The Set of p-Adic Continuous Functions not Satisfying the Luzin (N) Property

In this short note, we will study the existence of a vector space of continuous functions f:Zp→Qp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:{\mathbb {Z}}_p\rightarrow \mathbb Q_p$$\end{document}, where Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_p$$\end{document} and Qp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Q}}_p$$\end{document} are, respectively, the ring of p-adic integers and the field of p-adic numbers, such that each nonzero function does not satisfy the Luzin (N) property and the dimension of the vector space is the continuum.

and 46B87, lineability results can be traced back to a paper of 1941 by Levin and Milman [19]. Let us introduce below the basic notion of lineability. Let κ be a cardinal number, V a vector space and A ⊂ V .
• We say that A is κ-lineable if there is a vector space L ⊂ V such that L has dimension κ and L \ {0} ⊂ A.
The main point in the theory of lineability is the search for large algebraic structures composed of mathematical objects enjoying certain special (usually, pathological) properties. In this paper, we focus on the classic Luzin null-property (or, for short, (N)-property) introduced by Nikolai Luzin at the beginning of the twentieth century. This property plays an essential role in the theory of absolute continuity of real functions. Let us recall the definition.
Let (X, A, μ) be a measure space. We say that N ∈ A is a μ-null set provided that μ(N ) = 0. Definition 1.1 (Luzin (N) property, [20]). Let (X, A, μ) and (Y, B, ν) be measure spaces. We say that a function f : X → Y satisfies the Luzin (N) property with respect to the pair (μ, ν) if f (N ) ∈ B is a ν-null set for every μ-null set N ∈ A.
Luzin's Theorem has an extension to differentiable functions R n → R n , known as Sard's Theorem (see [23]). Interestingly, Luzin's Theorem is not exclusive to the real line as it can be seen in [22, theorem 73 ] is a μ-null set, where μ is the Haar measure over Z p ; where Z p and Q p are, respectively, the field of p-adic integers and p-adic numbers (in what follows we shall provide the definitions of these concepts).
Our aim in this article is to show that the set of continuous functions from Z p into Q p that do not satisfy the Luzin (N) property is c-lineable in the presence of a suitable measure. Thereby, we continue the study of continuous functions without the Luzin (N) property, no longer those defined on the real line, but on the non-Archimedean field of p-adic numbers, using the lineability approach. This line of research has been, recently, considered in [11,[16][17][18].
Let us now, briefly, present the necessary background of p-adic analysis and the set theoretical notions that are required for the main result. (For more information on p-adic analysis, see [22]; and for the study of lineability on p-adic analysis see, for instance, [2,10,[16][17][18]). We shall denote the set of rational numbers, the set of natural numbers including zero, the infinitely countable cardinal number and the cardinality of the continuum by Q, N 0 , ℵ 0 and c, respectively. The main result of the paper uses the existence of a family of almost disjoint subsets of N 0 of cardinality c. We say that a family N of subsets of an infinitely countable set N is almost disjoint if card(A ∩ B) < ℵ 0 for any distinct A, B ∈ N . It is well known that, given N an infinitely countable set, there exists a family N of almost disjoint subsets of N that has cardinality c and each A ∈ N has cardinality ℵ 0 . 1 A non-Archimedean field K is a field endowed with a function | · |: K → [0, +∞) (known as absolute value) that satisfies the following properties: (i) |x| = 0 if, and only if, x = 0; (ii) |xy| = |x||y| for every x, y ∈ K; and (iii) |x + y| ≤ max{|x|, |y|} for every x, y ∈ K. Condition (iii) is known as the strong triangular inequality. Observe that | − 1| = |1| = 1, |1 + · · · + 1| n times ≤ 1, and if |x| = |y|, then |x + y| = max{|x|, |y|}.
Moreover, the absolute value induces a topology on the field K and, if K is finite, the topology is in fact the discrete topology since the trivial absolute value: |x| = 0 if x = 0, and |x| = 1 otherwise; is the only absolute value that can be defined on a finite field. For the rest of this work, p will denote a fixed prime number. Let n be a nonzero integer and ord p (n) be the highest power of p that divides n. We define the p-adic absolute value | · | p over Q as: |n| p = p −ordp(n) , |0| p = 0 and n m p = p −ordp(n)+ordp (m) . Observe that | · | p is a norm on the field Q and, therefore, we can define the field Q p which is the completion of Q with respect to the p-adic absolute value. Ostrowski's Theorem states that every non-trivial absolute value on Q is equivalent (i.e., defines the same topology) to a p-adic absolute value or the usual absolute value (see [15]). Each For each a ∈ Q p and r > 0, the sets B(a, r) := {x ∈ Q p : |x − a| p < r} and B(a, r) := {x ∈ Q p : |x−a| p ≤ r} denote, respectively, the open and closed balls of center a and radius r. Finally, we define the ring of p-adic integers as the closed unit ball of Q p : Results Math

