Pseudo-Fubini Real-Entire Functions on the Plane

In this note, it is proved the existence of a c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {c}$$\end{document}-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∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document}-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.


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 × Y that 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 f on the product space. In [2] it is analyzed -among other questions-the algebraic size of the set of those measurable functions f : X × Y → R being 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 X and Y , there exists a c-dimensional vector space all of L. Bernal-González et al. MJOM whose nonzero members are pseudo-Fubini that is dense in the space of all measurable functions X × Y → R when endowed with its natural metrical topology. Here c denotes the cardinality of the continuum.
Turning to the more familiar setting of the Lebesgue measure (dx on the real line R, and dxdy on the plane R 2 ) as well as to functions with richer properties, in [3] it is exhibited an explicit c-dimensional vector space of analytic functions R 2 → R all of whose nonzero elements are pseudo-Fubini that is dense in the Fréchet space of all real continuous functions on R 2 when endowed with the compact-open topology. Recall that, if Ω is an open subset of R 2 , then a function f : Ω → R is said to be analytic on Ω whenever, given (x 0 , y 0 ) ∈ Ω, there are a neighborhood U of (x 0 , y 0 ) with U ⊂ Ω and a double sequence 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 R 2 → R that are pseudo-Fubini. Recall that a function f : R 2 → R is said to be entire if an absolutely convergent expansion f (x, y) = j,k≥0 a jk x j y k is valid on the whole plane. Note that any entire function on R 2 is analytic but the converse is false: consider, for instance, the function f (x, y) = 1 1+x 2 +y 2 . 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 2 will be devoted to fix some notation and provide a number of preliminary results. In Sect. 4 the problem of existence of closed infinite dimensional spaces made up of pseudo-Fubini smooth functions is considered.

Preliminaries and Notation
Let k, N ∈ N := {1, 2, . . . } and Ω be a nonempty open subset of R N . The case N = 2, Ω = R 2 will 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 Fréchet space when endowed with the topology of uniform convergence in compacta, see [9]. • C k (Ω) denotes the vector space of all functions Ω → R that are k times 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 Fréchet space when endowed with the topology of uniform convergence in compacta for all partial derivatives of all orders, see [9]. • C ω (Ω) will stand for the vector space of all functions Ω → R that are analytic on Ω. • E N denotes the vector space of all real entire functions on R N . Then • L 1 (Ω) represents the vector space of all functions Ω → R that are Lebesgue integrable on Ω. • For each multi-index α = (α 1 , . . . , α N ) ∈ (N ∪ {0}) N , we set |α| := α 1 + · · · + α N .
In addition, we denote by pF(R 2 ) the vector space of all pseudo-Fubini functions f : R 2 → R, meaning that each of such functions is Lebesgue measurable but not Lebesgue integrable and, in addition, both iterated integrals R R f (x, y) dx dy, R R f (x, y) dy dx exist as real numbers and have the same value.
We now turn to a different setting. If E is 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.
Under the previous terminology, the main result in [3] can be stated as follows.
The main result in this note (Theorem 3.1) tells us that the same assertion holds when one replaces C ω (R 2 ) by the much smaller family E 2 , and the space C(R 2 ) by the space C ∞ (R 2 ), 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 (k arbitrary, N = 1), resp.].

Theorem 2.3.
Assume that E is a metrizable separable topological vector space, that κ is an infinite cardinal number and that A, B are subsets of E satisfying the following properties:

