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