• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2002.05a
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• pp.277-280
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• 2002

In this paper, we characterize the Borel $\sigma$-field generated by the Hausdorff-Skorokhod metric on the space of normal and upper-semicontinuous fuzzy sets with compact support in the Ecleadean space R$\^$n/. As a result. we give a characterization of measurable fuzzy mappings .

• Communications of the Korean Mathematical Society
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• v.21 no.3
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• pp.429-432
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• 2006

We use Cauchy's functional equation to construct a new non-measurable set which is a (vector) subspace of \mathbb{R}$ and is of a codimensiion 1, considering \mathbb{R}$, the set of real numbers, as a vector space over a field \mathbb{Q}$ of rational numbers. Moreover, we show that \mathbb{R}$ can be partitioned into a countable family of disjoint non-measurable subsets.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1915-1922
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• 2013

We prove the admissibility of the space $L_0(\mathcal{A},X)$ of vector-valued measurable functions determined by real-valued finitely additive set functions defined on algebras of sets.

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.577-585
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• 2022

In this work, we introduced and study vector equilibrium problems for trifunction in measurable space (for short, VEPMS). The existence of solutions of (VEPMS) are obtained by employing Aumann theorem and Fan KKM lemma. As an application, we prove an existence result for vector variational inequality problem for measurable space. Our results in this paper are new which can be considered as significant extension of previously known results in the literature.

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1387-1401
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• 2019

Consider a finite measure space (${\Omega}$, ${\Sigma}$, ${\mu}$) and a Banach space $X({\mu})$ consisting of (equivalence classes of) real measurable functions defined on ${\Omega}$ such that $f{\chi}_A{\in}X({\mu})$ and ${\parallel}f{\chi}_A{\parallel}{\leq}{\parallel}f{\parallel}$, ${\forall}f{\in}({\mu})$, $A{\in}{\Sigma}$. We prove that if it satisfies the subsequence property, then it is an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm.

• The Pure and Applied Mathematics
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• v.4 no.1
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• pp.87-92
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• 1997

Some function of a complete finite measure space (for short, CFMS) into the duals and pre-duals of weakly compactly generated (for short, WCG) spaces are considered. We shall show that a universally weakly measurable function f of a CFMS into the dual of a WCG space has RS property and bounded weakly measurable functions of a CFMS into the pre-duals of WCG spaces are always Pettis integrable.

• Communications of the Korean Mathematical Society
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• v.18 no.4
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• pp.677-685
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• 2003

Let M be the vector space of all real S-measurable functions defined on a measure space (X, S, $\mu$). In this paper, we investigate some topological structure of T on M. Indeed, (M, T) becomes a topological vector space. Moreover, if $\mu$, is ${\sigma}-finite$, we can define a complete invariant metric on M which is compatible with the topology T on M, and hence (M, T) becomes a F-space.

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.425-434
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• 2007

We iteratively generate a sequence of measurable mappings and study necessary conditions for its convergence to a random fixed point of random nonexpansive operator. A random fixed point theorem for random nonexpansive operator, relaxing the convexity condition on the underlying space, is also proved. As an application, we obtained random fixed point theorems for Caristi type random operators.

• Honam Mathematical Journal
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• v.32 no.4
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• pp.633-642
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• 2010

In this note, we introduce the definition of the generalized analogue of Wiener measure on the space C[a, b] of all real-valued continuous functions on the closed interval [a, b], give several examples of it and investigate some important properties of it - the Fernique theorem and the existence theorem of scale-invariant measurable subsets on C[a, b].

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.607-618
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• 2017

In this paper, we introduce a quadratic stochastic operators on the set of all probability measures of a measurable space. We study the dynamics of the Lebesgue quadratic stochastic operator on the set of all Lebesgue measures of the set [0, 1]. Namely, we prove the regularity of the Lebesgue quadratic stochastic operators.