Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces

In this paper we study the Pettis integral of fuzzy mappings in arbitrary Banach spaces. We present some properties of the Pettis integral of fuzzy mappings and we give conditions under which a scalarly integrable fuzzy mapping is Pettis integrable

The Hahn-Banach Extension Theorem for Fuzzy Normed Spaces Revisited

Copyright © 2014 C. Alegre and S. Romaguera. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.This paper deals with fuzzy normed spaces in the sense of Cheng and Mordeson. We characterize fuzzy norms in terms of ascending and separating families of seminorms and prove an extension theorem for continuous linear functionals on a fuzzy normed space. Our result generalizes the classical Hahn-Banach extension theorem for normed spaces.The authors thank the reviewers for their valuable suggestions. They also acknowledge the support of the Spanish Ministry of Economy and Competitiveness under Grant MTM2012-37894-C02-01.Alegre Gil, MC.; Romaguera Bonilla, S. (2014). The Hahn-Banach Extension Theorem for Fuzzy Normed Spaces Revisited. Abstract and Applied Analysis. 2014:1-8. https://doi.org/10.1155/2014/151472S182014Bag, T., & Samanta, S. K. (2005). Fuzzy bounded linear operators. Fuzzy Sets and Systems, 151(3), 513-547. doi:10.1016/j.fss.2004.05.004Katsaras, A. K. (1984). Fuzzy topological vector spaces II. Fuzzy Sets and Systems, 12(2), 143-154. doi:10.1016/0165-0114(84)90034-4Felbin, C. (1992). Finite dimensional fuzzy normed linear space. Fuzzy Sets and Systems, 48(2), 239-248. doi:10.1016/0165-0114(92)90338-5Kaleva, O., & Seikkala, S. (1984). On fuzzy metric spaces. Fuzzy Sets and Systems, 12(3), 215-229. doi:10.1016/0165-0114(84)90069-1Schweizer, B., & Sklar, A. (1960). Statistical metric spaces. Pacific Journal of Mathematics, 10(1), 313-334. doi:10.2140/pjm.1960.10.313Alegre, C., & Romaguera, S. (2010). Characterizations of metrizable topological vector spaces and their asymmetric generalizations in terms of fuzzy (quasi-)norms. Fuzzy Sets and Systems, 161(16), 2181-2192. doi:10.1016/j.fss.2010.04.00

Equivalent Results in Minimax Theory

In this paper we review known minimax results with applications ingame theory and show that these results are easy consequences of thefirst minimax result for a two person zero sum game with finite strategysets published by von Neumann in 1928: Among these results are thewell known minimax theorems of Wald, Ville and Kneser and their generalizationsdue to Kakutani, Ky-Fan, König, Neumann and Gwinner-Oettli. Actually it is shown that these results form an equivalent chainand this chain includes the strong separation result in finite dimensionalspaces between two disjoint closed convex sets of which one is compact.To show these implications the authors only use simple propertiesof compact sets and the well-known Weierstrass Lebesgue lemma.convex analysis;game theory;finite dimensional separation of convex sets;generalized convexity;minimax theory

The Expectation Monad in Quantum Foundations

The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: algebras of the expectation monad are convex compact Hausdorff spaces, and are dually equivalent to so-called Banach effect algebras. These structures capture states and effects in quantum foundations, and also the duality between them. Moreover, the approach leads to a new re-formulation of Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.Comment: In Proceedings QPL 2011, arXiv:1210.029

A Note on Fuzzy Set--Valued Brownian Motion

In this paper, we prove that a fuzzy set--valued Brownian motion $B_t$, as defined in [1], can be handle by an $R^d$--valued Wiener process $b_t$, in the sense that B_t =\indicator{b_t}; i.e. it is actually the indicator function of a Wiener process