Pseudo-Fubini Real-Entire Functions on the Plane

Mediterr. J. Math. (2022) 19:218 https://doi.org/10.1007/s00009-022-02149-5 1660-5446/22/050001-11 published online September 9, 2022 c The Author(s) 2022 Pseudo-Fubini Real-Entire Functions on the Plane Luis Bernal-Gonz´alez, Mar´ıa del Carmen Calder´on-Moreno and Andreas Jung Abstract. In this note, it is proved the existence of a c-dimensional vector space of real-entire functions all of whose nonzero members are non-integrable in the sense of Lebesgue but yet their two iterated integrals exist as real numbers and coincide. Moreover, it is shown that this vector space can be chosen to be dense in the space of all real C∞-functions on the plane endowed with the topology of uniform convergence on compacta for all derivatives of all orders. If the condition of being entire is dropped, then a closed infinite dimensional subspace satisfying the same properties can be obtained. Mathematics Subject Classification. 15A03, 26B15, 26E05, 28A35, 46B87. Keywords. Fubini’s theorem, real entire functions, iterated integrals, dense lineability. 1. Introduction According to Fubini’s theorem (see e.g. [11, Chapter 17]), if a real function f, which is defined on a measure space X×Ythat is the product of two σfinite product spaces, is integrable, then its two iterated integrals exist as real numbers and coincide, and in fact their common value is the integral of fon the product space. In [2] it is analyzed –among other questions– the algebraic size of the set of those measurable functions f:X×Y→Rbeing nonintegrable but still satisfying the conclusion of Fubini’s theorem, that is, its two iterated integrals exist as real numbers and are equal. These functions are called pseudo-Fubini functions, and they abound in a topological-algebraic sense. Specifically, it is shown in [2] that, under appropriate soft conditions on the measure spaces Xand Y, there exists a c-dimensional vector space all of This research has been supported by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127 Grant P20-00637 and FEDER US-138096 and by MCINN Grant PGC2018-098474-B-C21. 218 Page 2 of 11 L. Bernal-González et al. MJOM whose nonzero members are pseudo-Fubini that is dense in the space of all measurable functions X×Y→Rwhen endowed with its natural metrical topology. Here cdenotes the cardinality of the continuum. Turning to the more familiar setting of the Lebesgue measure (dxon the real line R, and dxdyon the plane R2) as well as to functions with richer properties, in [3] it is exhibited an explicit c-dimensional vector space of analytic functions R2→Rall of whose nonzero elements are pseudo-Fubini that is dense in the Fr´echet space of all real continuous functions on R2when endowed with the compact-open topology. Recall that, if Ω is an open subset of R2, then a function f:Ω→Ris said to be analytic on Ω whenever, given (x0,y 0)∈Ω, there are a neighborhood Uof (x0,y 0) with U⊂Ω and a double sequence {ajk}j,k≥0⊂Rsuch that f(x, y)=j,k≥0ajk(x−x0)j(y−y0)kfor every (x, y)∈U, the convergence being absolute (see, e.g., [5, Chap. 4] for background on real or complex analytic functions of several variables). Going one step further, in [3] it is posed the problem of the existence of entire functions R2→Rthat are pseudo-Fubini. Recall that a function f: R2→Ris said to be entire if an absolutely convergent expansion f(x, y)= j,k≥0ajkxjykis valid on the whole plane. Note that any entire function on R2is analytic but the converse is false: consider, for instance, the function f(x, y)= 1 1+x2+y2. In this short note, we solve in the affirmative the last problem. In fact, it is proved that the family of pseudo-Fubini entire functions is, again, rather large in both algebraic and topological senses. This will be done in Sect. 3. Section 2will be devoted to fix some notation and provide a number of preliminary results. In Sect. 4the problem of existence of closed infinite dimensional spaces made up of pseudo-Fubini smooth functions is considered. 