The Radon-Nikod$\acute{Y}$m property of $\mathbb{L}$-Banach spaces and the dual representation theorem of $\mathbb{L}$-Bochner function spaces

Abstract

In this paper, we first introduce $\mathbb{L}$-$\mu$-measurable functions and $\mathbb{L}$-Bochner integrable functions on a finite measure space $(S,\mathcal{F},\mu),$ and give an $\mathbb{L}$-valued analogue of the canonical $L^{p}(\Omega,\mathcal{F},\mu).$ Then we investigate the completeness of such an $\mathbb{L}$-valued analogue and propose the Radon-Nikod$\acute{y}$m property of $\mathbb{L}$-Banach spaces. Meanwhile, an example constructed in this paper shows that there do exist an $\mathbb{L}$-Banach space which fails to possess the Radon-Nikod$\acute{y}$m property. Finally, based on above work, we establish the dual representation theorem of $\mathbb{L}$-Bochner integrable function spaces, which extends and improves the corresponding classical result.