non-trivial (and perhaps interesting) classes of spaces. Lastly, the space constructed in Theorem 3.0.6 has the following property: for every open set I [subset] [0,1], there exist only finitely many measures in [{[[mu].sub.n]}.sub.n[member of]N] such that I intersects the supports of these measures. This leads to the following question: Suppose we have a closed infinitely-dimensional subspace A [subset] M(X), A = [bar.span] {[[mu].sub.[gamma]], [gamma] [member of] [GAMMA]} such that for every non-empty open set G [subset] X there exist infinitely many measures in the set [{[[mu].sub.[lambda]]}.sub.[gamma][member of][GAMMA]] such that the supports of these measures intersect G. Does A then have to have humps?

Acknowledgements

The author would like to thank the referee for bringing the paper [Bernal-Gonzalez and Cabrera(2014)] to his attention, thus motivating him to formulate the results in Section 5. The author is also grateful for several other comments and suggestions which helped to improve the overall quality of the text.

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Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovska 83

email:petr.petracek@yahoo.com

Received by the editors in January 2014 - In revised form in April 2014.

Communicated by F. Bastin.

2010 Mathematics Subject Classification : 46E27, 28A33.
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