2. Preliminaries and Notation Let k,N ∈N:= {1,2,...}and Ω be a nonempty open subset of RN.The case N=2,Ω=R2will be mostly considered. Throughout this note, we shall use the following –mostly standard– notation: •C(Ω) will stand for the set of all real continuous functions on Ω. This set becomes a Fr´echet space when endowed with the topology of uniform convergence in compacta, see [9]. •Ck(Ω) denotes the vector space of all functions Ω →Rthat are ktimes differentiable on Ω. •C∞(Ω) represents the set of all real functions on Ω that are infinitely times differentiable on Ω. This set becomes a Fr´echet space when endowed with the topology of uniform convergence in compacta for all partial derivatives of all orders, see [9]. •Cω(Ω) will stand for the vector space of all functions Ω →Rthat are analytic on Ω. •E Ndenotes the vector space of all real entire functions on RN. Then EN⊂Cω(RN)⊂C∞(RN). MJOM Pseudo-Fubini Real-Entire Page 3 of 11 218 •L 1(Ω) represents the vector space of all functions Ω →Rthat are Lebesgue integrable on Ω. •For each multi-index α=(α1,...,α N)∈(N∪{0})N,weset |α|:= α1+···+αN. In addition, we denote by pF(R2) the vector space of all pseudo-Fubini functions f:R2→R, meaning that each of such functions is Lebesgue measurable but not Lebesgue integrable and, in addition, both iterated integrals RRf(x, y)dxdy,RRf(x, y)dydxexist as real numbers and have the same value. We now turn to a different setting. If Eis a vector space, we can study the algebraic size of a subset, which becomes more interesting if such a subset is not a vector space in itself (see [1] for background on this line of research, called lineability). The basic concepts that are relevant to this note are contained in the following definition. Definition 2.1. Assume that Eis a vector space. Let A⊂Eand αbe a cardinal number. Then Ais called lineable if A∪{0}contains some infinite dimensional vector space. If, in addition, Eis a topological vector space and A∪{0}contains some dense (some (dense) α-dimensional, some closed infinite dimensional, resp.) vector subspace of E, then Ais said to be dense lineable (α-(dense) lineable, spaceable, resp.) in E.And Ais called maximal dense lineable in Eif it is dim(E)-dense lineable. Under the previous terminology, the main result in [3] can be stated as follows. Theorem 2.2. The set pF∩Cω(R2)is maximal dense lineable in C(R2). The main result in this note (Theorem 3.1) tells us that the same assertion holds when one replaces Cω(R2) by the much smaller family E2, and the space C(R2) by the space C∞(R2), whose topology is much stronger than that inherited from the former space. With this aim, we shall make use of the next two assertions. The first one enables us to extract dense lineability from mere lineability and can be found in [1, Chapter 7], while the second one is an approximation result due to Frih and Gauthier [6], which is a strengthening of corresponding results due to Carleman [4], Scheinberg [12] and Hoischen [8][cases(k=0,N = 1), (k=0,N arbitrary) and (karbitrary, N= 1), resp.]