Weak convergence of masses: some neighborhood systems

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1 Weak convergence of masses: some neighborhood systems Bruno Girotto and Silvano Holzer Dipartimento di Matematica Applicata B. de Finetti Università di Trieste, Italy Piazzale Europa 1, 3417 Trieste, Italy Abstract. The purpose of the paper is to supply, in a finitely additive setting, some different neighborhood systems for the topology of the weak convergence of masses. Keywords. Mass, weak convergence, Lévy-topology, neighborhood system. M.S.C. classification: primary 60B05, 60B10; secondary 8A33, 8C15. J.E.L. classification: C10. 1 Introduction Many fundamental theorems in the classical countably additive probability theory are related with the notion of weak convergence of measures as, for instance, asymptotic results, Prokhorov theorems and metrizability conditions for the space of probability measures. In a finitely additive setting, after the pioneering work of Alexandroff ( ), Masani (198) first extended the direct Prokhorov theorem to the space of outer regular masses on normal Hausdorff topological spaces and Girotto and Holzer (1993) extended the Portmanteau theorem and Masani results to the space of masses (outer regular or not) on normal topological spaces. Moreover, Girotto and Holzer ( ) introduced the notion of weak convergence of bounded monotone set functions (additive or not) in order to extend Portmanteau and Prokhorov theorems in this general framework. From a topological point of view, the weak convergence can be seen as the convergence w.r.t. a suitable topology, usually called the Lévy-topology. Some different neighborhood systems for this topology are introduced, for example, by Blau (1951), Billingsley (1968) (in a countably additive setting) and by Varadarajan (1965), Masani (198) (in an outer regular finitely additive setting). The purpose of this paper, which can be regarded as a continuation of Girotto and Holzer [6], is to supply some different neighborhood systems for the Lévytopology on the space of all masses (outer regular or not). Partially supported by PRIN Analisi reale e teoria della misura and by PRIN Imprese di assicurazione e fondi pensione. Modelli per la valutazione, per la gestione e per il controllo.

2 158 Neighborhood systems for the Lévy-topology We adopt usual set theoretic and topological notation, as in []. In the sequel, Ω is a normal topological space (i.e. a topological space, not necessarily Hausdorff, for which any disjoint pair of closed sets can be separated by disjoint open sets) and F is any field on Ω including all open sets. Henceforth, U, C and F (with or without indices) are open sets, closed sets and elements of F, respectively. The symbol µ (with or without indices) always denotes a mass (i.e. a positive bounded charge) on F. For any µ, the µ-outer regularity field R µ is the field of sets F such that µ (F ) sup{µ(c) : C F } = µ(f ) = inf{µ(u) : U F } µ (F ); moreover, the µ-strong regularity field R µ is the field of sets F such that F R µ and µ( F ) = 0. Now, we introduce in the set ba + (Ω, F) of all masses on F the Lévy-topology, i.e. the topology having as base at any mass µ the family B(µ) of neighborhoods of µ of the form: N,F1...,F h {µ : µ (F i ) µ(f i ) < (i = 1,..., h)}, with > 0, h natural number and F 1,..., F h in R µ. We recall that, by Theorem 5. and Remark 5.4 in [5], any net of masses {µ d ; d D} converges to µ under the Lévy-topology iff the net {µ d ; d D} weakly converges to µ (i.e. for any f in the set C(Ω) of all bounded continuous real functions on Ω, the real net { f dµ d ; d D} converges to f dµ; here and in the sequel, the integrals are S-integrals as defined in []). Finally, for any µ and > 0, consider the following subsets in ba + (Ω, F): },f {µ : f dµ f dµ <, with f C(Ω) N (),U {µ : µ (Ω) < µ(ω) +, µ (U) > µ(u) }, with U R µ N (3),C {µ : µ (Ω) > µ(ω), µ (C) < µ(c) + }, with C R µ N (4),U,C {µ : µ (U) > µ(u), µ (C) < µ(c) + }, with U, C R µ and, given h, k natural numbers, let,f 1,...,f h N (3),C 1,...,C h,f i (µ), N (),U 1,...,U h N (3),C i (µ), N (4),U 1,...,U h,c 1,...,C k N (),U i (µ), k j=1 N (4),U i,c j (µ). Now, for any µ, let B (1) (µ), B () (µ), B (3) (µ) and B (4) (µ) be the families of basic neighborhoods at µ whose elements are, respectively, all sets of the type,f 1,...,f h (µ), N (),U 1,...,U h (µ), N (3),C 1,...,C h (µ) and N (4),U 1,...,U h,c 1,...,C k (µ). The following theorem assures that these families supply different neighborhood systems for the Lévy-topology, so that it can be seen as a topological

