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The Brooks-Jewett theorem for \(k\)-triangular functions. (English) Zbl 0986.03048
Let \(L\) be an orthomodular poset (OMP). A function \(\psi :L \rightarrow [0, \infty)\) for \(k\leq 1\) is a \(k\)-triangular function [E. Pap, Null-additive set functions, Kluwer, Dordrecht (1995; Zbl 0856.28001)] if it satisfies \(\psi(0) =0\) and \([\psi(a)-k \psi(b) \leq \psi(a \vee b) \leq \psi(a) + k \psi(b)]\) for all \(a,b \in L\) with \(a \bot b.\)
The main result is a generalization of the Brooks-Jewett convergence theorem (Theorem 6) for \(k\)-triangular functions defined on OMP \(L\) with subsequential completeness property, i.e., for every orthogonal sequence in \(L\) there exists a subsequence which has supremum in \(L.\) The proof is based on a diagonal method (close to the sliding hump method) [see E. Pap, loc. cit.] and a modification of H. Weber [“Compactness in spaces of group-valued contents, the Vitali-Hahn-Saks theorem and Nikodým’s boundedness theorem”, Rocky Mt. J. Math. 16, 253-275 (1986; Zbl 0604.28006)].
Theorem 6 is extended to functions with values in a commutative semigroup with a neutral element \(0\) endowed with a function \(f\: S \to [0, \infty)\) satisfying \(f(0)=0\) and \([|f(x+y)-f(x) |\leq f(y)\) for all \(x,y \in S],\) see E. Pap [“A generalization of the diagonal theorem on a block-matrix”, Mat. Vesn., N. Ser. 11(26), 66-71 (1974; Zbl 0321.22001)].

MSC:
03G12 Quantum logic
28A33 Spaces of measures, convergence of measures
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[1] AGAFONOVA L. V.- KLIMKIN V. M.: A Nikodym theorem for triangular set functions. Siberian Math. J. 15 (1974), 669-674. · Zbl 0281.28005
[2] ANTOSIK P.- PAP E.: A Simplification of the proof of Rosenthaľs lemma for measures on fields. Convergence methods in analusis, Proc. 2nd Conf., Szczurk/Pol., 1981, pp. 26-31.
[3] ANTOSIK P.-SAEKI S.: A lemma on set functions and its applications. Dissertationes Math. (Rozprawu Mat.) 340 (1995), 13-21. · Zbl 0837.28011
[4] ANTOSIK P.-SWARTZ C.: Matrix Methods in Analysis. Lecture Notes in Math. 1113, Springer Verlag, New York, 1985. · Zbl 0564.46001
[5] ANTOSIK P.-SWARTZ, C: A theorem on matrices and its applications in functional analysis. Studia Math. 77 (1984), 197-205. · Zbl 0538.46031
[6] AVALLONE A.-LEPELLERE M. A.: Modular functions: uniform boundedness and compactness. Rend. Circ. Mat. Palermo (2) · Zbl 0931.28009
[7] BERAN L.: Orthomodular Lattices, Algebraic Approach. Academia Prague, Reidel, Dordrecht, 1984. · Zbl 0558.06008
[8] BIRKHOFF G.-VON NEUMANN J.: The logic of quantum mechanics. Ann. of Math. (2) 37 (1936), 823-843. · Zbl 0015.14603
[9] CONSTANTINESCU C.: Some properties of spaces of measures. Atti Sem. Mat. Fis. Univ. Modena Suppl. 35 (1989). · Zbl 0696.46027
[10] D’ANDREA A. B.-DE LUCIA P.: The Brooks-Jewett theorem on an orthomodular lattice. J. Math. Anal. Appl. 154 (1991), 507-522. · Zbl 0727.28008
[11] DE LUCIA P.-PAP E.: Nikodym convergence theorem for uniform space valued functions defined on D-posets. Math. Slovaca 45 (1995), 367-376. · Zbl 0856.28008
[12] DE LUCIA P.-SALVATI S.: A Caficro characterization of uniform s-boundedn ss. Rend. Circ. Mat. Palermo 40 (199 4), 121-128.