. Theorem 2.3. Assume that Eis a metrizable separable topological vector space, that κis an infinite cardinal number and that A,Bare subsets of Esatisfying the following properties: •Ais κ-lineable, •Bis dense lineable, •A+B⊂A,and •A∩B=∅. Then Ais κ-dense lineable in E. 218 Page 4 of 11 L. Bernal-González et al. MJOM Theorem 2.4. Assume that N,k ∈N, that f∈Ck(RN)and that ε:RN→ (0,+∞)is a continuous function. Then there is an entire function g:RN→ Rsuch that |Dαf(x)−Dαg(x)|<ε(x) for all x∈RNand all α∈(N∪{0})Nwith |α|≤k. 3. A Large Vector Space of Pseudo-Fubini Entire Functions This section is devoted to prove the following theorem. Theorem 3.1. The set pF∩E 2is maximal dense lineable in C∞(R2). Proof. It is well-known (an easy proof follows, for instance, from the Baire category theorem) that dim C∞(R2)=c. Hence, we have to prove that pF∩ E2is c-dense lineable in C∞(R2). Our first task is to locate a pseudo-Fubini C∞-function on R2.For this, denote by I0the open unit interval (0,1) and consider the function ϕ0:R→Rdefined as ϕ0(t)=e1 t(t−1) if t∈I0 0 otherwise. It is well known that ϕ0∈C∞(R). For each a∈R,set Ia:= a+I0= (a, a+1), and let ϕa:R→Rbe the C∞-function given by ϕa(t)=ϕ0(t−a). Observe that, for every pair a, b of reals, the function (x, y)→ ϕa(x)ϕb(y) belongs to C∞(R2) and vanishes exactly outside Ia×Ib. Now, define the function Φ0:R2→Rby Φ0(x, y)=ϕ1(x)ϕ1(y)+ ∞  n=1 ϕ2n(x)ϕ2n+1(y)+ ∞  n=1 ϕ2n+1(x)ϕ2n(y) − ∞  n=1 ϕ2n(x)ϕ2n−1(y)− ∞  n=1 ϕ2n−1(x)ϕ2n(y). Observe that Φ0is well defined because the products of intervals Ia×Ib where the functions ϕa(x)ϕb(y) participating in the sum do not vanish are mutually disjoint and so, given a point of R2,thereisatmostoneterm of the series that is not zero at it. The same argument and the fact that differentiability is a local property yields that Φ0∈C∞(R2). Since ϕ0>0onI0,weget α:=  I2 0 ϕ0(x)ϕ0(y)dxdy=I0 ϕ0(t)dt2>0, and the change of variables rule together with Fubini’s theorem gives  R2 ϕa(x)ϕb(y)dxdy= Ia×Ib ϕa(x)ϕb(y)dxdy =Ia ϕa(x)dxIb ϕb(x)dx=I0 ϕ0(x)dx2 =α. MJOM Pseudo-Fubini Real-Entire Page 5 of 11 218 Therefore (note that for nonnegative functions it is always possible to interchange the integral with the series) we get  R2 |Φ0(x, y)|dxdy≥ R2∞  n=1 ϕ2n(x)ϕ2n+1(y)dxdy = ∞  n=1  R2 ϕ2n(x)ϕ2n+1(y)dxdy= ∞  n=1 α=+∞, which shows that Φ0∈L 1(R2). To see that Φ0∈pF, observe that if y0∈R\∞ n=1 Inthen Φ0(·,y 0)≡ 0, and so RΦ0(x, y0)dx=0.Andif y0belongs to some (necessarily unique) In, then Φ0(·,y 0)=cy0ϕpχIp−ϕqχIqfor some different p, q ∈Nand a constant cy0∈R. Therefore R Φ0(x, y0)dx=cy0Ip ϕp(x)dx−Iq ϕq(x)dx =cy0I0 ϕ0(x)dx−I0 ϕ0(x)dx=0 too. Thus, the iterated integral RRΦ0(x, y)dxdyexists in the Lebesgue sense as a real number, with value 0. Now the symmetry property Φ0(x, y)= Φ0(y,x) yields the same conclusion for RRΦ0(x, y)dydx, which shows that Φ0is pseudo-Fubini. Next, consider for each k∈Nthe translation τk:(x, y)∈R2−→ (x+4k,y +k)∈R2. For every a, b ∈Rwe have τk(Ia×Ib)=Ia+4k×Ib+k. Observe that if k,l ∈N (with k=l)andIm×Inand Ip×Iqare two different products of intervals where Φ0does not vanish, then we have not only (Im+4k×In+k)∩(Ip+4k× Iq+k)=∅but also (Im+4k×In+k)∩(Ip+4l×Iq+l)=∅. In particular, the supports of the functions Φk:(x, y)∈R2−→ (Φ0◦τ−1 k)(x, y)=Φ 0(x−4k,y −k)∈R(k=1,2,...) are mutually disjoint. The fact that the mappings x→ x−4k,y→ y−kare C∞-smooth on Rtogether with the rule of change of variables (for integrals on Rand R2) shows that Φk∈C∞(R2)∩pFfor every k∈N. Let us choose an almost disjoint family Nof subsets of N,thatis,N satisfies the following properties (see [10, Chapter 8]): each S∈N is infinite, card (N)=cand S∩Sis finite for every pair of different S, S∈N.Let us define, for every S∈N, the function fS:R2→Rby fS:=  k∈S Φk. From the facts that the supports of the Φk’s are mutually disjoint and Φk∈C∞(R2) we obtain fS∈C∞(R2). It is also obtained that |fS|= k∈S|Φk|. Hence selecting any k0∈Swe get R2 |fS(x, y)|dxdy≥R2 |Φk0(x, y)|dxdy=R2 |Φ0(x, y)|dxdy=+∞, 218 Page 6 of 11 L. Bernal-González et al. MJOM and so fS∈L 1(R2). Finally, the fact that the supports of the Φk’s are steadily moving up and right when k→∞ yields that, given x0∈R,the function y∈R→ fS(x0,y)∈Ris a (possibly empty) sum of finitely many functions of the form c(ϕaχIa−ϕbχIb) (with c∈Rand a, b ∈N,a=b), whose integrals over Rare zero. An analogous property happens if we fix y0∈Rand consider the function x∈R→ fS(x, y0)∈R. Therefore both iterates integrals of fSexist in the sense of Lebesgue and share the value 0. Summarizing, we have fS∈C∞(R2)∩pFfor each S∈N. Now, it is the turn of approximation. Thanks to Theorem 2.4, for every S∈N we can find gS∈E 2such that |fS(x, y)−gS(x, y)|<1 (1 + x2)(1 + y2)for all (x, y)∈R2.