A Large Vector Space of Pseudo-Fubini Entire Functions
This section is devoted to prove the following theorem.
Proof. It is well-known (an easy proof follows, for instance, from the Baire Our first task is to locate a pseudo-Fubini C ∞ -function on R 2 . For this, denote by I 0 the open unit interval (0, 1) and consider the function It is well known that ϕ 0 ∈ C ∞ (R). For each a ∈ R, set I a := a + I 0 = (a, a+1), and let ϕ a : R → R be the C ∞ -function given by ϕ a (t) = ϕ 0 (t−a). Observe that, for every pair a, b of reals, the function (x, y) → ϕ a (x)ϕ b (y) belongs to C ∞ (R 2 ) and vanishes exactly outside Observe that Φ 0 is well defined because the products of intervals I a ×I b where the functions ϕ a (x)ϕ b (y) participating in the sum do not vanish are mutually disjoint and so, given a point of R 2 , there is at most one term of the series that is not zero at it. The same argument and the fact that differentiability is a local property yields that Φ 0 ∈ C ∞ (R 2 ).
Since ϕ 0 > 0 on I 0 , we get α := and the change of variables rule together with Fubini's theorem gives Therefore (note that for nonnegative functions it is always possible to interchange the integral with the series) we get To see that Φ 0 ∈ pF, observe that if y 0 ∈ R \ ∞ n=1 I n then Φ 0 (·, y 0 ) ≡ 0, and so R Φ 0 (x, y 0 ) dx = 0. And if y 0 belongs to some (necessarily unique) I n , then Φ 0 (·, y 0 ) = c y0 ϕ p χ Ip − ϕ q χ Iq for some different p, q ∈ N and a constant c y0 ∈ R. Therefore too. Thus, the iterated integral R R Φ 0 (x, y) dx dy exists in the Lebesgue sense as a real number, with value 0. Now the symmetry property Φ 0 (x, y) = Φ 0 (y, x) yields the same conclusion for R R Φ 0 (x, y) dy dx, which shows that Φ 0 is pseudo-Fubini.
Next, consider for each k ∈ N the translation For every a, b ∈ R we have τ k (I a ×I b ) = I a+4k ×I b+k . Observe that if k, l ∈ N (with k = l) and I m × I n and I p × I q are two different products of intervals where Φ 0 does not vanish, then we have not only (I m+4k × I n+k ) ∩ (I p+4k × I q+k ) = ∅ but also (I m+4k × I n+k ) ∩ (I p+4l × I q+l ) = ∅. In particular, the supports of the functions . . ) are mutually disjoint. The fact that the mappings x → x − 4k, y → y − k are C ∞ -smooth on R together with the rule of change of variables (for integrals on R and R 2 ) shows that Φ k ∈ C ∞ (R 2 ) ∩ pF for every k ∈ N.
Let us choose an almost disjoint family N of subsets of N, that is, N satisfies the following properties (see [10,Chapter 8]): each S ∈ N is infinite, card (N ) = c and S ∩ S is finite for every pair of different S, S ∈ N . Let us define, for every S ∈ N , the function f S : R 2 → R by From the facts that the supports of the Φ k 's are mutually disjoint and Φ k ∈ C ∞ (R 2 ) we obtain f S ∈ C ∞ (R 2 ). It is also obtained that |f S | = k∈S |Φ k |. Hence selecting any k 0 ∈ S we get |Φ 0 (x, y)| dxdy = +∞, L. Bernal-González et al. MJOM and so f S ∈ L 1 (R 2 ). Finally, the fact that the supports of the Φ k 's are steadily moving up and right when k → ∞ yields that, given x 0 ∈ R, the function y ∈ R → f S (x 0 , y) ∈ R is a (possibly empty) sum of finitely many functions of the form c (ϕ a χ Ia − ϕ b χ I b ) (with c ∈ R and a, b ∈ N, a = b), whose integrals over R are zero. An analogous property happens if we fix y 0 ∈ R and consider the function x ∈ R → f S (x, y 0 ) ∈ R. Therefore both iterates integrals of f S exist in the sense of Lebesgue and share the value 0. Summarizing, we have f S ∈ C ∞ (R 2 ) ∩ pF for each S ∈ N . Now, it is the turn of approximation. Thanks to Theorem 2.4, for every S ∈ N we can find g S ∈ E 2 such that for all (x, y) ∈ R 2 . Then M is a vector space which is contained in E 2 . If we showed that any nontrivial finite linear combination of functions from M is pseudo-Fubini (and, hence, in particular non-integrable) then we would prove at one stroke that the g S '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 α 1 , . . . , α p and pairwise different sets S 1 , . . . , S p ∈ N such that G = α 1 g S1 + · · · α p g Sp and not all the α i 's are zero (we may assume without loss of generality that α 1 = 0). From (3.1), the comparison test and the fact that 1 (1+x 2 )(1+y 2 ) ∈ L 1 (R 2 ), it follows that the functions are integrable C ∞ -functions from R 2 to R. Then every h i satisfies the conclusion of Fubini's theorem and, as f Si also does, the linearity of the integral shows that for the function g Si and, consequently, for the function G, both iterated integrals exist as real numbers in the sense of Lebesgue and coincide. Now, since the family N is almost disjoint, the set S 1 \ p j=2 S j ) is not empty, so that we can select an element k 0 in it. Consider the set A := ∞ n=1 I 2n+4k0 × I 2n+1+k0 , which has empty intersection with each of the supports of the functions f Si (i = 2, . . . , p). Therefore, G is not Lebesgue integrable on R 2 because if G were Lebesgue integrable on R 2 , it would be integrable on A, but we have (1+x 2 )(1+y 2 ) dxdy < ∞, that is absurd. Consequently, G ∈ pF and so the family A := pF ∩ E 2 is maximal lineable in C ∞ (R 2 ).
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 ∞ (R 2 ) and κ := c. According to Theorem 2.3, and taking into account that B is itself a vector space, it only remains to prove that B is dense in E. To this end, observe first that the formula defines an increasing sequence of seminorms generating the natural Fréchet topology of our space E. In fact, C ∞ (R 2 ) endowed with multiplication of functions is a Fréchet 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 {C k } k≥1 ⊂ (0, +∞) such that 2 with i + j ≤ k and all (x, y) ∈ R 2 . Consider the sequence {P n } n≥0 of Taylor polynomials of g at the origin, that is, i!j! x i y j (n = 0, 1, 2, . . . ).
Then (see, e.g., [5]) for each 2-index (i, j) ∈ (N ∪ {0}) 2 the sequence {D ij P n } n≥0 converges uniformly to D ij g on every compact subset of R 2 . In particular, we can find n 0 ∈ N such that |D ij P n0 (x, y) − D ij g(x, y)| < ε/2 for all (i, j) ∈ (N ∪ {0}) 2 with i + j ≤ k and all (x, y) ∈ [−k, k] 2 . Let us set h := P n0 . Thus, according to (3.2) we obtain Consequently, h ∈ U . This tells us that the set P of polynomials in two real variables with real coefficients is dense in E. Finally, consider the multiplication operator that is linear and surjective (because Ψ 1 never vanishes and 1/Ψ 1 ∈ E). Thanks to (3.3), we have S(f ) k ≤ C k Ψ 1 k f k for all f ∈ E and all k ∈ N, which proves that S is continuous. Then B = S(P) is dense in S(E) = E. The proof is complete.

