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

TL;DR

This paper develops an $ ext{L}$-valued analogue of classical $L^p$ spaces, investigates the Radon-Nikodým property in $ ext{L}$-Banach spaces, and establishes a dual representation theorem for $ ext{L}$-Bochner integrable functions, extending classical results.

Contribution

It introduces $ ext{L}$-measurable and $ ext{L}$-Bochner integrable functions, studies their properties, and proves a dual representation theorem, extending classical Banach space theory.

Findings

Existence of an $ ext{L}$-Banach space without the Radon-Nikodým property.

Established dual representation theorem for $ ext{L}$-Bochner spaces.

Extended classical $L^p$ space results to the $ ext{L}$-valued setting.

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.