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Representation of additive functionals on Musielak-Orlicz space of vector valued functions. (English) Zbl 0618.46034

Let (T,\(\Sigma\),\(\mu)\) be a space with a complete, and \(\sigma\)-finite measure and X be a real separable Banach space. Moreover, let \(E_ M\) be a subspace of finite elements of non-solid Musielak-Orlicz space \(L_ M\) of functions with values in the space X. A functional \(\Phi:E_ M\to {\bar {\mathbb{R}}}\) is called to be additive if \(\Phi (f+g)=\Phi (f)+\Phi (g)\) for each \(f,g\in E_ M\) such that \(\mu\) (Supp \(f\cap Supp g)=0\). We will establish integral representations of the form \[ \Phi (f)=\int_{T}\phi (t,f(t))d\mu \] with certain kernel function \(\phi\) :T\(\times X\to {\bar {\mathbb{R}}}\). Representation theorems have been obtained for continuous and lower semicontinuous additive functionals.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E40 Spaces of vector- and operator-valued functions
47B38 Linear operators on function spaces (general)
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