• 충청수학회지
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• 제4권1호
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• pp.121-126
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• 1991

In this paper, we show that uniform integrability is equivalent to convergence to a ${\mu}$-integrable function f in $L_1$ for ${\mu}$-integrable functions in the sense of the integral defined by Lewis.

• East Asian mathematical journal
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• 제17권2호
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• pp.253-264
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• 2001

• Korean Journal of Mathematics
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• 제7권1호
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• pp.123-130
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• 1999

We characterize the compactness, weak precompactness and weak compactness in $L^P({\mu},X)$ and in more general space $P^c({\mu},X)$. Moreover, we present this characterization in terms of extremal structure in X.

• 충청수학회지
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• 제5권1호
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• pp.159-165
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• 1992

In 1986, R. Huff [3] showed that a Dunford integrable function is Pettis integrable if and only if T : $X^*{\rightarrow}L_1(\mu)$ is weakly compact operator and {$T(K(F,\varepsilon))|F{\subset}X$, F : finite and ${\varepsilon}$ > 0} = {0}. In this paper, we introduce the notion of Bourgain property of real valued functions formulated by J. Bourgain [2]. We show that the class of pettis integrable functions is linear space and if lis bounded function with Bourgain property, then T : $X^{**}{\rightarrow}L_1(\mu)$ by $T(x^{**})=x^{**}f$ is $weak^*$ - to - weak linear operator. Also, if operator T : $L_1(\mu){\rightarrow}X^*$ with Bourgain property, then we show that f is Pettis representable.

• 충청수학회지
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• 제8권1호
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• pp.111-117
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• 1995

A function $f:{\Omega}{\rightarrow}X$ is called intrinsically-separable valued if there exists $E{\in}{\Sigma}$ with ${\mu}(E)=0$ such that $f({\Omega}-E)$ is a separable in X. For a given Dunford integrable function $f:{\Omega}{\rightarrow}X$ and a weakly compact operator T, we show that if f is intrinsically-separable valued, then f is Pettis integrable, and if there exists a sequence ($f_n$) of Dunford integrable and intrinsically-separable valued functions from ${\Omega}$ into X such that for each $x^*{\in}X^*$, $x^*f_n{\rightarrow}x^*f$ a.e., then f is Pettis integrable. We show that a function f is Pettis integrable if and only if for each $E{\in}{\Sigma}$, F(E) is $weak^*$-continuous on $B_{X*}$ if and only if for each $E{\in}{\Sigma}$, $M=\{x^*{\in}X^*:F(E)(x^*)=O\}$ is $weak^*$-closed.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제9권1호
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• pp.73-79
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• 2002

In this paper, We Characterize the Pettis integrability for the Dunford integrable functions on a perfect finite measure space.

• 대한수학회보
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• 제32권2호
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• pp.205-209
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• 1995

Throughout this paper X is a Banach space and $\mu$ is the Lebesgue measure on [0, 1] and all operators are assumed to be bounded and linear. $L^1(\mu)$ is the Banach space of all (classes of) Lebesgue integrable functions on [0, 1] with its usual norm. Let $T : L^1(\mu) \to X$ be an operator.

• 충청수학회지
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• 제8권1호
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• pp.161-166
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• 1995

Let (${\Omega}$, ${\Sigma}$, ${\mu}$) be a finite perfect measure space, and let $f:{\Omega}{\rightarrow}X$ be strongly measurable. f is Pettis integrable if and only if there is a sequence ($f_n$) of Pettis integrable functions from ${\Omega}$ into X such that (a) there is a positive increasing function ${\phi}$ defined on [0, ${\infty}$) such that ${\lim}_{t{\rightarrow}{\infty}}\frac{{\phi}(t)}{t}={\infty}$ and sup $f_{\Omega}{\phi}({\mid}x^*f_n{\mid})d{\mu}$ < ${\infty}$ for each $x^*{\in}B_{X*}$,$n{\in}N$, and (b) for each $x^*{\in}X^*$, $lim_{n{\rightarrow}{\infty}}x^*f_n=x^*fa.e.$.

• International Journal of Fuzzy Logic and Intelligent Systems
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• 제9권2호
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• pp.71-75
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• 2009

The purpose of this paper is to prove a Jensen type inequality for Choquet integrals with respect to a non-additive measure which was introduced by Choquet [1] and Sugeno [20]; $$\Phi((C)\;{\int}\;fd{\mu})\;{\leq}\;(C)\;\int\;\Phi(f)d{\mu},$$ where f is Choquet integrable, ${\Phi}\;:\;[0,\;\infty)\;\rightarrow\;[0,\;\infty)$ is convex, $\Phi(\alpha)\;\leq\;\alpha$ for all $\alpha\;{\in}\;[0,\;{\infty})$ and ${\mu}_f(\alpha)\;{\leq}\;{\mu}_{\Phi(f)}(\alpha)$ for all ${\alpha}\;{\in}\;[0,\;{\infty})$. Furthermore, we give some examples assuring both satisfaction and dissatisfaction of Jensen type inequality for the Choquet integral.

• 대한수학회지
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• 제43권2호
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• pp.357-371
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• 2006

Let $X\;=\;(X_t,\;t{\in}[0, T])$ be a generalized Brownian motion(gBm) determined by mean function a(t) and variance function b(t). Let $L^2({\mu})$ denote the Hilbert space of square integrable functionals of $X\;=\;(X_t - a(t),\; t {in} [0, T])$. In this paper we consider a class of nonlinear functionals of X of the form F(. + a) with $F{in}L^2({\mu})$ and discuss their analysis. Firstly, it is shown that such functionals do not enjoy, in general, the square integrability and Malliavin differentiability. Secondly, we establish regularity conditions on F for which F(.+ a) is in $L^2({\mu})$ and has its Malliavin derivative. Finally we apply these results to compute the price and the hedging portfolio of a contingent claim in our financial market model based on a gBm X.