[13] DE LUCIA P.-TRAYNOR T.: Non-commutative group valued measures on an orthomodular poset. Math. Japonica 40 (1994), 309-315. · Zbl 0812.28008
[14] DIESTEL J.-UHL J. J.: Vector Measures. Math. Surveys Monographs 15, Amer. Math. Soc, Providence, RI, 1977. · Zbl 0369.46039
[15] DOBRAKOV I.: On submeasures I. Dissertationes Math. (Rozprawy Mat.) 112 (1974), 1-35. · Zbl 0292.28001
[16] DREWNOWSKI L.: On the continuity of certain non-additive set functions. Colloq. Math. 38 (1978), 243-253. · Zbl 0398.28003
[17] GUARIGLIA E.: K-triangular functions on an ortho-modular lattice and the Brooks-Jewett theorem. Rad. Mat. 6 (1990), 241-251. · Zbl 0763.28006
[18] GUARIGLIA E.: Uniform boundedness theorems for k-triangular set functions. Acta Sci. Math. (Szeged) 54 (1990), 391-407. · Zbl 0726.28008
[19] GUSEL’NIKOV N. S.: Triangular set functions and Nikodym’s theorem on the uniform boundedness of a family of measures. Mat. Sb. 35 (1979), 19-33. · Zbl 0418.28003
[20] GUSEL’NIKOV N. S.: Extension of quasi-Lipschitz set functions. Math. Notes 17 (1975), 14-19. · Zbl 0355.28001
[21] HABIL E. D.: The Brooks-Jewett theorem for k-triangular functions on difference posets and orthoalgebras. Math. Slovaca 47 (1997), 417-428. · Zbl 0961.28003
[22] KALMBACH G.: Orthomodular Lattices. Academic Press, London-New York, 1983. · Zbl 0528.06012
[23] KUPKA J.: A short proof and generalization of a measure theoretic disjointization lemma. Proc. Amer. Math. Soc. 45 (1974), 70-72. · Zbl 0291.28004
[24] PAP E.: The Vitali-Hahn-Saks theorems for k-triangular set functions. Atti Sem. Mat. Fis. Univ. Modena 35 (1987), 21-32. · Zbl 0626.28001
[25] PAP E.: A generalization of a theorem of Dieudonne for k-triangular set functions. Acta Sci. Math. (Szeged) 50 (1986), 159-167. · Zbl 0609.28002
[26] PAP E.: Null-Additive Set Functions. Math. Appl. 337, Kluwer Acad. Publ., Dordrecht, 1995. · Zbl 0968.28010
[27] PAP E.: Funkcionalna analiza. Institut za matematiku, Novi Sad, 1982. · Zbl 0496.46001
[28] PTAK P.-PULMANNOVA S.: Orthomodular Structures as Quantum Logics. Kluwer Acad. Publ., Dordrecht, 1991. · Zbl 0743.03039
[29] SALVATI S.: Teoremi di convergenza in teoria della misura non commutativa. PhD Thesis, 1997.
[30] SWARTZ C.: An Introduction to Functional Analysis. Dekker, New York, 1992. · Zbl 0751.46002
[31] VON NEUMANN J.: Matematische Grendlagen der Quantunmechanik. Springer Verlag, Berlin, 1932 [
[32] WEBER H.: A diagonal theorem. Answer to a question of Antosik. Bull. Polish Acad. Sci. Math. 41 (1993), 95-102. · Zbl 0799.40005
[33] WEBER H.: Compactness in spaces of group-valued contents, the Vitali-Hahn-Saks theorem and Nikodym’s boundedness theorem. Rocky Mountain J. Math. 16 (1986), 253-275. · Zbl 0604.28006
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