3 version of the finitely additive Portmanteau-type theorem stated, as usual, in a sequential form in [5; Theorem 3.3]. 159 Theorem 1. Each collection {B (i) (µ)} µ ba+ (Ω,F) (i = 1,, 3, 4) is a neighborhood system for the Lévy-topology. Proof. Plainly, each of these collections is a neighborhood system for a topology on ba + (Ω, F). The proof that they all determine the Lévy-topology is carried out in the following steps. 1. Given > 0 and f C(Ω), we claim that,f in B(µ). Let M be a real number such that f < M. Moreover, let C x = f 1 ({x}) for any real number x and note that the sets C x are closed and pairwise disjoint. Consequently, by Theorem 3. in [5] and a well known property, there is at most a countable number of values of x such that C x R µ or µ(c x ) > 0. First, assume µ µ(ω) > 0. Then, we can choose x 0, x 1,..., x m with M = x 0 < x 1 <... < x m = M and x i x i 1 <, with i = 1,..., m, 4 µ µ(c xi ) = 0, C xi R µ, with i = 0, 1,..., m. Let U i = {ω Ω : f(ω) < x i } for i = 0, 1,..., m. Then U i is an open set; moreover, on noting that U i C xi, we get U i R µ, µ( U i ) = 0 and hence U i R µ. Therefore, by Theorem.5(i) in [5], we have F i = U i U i 1 R µ for i = 1,..., m. Letting then g = m x i I i (where I i is the indicator function of the set F i for all i), we have f g < 4 µ. Now, let = min( 4Mm, µ ). To verify N,Ω,F 1...,F m (µ),f (µ), choose µ N,Ω,F 1...,F m (µ). Since µ µ = µ (Ω) µ(ω) <, we get (f g ) dµ (f g ) dµ f g dµ + f g dµ 4 µ ( µ + µ ) < 4 µ ( µ + ) + 4 µ + 4, g dµ g dµ = m m x i µ m (F i ) x i µ(f i ) x i µ (F i ) µ(f i ) < Mm 4 and hence f dµ f dµ (f g ) dµ (f g ) dµ + g dµ g dµ < =,

4 160 i.e. µ,f (µ). This establishes the desired inclusion. Finally, assume µ = 0. To verify N M,Ω (µ),f (µ), choose µ N M,Ω (µ). Then, we get f dµ f dµ = f dµ f dµ M µ (Ω) < M M =, i.e. µ,f. Given > 0 and U R µ, we claim that N (),U in B (1) (µ). Since U R µ, there is C U such that µ(u) µ(c) <. Moreover, by Urysohn Lemma, we can consider a continuous function f : Ω [0, 1] such that f(ω) = 1, if ω C, and f(ω) = 0, if ω U. Consider now,f,g(µ), where g is the indicator function of Ω. To verify (),f,g(µ) N,U (µ), choose µ,f,g(µ). Then, we have µ(u) < µ(c) + f dµ + < f dµ + + µ (U) +, µ (Ω) µ(ω) = g dµ g dµ <, i.e. µ N (),U 3. Given > 0 and F R µ, we claim that N,F in B () (µ). Since F R µ, by Theorem.5 in [5], we get F, F, F R µ R µ and µ( F ) = µ(f ) = µ(f ). To verify N () Then we have, F c(µ) N,F (µ), choose µ N (),F, F c(µ).,f µ (F ) µ (F ) = µ (Ω) µ (F c ) < µ(ω) + [ µ(f c ) ] = µ(f ) + = µ(f ) +, µ (F ) µ ( F ) > µ( F ) = µ(f ), i.e. µ N,F 4. Given > 0 and U R µ, we claim that N (),U in B (3) (µ). To verify N (3) (),U c,ω(µ) N,U (µ), choose µ N (3),U c,ω(µ). Then we have µ (Ω) < µ(ω) + and µ (Ω) > µ(ω), µ (U c ) < µ(u c ) +, so that µ (U) = µ (Ω) µ (U c ) > µ(ω) [ µ(u c ) + ] = µ(u),

5 161 i.e. µ N (),U 5. Given > 0 and C R µ, the proof that N (3),C in B () (µ) is similar to the previous one. 6. Given > 0 and U, C R µ, we claim that N (4),U,C in B () (µ). To verify N () (4),U,Cc(µ) N,U,C (µ), choose µ N (),U,Cc(µ). Then we have µ (U) > µ(u) and µ (Ω) < µ(ω) +, µ (C c ) > µ(c c ), so that µ (C) = µ (Ω) µ (C c ) < µ(ω) + [ µ(c c ) ] = µ(c) +, i.e. µ N (4),U,C 7. Given > 0 and U R µ, we have N (),U (4) () N,U,Ω (µ) and hence N,U (µ) contains a neighborhood in B (4) (µ). 8. Since any neighborhood in B(µ), B (i) (µ) (i = 1,, 3, 4) is, respectively, a finite intersection of the corresponding elementary neighborhoods considered in the previous steps, the five neighborhood systems all generate the same topology, i.e. the Lévy-topology. References 1. Alexandroff, D.: Additive set functions in abstract spaces. Matematicheskij Sbornik 8 (1940) ; 9 (1941) ; 13 (1943) Bhaskara Rao, K.P.S., Bhaskara Rao, M.: Theory of charges. Academic Press, New York (1983) 3. Billingsley, P.: Convergence of probability measures. John Wiley, New York (1968) 4. Blau, J.H.: The space of measures on a given set. Fundamenta Mathematicae 38 (1951) Girotto, B., Holzer, S.: Weak convergence of masses on normal topological spaces. Sankhyā A 55 (1993) Girotto, B., Holzer, S.: Weak convergence of masses: topological properties. Atti del Seminario Matematico e Fisico dell Università di Modena 48() (000) Girotto, B., Holzer, S.: Weak convergence of bounded, monotone set functions in an abstract setting. Real Analysis Exchange 6(1) (000/001) Masani, P.: The outer regularization of finitely-additive measures over normal topological spaces. In: Proc. Measure Theory Conf., Lecture Notes in Mathematics, Oberwolfach, 1981, vol Springer-Verlag, Berlin (198) Varadarajan, V.S.: Measures on topological spaces. American Mathematical Society Translations 48 (1965) 161-8

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