(3.1) We define the family Mas M:= span {gS:S∈N}. Then Mis a vector space which is contained in E2. If we showed that any nontrivial finite linear combination of functions from Mis pseudo- Fubini (and, hence, in particular non-integrable) then we would prove at one stroke that the gS’s are linearly independent (hence dim(M)=c) and that M⊂(pF∩E 2)∪{0}, which would show the desired maximal lineability. Assume that G∈M is one of such combinations, so that there are p∈N, reals α1,...,α pand pairwise different sets S1,...,S p∈N such that G=α1gS1+···αpgSpand not all the αi’s are zero (we may assume without loss of generality that α1= 0). From (3.1), the comparison test and the fact that 1 (1+x2)(1+y2)∈L 1(R2), it follows that the functions hi:= gSi−fSi(i=1,...,p) are integrable C∞-functions from R2to R. Then every hisatisfies the conclusion of Fubini’s theorem and, as fSialso does, the linearity of the integral shows that for the function gSiand, consequently, for the function G, both iterated integrals exist as real numbers in the sense of Lebesgue and coincide. Now, since the family Nis almost disjoint, the set S1\p j=2 Sj) is not empty, so that we can select an element k0in it. Consider the set A:= ∞ n=1 I2n+4k0×I2n+1+k0, which has empty intersection with each of the supports of the functions fSi(i=2,...,p). Therefore, Gis not Lebesgue integrable on R2because if Gwere Lebesgue integrable on R2, it would be integrable on A, but we have MJOM Pseudo-Fubini Real-Entire Page 7 of 11 218  A |G(x, y)|dxdy= A p  i=1 αigSi(x, y) dxdy ≥ A |α1||gS1(x, y)|dxdy− A p  i=2 |αi||gSi(x, y)|dxdy ≥ A |α1|(|fS1(x, y)|−|gS1(x, y)−fS1(x, y)|)dxdy − A p  i=2 |αi|(|fSi(x, y)|+|gSi(x, y)−fSi(x, y)|)dxdy = A |α1||fS1(x, y)|dxdy− p  i=1 |αi| A |gSi(x, y)−fSi(x, y)|dxdy ≥ A |α1||fS1(x, y)|dxdy− p  i=1 |αi| A 1 (1 + x2)(1 + y2)dxdy =|α1| A |Φk0(x, y)|dxdy−C =|α1| ∞  n=1  I2n+4k0×I2n+1+k0 |Φk0(x, y)|dxdy−C =|α1| ∞  n=1  I2n×I2n+1 |Φ0(x, y)|dxdy−C =|α1| ∞  n=1  I2n×I2n+1 ϕ2n(x)ϕ2n+1(y)dxdy−C =|α1| ∞  n=1 α−C=+∞, since α>0andC:= p i=1 |αi|A 1 (1+x2)(1+y2)dxdy<∞, that is absurd. Consequently, G∈pFand so the family A:= pF∩E 2is maximal lineable in C∞(R2). Next, we consider the family B:= {P·Ψ1:Pis a polynomial of two real variables with coefficients in R}, where Ψ1is the Gaussian function Ψ1(x, y)=e−x2−y2, that is integrable on the plane (with integral equal to π) together with every function Ψλ(x, y):= e−λ(x2+y2)with λ>0. Plainly, Bis a vector space contained in E2.Onthe one hand, each function h:= P·Ψ1∈B belongs to L1(R2). Indeed, since Pis a polynomial, there are a constant M∈(0,+∞)andanN∈N such that |P(x, y)|≤M(1 + (x2+y2)N) for all (x, y)∈R2. And since limt→+∞M(1+tN) et/2= 0, we derive the existence of a constant γ∈(0,+∞) such that |P(x, y)|≤γ·ex2+y2 2on R2, and therefore |h|≤γ·Ψ1/2on R2. The comparison test yields the integrability of h. In particular, Fubini’s theorem implies that 218 Page 8 of 11 L. Bernal-González et al. MJOM RR h(x, y)dxdy= R2 h(x, y)dxdy=RR h(x, y)dydx for all h∈B. Since B⊂L 1(R2), we have A∩B=∅. On the other hand, we also have A+B⊂A. This is a consequence of the facts that the sum of an integrable function with a non-integrable one is non-integrable and that both finiteness and coincidence