Main Result
Our main result is Theorem 2.5. Before stating and proving it, let us recall the following definition and theorem from Probability Theory, and the definition of a Haar measure and Haar's Theorem (see [13, theorem 2.10]).
Definition 2.1. Let (Ω, F, P ) be a probability measure and X a real-valued function on Ω. We say that X is a simple random variable if X has finite range and {ω : Let us recall a crucial theorem on locally compact Hausdorff topological groups. Proof. Let N be a family of almost disjoint subsets of N 0 of cardinality c.
Recall that each A ∈ N has cardinality ℵ 0 . For every A ∈ N , let π A : N 0 → A be an increasing bijection, and let us define f A : Z p → Q p as: x πA(n) p n , Vol. 78 (2023) The Set of p-Adic Continuous Functions Page 5 of 10 157 for every x = ∞ n=0 x n p n ∈ Z p . Let us prove that the family F := {f A : A ∈ N } is a family of linearly independent continuous functions over Q p such that each f ∈ span Qp F does not satisfy Luzin (N) property.
(Trivially each f ∈ span Qp F is continuous provided that the functions in F are continuous.) We will prove first that the functions in F are continuous. Take any A ∈ N . Fix ε > 0 arbitrary and take x = ∞ n=0 x n p n ∈ Z p . Let n 0 ∈ N 0 be such that p −n0 < ε. Then, by taking y = ∞ n=0 y n p n ∈ Z p such that |x − y| p < p −πA(n0) we have that y n = x n for every n ∈ {0, . . . , π A (n 0 )}. In particular, y satisfies that x πA(n) = y πA(n) for every n ∈ {0, . . . , n 0 } since π A is increasing. Therefore, notice that Let us prove now that x n p n ∈ Z p : x n = 0 for every n / ∈ A is a μ-null set. For every n ∈ N 0 , define the random variable X n : Z p → {0, . . . , p − 1} as X n (x) = x n . Notice that the random variables (X n ) n∈N0 are independent, identically distributed and E[X n ] = 1 p for every n ∈ N 0 . Hence, by Theorem 2.2, we have that Observe that If N A were a Borel set, then it would be clear that N A is a μ-null set. However, since the Haar measure is not complete on (Z p , B), it is not obvious that N A is a Borel set. So let us prove that N A ∈ B. Let F p denote the finite field of p elements {0, 1, . . . , p − 1} endowed with the trivial absolute value, i.e., F p is a discrete topological space. For any n ∈ N 0 , let us define π n : Z p → F p as π n (x) = x n for every x = ∞ n=0 x n p n ∈ Z p . Let n ∈ N 0 , fix ε > 0 arbitrary and take x ∈ Z p . Choose any integer m > n. For every y ∈ Z p such that |x − y| p < p −m we have that π n (x) = π n (y). Therefore, |π n (x) − π n (y)| = 0 < ε, i.e., π n is continuous. Now observe that N A = n∈N\A π −1 n ({0}), where π −1 n ({0}) is closed since π n is continuous and {0} is closed in F p . Therefore, the set N A is a Borel set since N A is the countable intersection of closed sets (in fact, N A is closed). Take any A ∈ N .
Recall that the open balls B(x, p −n ) (where x ∈ Z p and n ∈ N 0 ) are Borel sets. Therefore, we have that the set where n A = card (A ∩ {0, . . . , n}). Hence, since μ is a Haar measure, i.e., f A does not satisfy Luzin (N) property for the pair (μ Z p , μ). Observe that αf A is continuous and does not satisfy Luzin (N) property for the pair (μ Z p , μ) for any α ∈ Q p \ {0} and A ∈ N .
Finally, let us prove that the functions in F are linearly independent over Q p and any nonzero linear combination over Q p does not satisfy Luzin (N) property. Take A 1 , . . . , A m ∈ N distinct (where m ∈ N\{1}), λ 1 , . . . , λ m ∈ Q p , and consider the linear combination F : where n A1 = card(A 1 ∩ {0, . . . , n 0 }). Thus λ 1 = 0. By applying analogous arguments, we arrive at λ i = 0 for every i ∈ {1, . . . , m}, that is, the functions in F are linearly independent over Q p . Assume for the rest of the proof that λ i = 0, for every i ∈ {1, . . . , m}, in order to prove that F does not satisfy Luzin (N) property. Once again, consider n 0 ∈ N 0 such that x n p n ∈ N A1 : x n = 0 for every n ∈ {0, . . . , n 0 } .
Observe that N A1 is a μ-null set. Indeed, this is an immediate consequence of the fact that N A1 = N A1 ∩B(0, p −n0 ) is a Borel set and N A1 is contained in the μ-null set N A1 . Moreover, by construction, N A1 ∩ B(0, p −n ) = N A1 ∩ B(0, p −n ) for any n > n 0 , and, for every x ∈ N A1 , we have In particular, if n > n 0 , then Remark 2.6. The functions that satisfy the assumptions of Theorem 2.5 are not differentiable. This is an immediate consequence of [22, theorem 73.2 (ii)], which reads as follows:

Conflict of interests
The authors have no relevant financial or non-financial interests to disclose.
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