On Generalized Probabilistic Normed Spaces

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1 International Mathematical Forum, Vol. 6, 2011, no. 42, On Generalized Probabilistic Normed Spaces Ioan Goleţ Department of Mathematics, Politehnica University Timişoara, Romania Abstract In this paper, generalized probabilistic n-normed spaces are studied, topological properties of these spaces are given. As examples, spaces of random variables are considered. Connections with generalized deterministic n-normed spaces are also given. Mathematics Subject Classification: 60H10, 47H10 Keywords: probabilistic n-normed space, random variable 1 Introduction In [10] K. Menger proposed the probabilistic concept of distance by replacing the number d(p, q), as distance between points p, q by a distribution function F p,q. This idea led to a large development of probabilistic analysis [8],[10]. Probabilistic normed spaces were first defined by Serstnev in [12]. So, a fruitful theory concordant with that of ordinary normed spaces and with that of probabilistic metric spaces was initiated. The theory of probabilistic normed spaces is important as a random generalization of deterministic linear normed space theory. In the same time it gives also new tools in the study of random operator equations. For important results of probabilistic functional analysis we refer to [8],[10]. The linear 2-normed spaces were first introduced by S. Gähler in [3]. The study of n-normed spaces was also introduced and enhanced [7], [9]. In this paper we consider a class of probabilistic n-normed spaces in a quite general version in concordance with the definition of probabilistic normed spaces given in [4], [5]. Topological properties of these spaces and their connections with deterministic n-normed spaces and probabilistic normed spaces are considered. Examples of probabilistic n-normed spaces are also given.

2 2074 I. Goleţ 2 Preliminary Notes As usual R denotes the set of real numbers, R + = {x R : x 0} and I =[0, 1] the closed unit interval. Let Δ + denotes the set of distance distribution functions (briefly, d.d.f), i.e. the set of non decreasing, left-continuous functions F : R I with F (0) = 0. The set Δ + will be endowed with the topology given by the modified Levy metric d L [11]. Let D + denotes the set of those functions F Δ + for that lim F (x) =1. Let F, G be in Δ +, then x we write F G if F (t) G(t) for all t R.Ifa R + then H a will be the function of Δ +, defined by H a (t) =0ift a and H a (t) =1 if t>a.itis obvious that H 0 F, for all F Δ +. Let ϕ be a function defined on the real field R into itself with the following properties : (1) ϕ( t) =ϕ(t), for every t R; (2) ϕ(1) = 1; (3) ϕ is strict increasing and continuous on (0, ); (4) lim ϕ(α) = 0 and lim ϕ(α) =. α 0 α As examples of such functions we have : ϕ(α) = α ; ϕ(α) = α p,p R + ; ϕ(α) = 2α2n α +1 N+. The functions ϕ help us to give an easy generalization of n-normed spaces defined in [7], but their importance increase in the development of probabilistic versions. Definition 2.1 The n-normed space is a pair (L,,,,,) where L is a linear space of a dimension greater than n and,,, is a real valued mapping on L n such that the following conditions be satisfied: (5) x 1,x 2,,x n =0if, and only if, x 1,x 2,,x n are linearly dependent; (6) x 1,x 2,,x n is invariant under any permutation of x 1,x 2,,x n ; (7) x 1,x 2,,αx n = ϕ(α) x 1,x 2,,x n, whenever x 1,x 2,,x n L and α R; (8) x 1,x 2,,x n 1,y+ z x 1,x 2,,x n 1,y + x 1,x 2,,x n 1,z, for all x 1,x 2,,x n 1,y,z L. Recall that a t-norm is a two variables mapping T : I I I that is associative, commutative, non decreasing in each variable and such that T (a, 1) = a, for all a [0, 1]. A mapping τ :Δ + Δ + Δ + is a triangle function if it is commutative, associative and it has the H 0 is the identity, i.e., τ(f, H 0) )=F, for every F Δ +. Note that if T is a left continuous t-norm and τ T is defined by τ T (F, G)(t) = sup T (F (t 1 ),G(t 2 )), t > 0, then τ T is a triangle function. t 1 +t 2 <t We mention that the terminology and notations are standard as in [8],[10].

