About the L p space of intuitionistic fuzzy observables

: The aim of this paper is to define an L p space of intuitionistic fuzzy observables. We work in an intuitionistic fuzzy space ( F , m ) with product, where F is a family of intuitionistic fuzzy events and m is an intuitionistic fuzzy state. We prove that the space L p with corresponding intuitionistic fuzzy pseudometric ρ IF is a pseudometric space.


Introduction
In paper [7], B. Riečan studied L p space of fuzzy sets M.He proved that this L p space is a complete pseudometric space.A more general situation was studied in paper [8].There, an L p space was constructed for the observables of MV-algebra with product.In this case L p is a complete pseudometric space, too.
In this paper, we define an L p space of intuitionistic fuzzy observables and we prove that the space L p with corresponding intuitionistic fuzzy pseudometric ρ IF is a pseudometric space.Since the notion of intuitionistic fuzzy observable x : B(R) → F is a generalization of the notion of random variable ξ : Ω → R (more precisely ξ : (Ω, S, P ) → (R, B(R), P ξ )), we are inspired by L p space of random variables.There The distance in the L p space of random variables is defined by the formula Remark that in the whole text we use the abbreviation "IF" for the term "intuitionistic fuzzy".

Preliminaries and auxiliary notions
The year 2023 is the 40-th anniversary of the invention of the concept and theory of intuitionistic fuzzy sets by K. T. Atanassov in the paper [1].As an IF-set A on Ω he understands a pair In this paper we will work with a family of intuitionistic fuzzy events on (Ω, S) denoted by F.
On this family we use the Łukasiewicz binary operations ⊕, ⊙ given by The partial ordering is given by In the papers [9,11], B. Riečan defined the notion of an IF-state as a mapping m : F → [0, 1] with the following three conditions: and he defined the notion of an IF-observable as a mapping x : B(R) → F satisfying the following conditions: where B(R) is a σ-algebra of the family J of all intervals in R of the form Similarly, we can formulate the notion of an n-dimensional IF-observable as a mapping x : B(R n ) → F with the following conditions: If n = 1, we simply say that x is an IF-observable.
Remark that the composition of an IF-state m and an IF-observable x is a probability measure denoted m x , i.e., m x (C) = m(x(C)) for each C ∈ B(R).
In [10], B. Riečan defined the notion of a joint IF-observable and proved its existence.The joint IF-observable of the IF-observables x, y is a mapping h : B(R 2 ) → F satisfying the following conditions: There • is a product operation on the family of IF-events F introduced in [6].It is defined by If we have several IF-observables and a Borel measurable function, we can define the IF-observable, which is the function of several IF-observables.Regarding this, we provide the following definition, see [5].
Let x 1 , . . ., x n : B(R) → F be IF-observables, h n their joint IF-observable and g n : R n → R a Borel measurable function.Then we define the IF-observable g n (x 1 , . . ., x n ) : B(R) → F by the formula

L p space of IF-observables
In this section, we formulate L p space of IF-observables.We can consider an IF-observable x instead of a random variable and a joint IF-observable h instead of a random vector.
Consider the probability space (R 2 , B(R), P = m • h xy ) and the random variables ξ, η : Evidently, and Since x, y ∈ L p m , i.e., the integrals R |t| p dm x (t), R |t| p dm y (t) exist, then by ( 1), (2) we have Therefore, the random variables ξ, η belong to L p P and the random variable ξ − η belong to ), then we have where h xy : B(R 2 ) → F is the joint IF-observable of IF-observables x, y and the Borel measurable function g : R → R is given by g(u, v) = u − v.
can be rewritten in the following form If we put g(u, v) = u − v and ψ(w) = −w, then we obtain Next we prove the triangle inequality.Let x, y, z : B(R) → F be three different IF-observables.Consider a joint IF-observable h xyz : B(R 3 ) → F of IF-observables x, y, z.Then Consider the probability space (R 3 , B(R 3 ), P = m • h xyz ).Then the mappings ξ, η, ζ : R 3 → R defined by ξ(u, v, w) = u, η(u, v, w) = v, ζ(u, v, w) = w are the random variables and (3) Similarly, for each A ∈ B(R).Using (3), ( 4) and x, y, z ∈ L p m , we obtain that ξ, η, ζ ∈ L . Analogously, we obtain Therefore, the IF-space (L p m , ρ IF ) is a pseudometric space.

Conclusion
The paper is devoted to an L p space of IF-observables with respect the IF-state m.We proved that (L p m , ρ IF ) is a pseudometric space.The presented results are the generalization of the results in [7], because if µ A : Ω −→ [0, 1] is a fuzzy set, then A = (µ A , 1 − µ A ) : Ω → [0, 1] 2 is the corresponding intuitionistic fuzzy set.The Definition 3.1 generalizes the notion of integrable and square integrable IF-observable introduced in [4].

Definition 3 . 1 .
Fix a real number p ≥ 1.Let (F, m) be an IF-space with product.We say that an IF-observable x : B(R) → F belongs to L p m if there exists the integral R |t| p dm x (t).If x, y : B(R) → F are the IF-observables and h xy : B(R 2 ) → F is their joint IF-observable, then we define the IF-observable x − y : B(R) → F by the formula(x − y)(A) = h xy g −1 (A)for each A ∈ B(R), where g : R 2 → R is a Borel measurable function defined by g(u, v) = u−v.Proposition 3.1.Let (F, m) be an IF-space with product.If the IF-observables x, y : B(R) → F are in L p m , then the IF-observable x − y : B(R) → F is in L p m .Proof.From definition of IF-observable x − y we have

R 2 |ξ
− η| p dP .But ξ − η ∈ L p P , i.e., the integral R 2 |ξ − η| p dP exists, hence the integral R |t| p dm x−y (t) exists and x − y ∈ L p m .Definition 3.2.Let (F, m) be an IF-space with product.For each IF-observables x, y ∈ L p m define the map ρ

Proposition 3 . 2 .
The IF-space (L p m , ρ IF ) is a pseudometric space.Proof.By the Definition 3.2, we have ρ IF (x, x) = 0 and ρ IF (x, y) ≥ 0. Now, we prove the symmetry.Consider any different IF-observables x, y ∈ L p m .Let h xy be the joint IF-observable of IF-observables x, y and h yx be the joint IF-observable of IF-observables y