Abstract
The interest devoted over the last few decades to set valued mappings — multifunctions — is seen also in the theory of measurability and integrability. Therefore we present here the foundations of measure and integration theory.
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The results of the theory of multifunctions can be applied, e.g., in the theory of approximation. Interesting problems concerning various metrics are studied in the papers by Beer [1–3]. The paper Penkov and Sendov [1] deals with approximation as do the papers by Holá [1], [6] and Holá and Neubrunn [1], [2].
Besides measurability and weak measurability of multifunctions many other types of measurability are studied. They are also studied in Castaing and Valadier [1], Himmelberg [1], Nishiura [1], Toma [1]. Connections betweeen measurability and continuity of multifunctions, and some versions of the Lusin theorem for measurable multifunctions have been studied, e.g., Jacobs [1] and the unpublished dissertation of Malik (see Holý [1]). Concerning measurable selectors recall the paper Wagner [1]. With respect to integration let us mention the fundamental paper Aumann [1] and the paper Holý [1]. Some applications in differential equations are studied in Jarník and Kurzweil [1]. Finally recall some recent results by Holá [5], Holá and Holý [1], Holá, Di Maio and Holá [1].
The theory of random sets has been developed in two directions. A random set can be understood as a multifunction from a measurable space to a measurable space (Štěpán [1]) or as a multifunction from a measurable space to a topological space (as is our case). Some probabilistic results were achieved here (e.g., Bán [4], [5], Hiai [1], [2], Hiai and Umegaki [1], Puri and Ralescu [2–4]).
The theory of fuzzy random variables is another application of the theory of multifunctions (Kwakernaak [1], Ma Jifeng and Zuang Wanziu [1], Nahmias [1]) with many results of the probabilistic character (Bán [1–6], Miyakoshi and Shimbo [1], [2], Puri and Ralescu [1], Klement [3]).
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© 1997 Beloslav Riecan and Tibor Neubrunn
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Riečan, B., Neubrunn, T. (1997). Measurability and integrability of multifunctions. In: Integral, Measure, and Ordering. Mathematics and Its Applications, vol 411. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8919-2_11
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DOI: https://doi.org/10.1007/978-94-015-8919-2_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4855-4
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