of the iterated integrals are preserved under finite summations. Take E:= C∞(R2)andκ:= c. According to Theorem 2.3, and taking into account that Bis itself a vector space, it only remains to prove that B is dense in E. To this end, observe first that the formula fk:= max 0≤i+j≤ksup (x,y)∈[−k,k]2 |Dijf(x, y)|(f∈C∞(R2); k=1,2,...) (3.2) defines an increasing sequence of seminorms generating the natural Fr´echet topology of our space E. In fact, C∞(R2) endowed with multiplication of functions is a Fr´echet algebra (see, e.g., [7, Chapter 1] for background about this structure) because the Leibniz rule for the derivative of a product gives the existence of a sequence {Ck}k≥1⊂(0,+∞) such that f·gk≤Ckfkgk(f,g ∈E;k∈N).(3.3) Now, fix a nonempty open subset Uof E. Then there are ε>0, f∈E and k∈Nsuch that U⊃{h∈E:h−fk<ε}. Define ε(x, y):=ε/2. By Theorem 2.4, there is g∈E 2such that |Dijf(x, y)−Dijg(x, y)|<ε/2 for all (i, j)∈(N∪{0})2with i+j≤kand all (x, y)∈R2. Consider the sequence {Pn}n≥0of Taylor polynomials of gat the origin, that is, Pn(x, y)=  0≤i+j≤n Dijg(0) i!j!xiyj(n=0,1,2,...). Then (see, e.g., [5]) for each 2-index (i, j)∈(N∪{0})2the sequence {DijPn}n≥0converges uniformly to Dij gon every compact subset of R2.In particular, we can find n0∈Nsuch that |DijPn0(x, y)−Dijg(x, y)|<ε/2 for all (i, j)∈(N∪{0})2with i+j≤kand all (x, y)∈[−k,k]2. Let us set h:= Pn0. Thus, according to (3.2) we obtain h−fk≤Pn0−gk+f−gk<ε 2+ε 2=ε. Consequently, h∈U. This tells us that the set Pof polynomials in two real variables with real coefficients is dense in E. Finally, consider the multiplication operator S:f∈E−→ Ψ1·f∈E, that is linear and surjective (because Ψ1never vanishes and 1/Ψ1∈E). Thanks to (3.3), we have S(f)k≤CkΨ1kfkfor all f∈Eand all k∈N, which proves that Sis continuous. Then B=S(P) is dense in S(E)=E. The proof is complete.  MJOM Pseudo-Fubini Real-Entire Page 9 of 11 218 4. Pseudo-Fubini Smooth Functions: Spaceability In this final section, we want to raise the question of existence of closed infinite dimensional spaces of entire pseudo-Fubini functions. Question. Is pF∩E 2spaceable in C∞(R2)? Or at least: Is pF∩Cω(R2) spaceable in C∞(R2)? We have not been able to give an answer to it. Therefore, we shall for the moment content ourselves with establishing the next assertion, which puts an end to this note. Theorem 4.1. The set pF∩C∞(R2)is spaceable in C∞(R2). Proof. Consider the functions Φk∈C∞(R2)∩pF(k∈N) defined in the proof of Theorem 3.1. Since their supports Skare pairwise disjoint, they are linearly independent. By the same reason, for each sequence α=(αk)∈RN the function ∞ k=1 αkΦkis well defined and belongs to C∞(R2). Let us define the family M:= ∞  k=1 αkΦk:α∈RN, which is plainly a vector subspace of C∞(R2). Moreover, it is infinite dimensional because Mcontains every Φk. The fact that every f∈M\{0} belongs to pFcan be seen as in the proof of Theorem 3.1, where one uses the property that the supports of the Φk’s move steadily up and right as k→∞. Hence, our unique task is to show that Mis closed in C∞(R2). To this end, assume that fj→fas j→∞in the topology of C∞(R2), where fj=∞ k=1 αj,kΦk∈M(k=1,2,...). It should be shown that f∈M. Since convergence in C∞(R2) implies pointwise convergence, we get limk→∞ fj(x)=f(x) for every x∈R2.If x∈ S:= k∈NSkthen Φk(x)=0 for all k,andso fj(x) = 0. Hence f(x) = 0 in this case. If x∈Sthen there is a (unique) k=k(x)∈Nwith x∈Sk. Then fj(x)=αjkΦk(x)→f(x) as j→∞. Therefore limj→∞ αjk =f(x) Φk(x). But this limit (say, βk)mustbe independent of x.Thus,f(x)=βkΦk(x) for all x∈Sk. Consequently, from the disjointness of the Sk’s, we get f=∞ k=1 βkΦk∈M, as required.  Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. 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