×

Nonstandard methods in measure theory. (English) Zbl 1474.28025

Summary: Ideas and techniques from standard and nonstandard theories of measure spaces and Banach spaces are brought together to give a new approach to the study of the extension of vector measures. Applications of our results lead to simple new proofs for theorems of classical measure theory. The novelty lies in the use of the principle of extension by continuity (for which we give a nonstandard proof) to obtain in an unified way some notable theorems which have been obtained by Fox, Brooks, Ohba, Diestel, and others. The methods of proof are quite different from those used by previous authors, and most of them are realized by means of nonstandard analysis.

MSC:

28E05 Nonstandard measure theory
26E35 Nonstandard analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Halmos, P. R., Measure Theory (1950), New York, NY, USA: D. van Nostrand Company, New York, NY, USA
[2] Dunford, N.; Schwartz, J. T., Linear Operators. I. General Theory (1958), New York, NY, USA: Interscience Publishers, New York, NY, USA
[3] Fox, G., Extension of a bounded vector measure with values in a reflexive Banach space, Canadian Mathematical Bulletin, 10, 525-529 (1967) · Zbl 0186.46501 · doi:10.4153/CMB-1967-052-1
[4] Găină, S., Extension of vector measures, Revue Roumaine de Mathématique Pures et Appliquées, 8, 151-154 (1963) · Zbl 0136.12201
[5] Dinculeanu, N., Vector Measures. Vector Measures, Hochschulbücher für Mathematik (1966), Berlin, Germany: VEB Deutscher, Berlin, Germany · Zbl 0142.10502
[6] Dinculeanu, N., On regular vector measures, Acta Scientiarum Mathematicarum, 24, 236-243 (1963) · Zbl 0117.33702
[7] Arsene, Gr.; Strătilă, Ş., Prolongement des mesures vectorielles, Revue Roumaine de Mathématique Pures et Appliquées, 10, 333-338 (1965) · Zbl 0138.38404
[8] Dinculeanu, N.; Kluvanek, I., On vector measures, Proceedings of the London Mathematical Society. Third Series, 17, 505-512 (1967) · Zbl 0195.34002
[9] Fox, G., Inductive extension of a vector measure under a convergence condition., Canadian Journal of Mathematics, 20, 1246-1255 (1968) · Zbl 0159.19001 · doi:10.4153/CJM-1968-120-x
[10] Kluvanek, I., On the theory of vector measures. II, Matematicko-Fyzikálny Časopis, Slovenskej Akadémie Vied, 16, 76-81 (1966) · Zbl 0145.05302
[11] Ohba, S., Extensions of vector measures, Yokohama Mathematical Journal, 21, 61-66 (1973) · Zbl 0288.28016
[12] Gould, G. G., Extensions of vector-valued measures, Proceedings of the London Mathematical Society, 3, 16, 685-704 (1966) · Zbl 0148.38102
[13] Sion, M., Outer measures with values in a topological group, Proceedings of the London Mathematical Society, 19, 89-106 (1969) · Zbl 0167.14503
[14] Drewnowski, L., Topological rings of sets, continuous set functions, integration I, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 20, 269-276 (1972) · Zbl 0249.28004
[15] Drewnowski, L., Topological rings of sets, continuous set functions, integration II, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 20, 277-286 (1972) · Zbl 0249.28005
[16] Drewnowski, L., Topological rings of sets, continuous set functions, integration III, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 20, 445 (1972) · Zbl 0249.28006
[17] Takahashi, M., On topological-additive-group-valued measures, Proceedings of the Japan Academy, 42, 330-334 (1966) · Zbl 0144.04703 · doi:10.3792/pja/1195522027
[18] Loeb, P. A., Conversion from nonstandard to standard measure spaces and applications in probability theory, Transactions of the American Mathematical Society, 211, 113-122 (1975) · Zbl 0312.28004 · doi:10.1090/S0002-9947-1975-0390154-8
[19] Albeverio, S.; Høegh-Krohn, R.; Fenstad, J. E.; Lindstrøm, T., Nonstandard Methods in Stochastic Analysis and Mathematical Physics, 122 (1986), Orlando, Fla, USA: Academic Press, Orlando, Fla, USA · Zbl 0605.60005
[20] Stroyan, K. D.; Bayod, J. M., Foundations of Infinitesimal Stochastic Analysis, 119 (1986), Amsterdam, The Netherlands: North-Holland Publishing Co., Amsterdam, The Netherlands · Zbl 0624.60052
[21] Luxemburg, W. A. J., A general theory of monads, Applications of Model Theory to Algebra, Analysis, and Probability, 18-86 (1969), New York, NY, USA: Holt, Rinehart and Winston, New York, NY, USA · Zbl 0207.52402
