Abstract

Suppose that X is a Banach space with continuous dual <TEX>$X^{**}$</TEX>, (<TEX>$\Omega$</TEX>, <TEX>$\Sigma$</TEX>, <TEX>${\mu}$</TEX>) is a finite measure space. f : <TEX>$\Omega\;{\longrightarrow}$</TEX> <TEX>$X^{*}$</TEX> is a weakly measurable function such that <TEX>$\chi$</TEX><TEX>$^{**}$</TEX> f <TEX>$\in$</TEX> <TEX>$L_1$</TEX>(<TEX>${\mu}$</TEX>) for each <TEX>$\chi$</TEX><TEX>$^{**}$</TEX> <TEX>$\in$</TEX> <TEX>$X^{**}$</TEX> and <TEX>$T_{f}$</TEX> : <TEX>$X^{**}$</TEX> \longrightarrow <TEX>$L_1$</TEX>(<TEX>${\mu}$</TEX>) is the operator defined by <TEX>$T_{f}$</TEX>(<TEX>$\chi$</TEX><TEX>$^{**}$</TEX>) = <TEX>$\chi$</TEX><TEX>$^{**}$</TEX>f. In this paper we study the properties of bounded <TEX>$X^{*}$</TEX> - valued weakly measurable functions and bounded <TEX>$X^{*}$</TEX> - valued weak＊ measurable functions.(omitted)