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.
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.
Proof. Consider the functions Φ k ∈ C ∞ (R 2 ) ∩ pF (k ∈ N) defined in the proof of Theorem 3.1. Since their supports S k are pairwise disjoint, they are linearly independent. By the same reason, for each sequence α = (α k ) ∈ R N the function ∞ k=1 α k Φ k is well defined and belongs to C ∞ (R 2 ). Let us define the family which is plainly a vector subspace of C ∞ (R 2 ). Moreover, it is infinite dimensional because M contains every Φ k . The fact that every f ∈ M \ {0} belongs to pF can be seen as in the proof of Theorem 3.1, where one uses the property that the supports of the Φ k 's move steadily up and right as k → ∞. Hence, our unique task is to show that M is closed in C ∞ (R 2 ).
To this end, assume that f j → f as j → ∞ in the topology of C ∞ (R 2 ), where f j = ∞ k=1 α j,k Φ k ∈ M (k = 1, 2, . . . ). It should be shown that f ∈ M. Since convergence in C ∞ (R 2 ) implies pointwise convergence, we get lim k→∞ f j (x) = f (x) for every x ∈ R 2 . If x ∈ S := k∈N S k then Φ k (x) = 0 for all k, and so f j (x) = 0. Hence f (x) = 0 in this case. If x ∈ S then there is a (unique) k = k(x) ∈ N with x ∈ S k . Then f j (x) = α jk Φ k (x) → f (x) as j → ∞. Therefore lim j→∞ α jk = f (x) Φ k (x) . But this limit (say, β k ) must be independent of x. Thus, f (x) = β k Φ k (x) for all x ∈ S k . Consequently, from the disjointness of the S k 's, we get f = ∞ k=1 β k Φ k ∈ M, as required.
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