[22] Henson, C. W.; Moore,, L. C., Nonstandard analysis and the theory of Banach spaces, Nonstandard Analysis—Recent Developments. Nonstandard Analysis—Recent Developments, Lecture Notes in Mathematics, 983, 27-112 (1983), Berlin, Germany: Springer, Berlin, Germany · doi:10.1007/BFb0065334
[23] Živaljević, R. T., Loeb completion of internal vector-valued measures, Mathematica Scandinavica, 56, 2, 276-286 (1985) · Zbl 0607.28010
[24] Osswald, H.; Sun, Y., On the extensions of vector-valued Loeb measures, Proceedings of the American Mathematical Society, 111, 3, 663-675 (1991) · Zbl 0723.28009 · doi:10.2307/2048403
[25] Diestel, J.; Faires, B., On vector measures, Transactions of the American Mathematical Society, 198, 253-271 (1974) · Zbl 0297.46034 · doi:10.1090/S0002-9947-1974-0350420-8
[26] Diestel, J.; Uhl,, J. J., Vector Measures. Vector Measures, Mathematical Surveys, 15 (1977), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0369.46039
[27] Lewis, P. W., Extension of operator valued set functions with finite semivariation, Proceedings of the American Mathematical Society, 22, 563-569 (1969) · Zbl 0191.43101
[28] Hurd, A. E.; Loeb, P. A., An Introduction to Nonstandard Real Analysis. An Introduction to Nonstandard Real Analysis, Pure and Applied Mathematics, 118 (1985), Orlando, Fla, USA: Academic Press, Orlando, Fla, USA · Zbl 0583.26006
[29] Orlicz, W., Absolute continuity of vector-valued finitely additive set functions. I, Studia Mathematica, 30, 121-133 (1968) · Zbl 0169.46703
[30] Rickart, C. E., Decomposition of additive set functions, Duke Mathematical Journal, 10, 653-665 (1943) · Zbl 0063.06492 · doi:10.1215/S0012-7094-43-01061-0
[31] Oberle, R. A., Characterization of a class of equicontinuous sets of finitely additive measures with an application to vector valued Borel measures, Canadian Journal of Mathematics, 26, 281-290 (1974) · Zbl 0296.28012 · doi:10.4153/CJM-1974-029-x
[32] Bogdanowicz, W. M.; Oberle, R. A., Decompositions of finitely additive vector measures generated by bands of finitely additive scalar measures, Illinois Journal of Mathematics, 19, 370-377 (1975) · Zbl 0337.28012
[33] Walker, H. D., Uniformly additive families of measures, Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, 18, 1-2, 217-222 (1974) · Zbl 0321.28003
[34] Gould, G. G., Integration over vector-valued measures, Proceedings of the London Mathematical Society, 3, 15, 193-225 (1965) · Zbl 0138.38403
[35] Davis, M., Applied Nonstandard Analysis. Applied Nonstandard Analysis, Pure and Applied Mathematics (1977), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0359.02060
[36] Brooks, J. K., On the existence of a control measure for strongly bounded vector measures, Bulletin of the American Mathematical Society, 77, 999-1001 (1971) · Zbl 0229.28008 · doi:10.1090/S0002-9904-1971-12834-3
[37] Bartle, R. G.; Dunford, N.; Schwartz, J., Weak compactness and vector measures, Canadian Journal of Mathematics, 7, 289-305 (1955) · Zbl 0068.09301 · doi:10.4153/CJM-1955-032-1
[38] Ohba, S., On vector measures. I, Proceedings of the Japan Academy, 46, 51-53 (1970) · Zbl 0197.11703 · doi:10.3792/pja/1195520509
[39] Porcelli, P., Two embedding theorems with applications to weak convergence and compactness in spaces of additive type functions, Journal of Mathematics and Mechanics, 9, 273-292 (1960) · Zbl 0090.32704
[40] Leader, S., The theory of \(L^p\)-spaces for finitely additive set functions, Annals of Mathematics, 58, 528-543 (1953) · Zbl 0052.11401 · doi:10.2307/1969752
[41] Brooks, J. K.; Dinculeanu, N., Weak compactness and control measures in the space of unbounded measures, Proceedings of the National Academy of Sciences of the United States of America, 69, 1083-1085 (1972) · Zbl 0238.28002 · doi:10.1073/pnas.69.5.1083
[42] Dieudonné, J., Sur les espaces de Köthe, Journal d’Analyse Mathématique, 1, 81-115 (1951) · Zbl 0044.11703 · doi:10.1007/BF02790084
[43] Traynor, T., \(S\)-bounded additive set functions, Vector and Operator Valued Measures and Applications, 355-365 (1973), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0297.28012
[44] Kluvánek, I., Intégrale vectorielle de Daniell, Matematicko-Fyzikálny Časopis, Slovenskej Akadémie Vied, 15, 146-161 (1965) · Zbl 0146.12301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.