Given a Banach space $X$ and a measure space $(\mathfrak{A}, \mu)$ One can form the Banach space $L_\infty(\mu, X)$ of all measurable, essentially bounded functions from $\mathfrak{A}$ to $X$. Is it obvious that one can identify $L_\infty(\mu, X)$ with the projective tensor product $L_\infty(\mu)\hat{\otimes}X$? If I am mistaken (that is, this is not true), can we find another tensor product to make such identification?
EDIT: Since the answer is 'no', I would appreciate any other 'reasonable' descriptions of the Banach space $L_\infty(\mu, X)$, if there are any.