3 On generalized probabilistic normed spaces Main Results Starting from the previous results in the theory of of a probabilistic normed [4-6] we define the following class of generalized probabilistic n-normed spaces. Definition 3.1 Let L be a real linear space of dimension greater than n, and let F be a mapping defined on the cartesian product of L by itself of n times L n into D + such that the following properties are satisfied: (9) F x1,x 2,,x n (t) is invariant under any permutation of x 1,x 2,,x n ; (10) F x1,x 2,,αx n (t) =F x1,x 2,,x n ( t ϕ(α) 1,x 2,,x n L and α R. (11) F x1,x 2,,x n 1,y+z τ(f x1,x 2,,x n 1,y,F x1,x 2,,x n 1,z), for every x 1,x 2,,x n 1,y,z L. The function F is called a probabilistic n-norm on L and the triple (L, F,τ) is called a probabilistic n-normed space. The triangle inequalities (11)) can be formulated by using a t-norm T. (12) F x1,x 2,,x n 1,y+z(t 1 +t 2 ) T (F x1,x 2,,x n 1,y(t 1 ),F x1,x 2,,x n 1,z(t 2 )), for every t 1,t 2 R +,x 1,x 2,,x n1,y,z L If (9), (10) and (12) are satisfied then the triple (L, F, T) is called a generalized probabilistic n-normed spaces of Menger type or simply Menger n-normed space. Proposition 3.2 If (L, F,T) is a Menger n-normed space then the probabilistic n-norm F has the following property: (13) F x1,x 2,,x n 1,θ(t) =H 0 (t), for all t>0 and x 1,x 2,,x n 1 L, where θ is the null vector in L. Prof. Indeed, F x1,x 2,,x n 1,θ(t) =F x1,x 2,,x n 1,αθ(t) =F x1,x 2,,x n 1,θ( t ), for ϕ(α) all α R {0}. Then F x1,x 2,,x n 1,θ(t) = lim α 0 F x1,x 2,,x n 1,θ( t ϕ(α) )=F x 1,x 2,,x n 1,θ( ) =H 0 (t) The probabilistic n-norm F induces a topology on the linear space L. Let A be the family of all finite and non-empty subsets of the linear space L n 1, A A, ε>0 and λ (0, 1). By a neighborhood of the null vector θ in the linear space L we mean a subset of L defined by V (ε, λ, A) ={x L : F x,a (ε) > 1 λ, a A}. Theorem 3.3 Let (L, F,T) be a probabilistic n-normed space under a continuous t-norm T, T T m, where T m = Max(Sum 1, 0). Then the family V = {V (ε, λ, A) :ε>0,λ (0, 1),A A} is a fundamental system of neighborhoods of the null vector in the linear space L.

4 2076 I. Goleţ Proof. Let V (ε k,λ k,a k ),k =1, 2beinV. We consider A = A 1 A 2,ε= min{ε 1,ε 2 },λ= min{λ 1,λ 2 }, then V (ε, λ, A) V (ε 1,λ 1,A 1 ) V (ε 2,λ 2,A 2 ). Let α R such that 0 α 1 and x αv (ε, λ, A), then x = αy, where y V (ε, λ, A). For every a A we have ε F x,a (ε) =F αy,a (ε) =F y,a ( ϕ(α) ) F y,a(ε) > 1 λ. The above inequalities shows us that x V (ε, λ, A). Hence αv (ε, λ, A) V (ε, λ, A). Let s show that, for every V Vand x L there exists β R, β 0 such that βx V. If V Vthen there exists ε>0, λ (0, 1) and A Asuch that V = V (ε, λ, A). Let x be arbitrarily fixed in L and α R,α 0, then F αx,a (ε) =F x,a ( ε ). ϕ(α) Since, lim F x,a ( ε ) = 1 it follows that, for each a A there exists α(a) R α 0 ϕ(α) ε such that F x,a ( ) > 1 λ. If we choose β = min{ α(a) : a A}, then we ϕ(α(a)) have ε F βx,a (ε) =F x,a ( ϕ(β) ) F ε x,a( ϕ(α(a)) ) > 1 λ, for all a A. So, βx V. Let us prove that, for any V Vthere exists V 0 Vsuch that V 0 + V 0 V. If V = V (ε, λ, A) and x V (ε, λ, A), then there exists η > 0 such that F x,a (ε) > 1 η>1 λ. If V 0 = V ( ε, η,a) and x, y V 2 2 0, a A then, by the triangle inequality (12) we have F x+y,a (ε) T (F x,a ( ε 2 ),F y,a( ε 2 )) T (1 η 2, 1 η 2 ) T m (1 η 2, 1 η ) > 1 η>1 λ. 2 The above inequalities show us that V 0 + V 0 V. Now, we show that V V and α R, α 0 implies αv V. Let us remark that αv = αv (ε, λ, A) ={αx : F x,a (ε) > 1 λ, a A) and F x,a (ε) > 1 λ F x,a ( ϕ(α)ε )=F ϕ(α) αx,a(ϕ(α)ε) > 1 λ. This shows that αv = V (αε, λ, A), hence αv V. The above statements show us that V is a base of neighborhoods of the origin for a topology on the linear space L. This is generated by the generalized probabilistic n-norm F and is named F-topology on L. Now, we consider the following example of generalized probabilistic n-normed space having as base space a set of random variables with values in a Banach algebra. The study of Banach algebra-valued random variables is of great importance in the theory of random equations since many of the Banach spaces encountered are also algebras.

