Spaces Of p-Tensor Integrable Functions and Related Banach Space Properties

Abstract

In [9] G. F. Stefansson has studied the Banach space $L_1(\nu, X, Y)$, the space of all tensor integrable functions $f : \Om \to X$ with respect to a countably additive vector valued measure $\nu : \Si \to Y$ and also the tensor integral of weakly $\nu$-measurable functions. In [1] we obtained some Banach space properties of $L_1(\nu, X, Y)$ and also of w-$L_1(\nu, X, Y)$, the space of all weakly tensor integrable functions. In the present paper, for $1 < p < \infty$, we define the spaces $L_p(\nu, X, Y)$ and w-$L_p(\nu, X, Y)$ of all $\check \otimes_p$-integrable functions and weakly $\check \otimes_p$-integrable functions respectively and discuss several basic properties of these spaces. We also study vector measure duality in $L_p(\nu, X, Y)$ for $1 < p < \infty$.