## 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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### References:

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