5 On generalized probabilistic normed spaces 2077 Let (X,. ) be a separable Banach space which is also an commutative algebra. Let (Ω, K,P) be a complete probability measure space and let (X, B) be the measurable space, where B is the σ -algebra of Borel subsets of the separable Banach algebra (X,. ). We denote by L the linear space of all random variables defined on (Ω, K,P) with values in (X, B). Since, in a Banach algebra, the operation of multiplication is continuous, the product of X-valued random variables x 1 (ω),x 2 (ω),,x n (ω) is a well-defined X-valued random variable. For all x 1,x 2,,x n L and t R,t>0 we define F x1,x 2,,x n (t) =P ({ω Ω: x 1 (ω) x 2 (ω) x n (ω) <t}) Theorem 3.4 Let L be the linear space of all classes of random variables equal with probability 1 defined on (Ω, K,P) with values in (X, B). Forϕ(α) = α the triple (L, F,T m ) is a generalized probabilistic n-normed space. Proof. We verify that the conditions of the Definition 2.1 are satisfied. The property (9) is true because the product of random variables is commutative. F x1,x 2,,αx n (t) =P ({ω Ω: x 1,x 2,,αx n <t}) =P ({ω Ω: α x 1 x 2 x n <t}) =P ({ω Ω: x 1 x 2,x n < t })=F α x 1,x 2,,x n ( t ). So, the the α conditions (10) is verified. For all x 1,x 2,,x n,y,z L and t 1,t 2 R + {0} we define the sets: A = {ω Ω: x 1 (ω) x 2 (ω),x n 1 (ω)y(ω) <t 1 }, B = {x 1 (ω) x 2 (ω),x n 1 (ω)z(ω) <t 2 }, C = {ω Ω: x 1 (ω) x 2 (ω),x n 1 (ω)[y(ω)+z(ω)]) <t 1 + t 2 } From the triangle inequality of a n-norm,, (8) it follows that A B C. By properties of the measure of probability P we have P (C) P (A B) P (A)+P (B) P (A B) P (A)+P (B) 1 Taking in account that P (A) =F x1,x 2,,x n 1,y(t 1 ) P (B) =F x1,x 2,,x n 1,z(t 1 ) and P (C) =F x1,x 2,,x n 1,y+z(t 1 + t 2 ) it follows that the conditions (12) from the definition of a generalized Menger n-normed space is satisfied. The proof is complete. Remark 3.5 The concept of probabilistic n-normed space developed in this paper can be a starting point for new developments: (a) the spaces of the product L n can be considered different; (b) the product of the vectors by scalars can be defined by different functions ϕ on different linear spaces. So, there are large possibilities for a mathematical frame of random phenomena.

6 2078 I. Goleţ References [1] A. T. Bharucha-Reid, Random integral equation,academic press (1972). [2] S. Gähler, 2-metrische Räume und ihr topologische structure, Math.Nachr., 26 (1963), [3] S. Gähler, Lineare 2-nomierte Raume Math. Nachr., 28 (1964), [4] I. Goleţ, On probabilistic 2-normed spaces, Journal of Mathematics, Novi Sad 35 (2005), [5] I. Goleţ, Some remarks on functions with values in probabilistic normed spaces, Math. Slovaca, 57 (2007) [6] I. Goleţ, On generalized probabilistic metric spaces, International Mathematical Forum, 5 (2010), [7] H. Gunawan and M. Mashadi, On n-normed spaces, Int. J. Math. Sci., 27 (2001), [8] O. Hadžić, Endre Pap, Fixed point theory in probabilistic metric spaces, Kluver Academic Publishers, Dordrecht, [9] I. H. Jebril, Bounded sets in random n-normed linear space, Int. J. of Acad. Research, 4 (2010), [10] K. Menger, Statistical metrics, Proc. Nat. Acad. Sci., USA, 28 (1942), [11] B. Schweizer, A. Sklar, Probabilistic metric spaces, North Holland, New York, Amsterdam, Oxford, (1983). [12] A.N.Serstnev, Random normed spaces : Problems of completeness, Kazan Gos. Univ. Ucen. Zap., , Received: February, 2011