Suppose $\Omega$ is a $\sigma$-finite measure space (I'm happy to take $\Omega = \mathbb{N}$) and let $X$ be a Banach space. It's pretty well known that the Banach dual of $L^\infty(\Omega)$ can be identified with the space of finitely additive measures of bounded variation on $\Omega$, but is there a corresponding result for the Bochner space $L^\infty(\Omega;X)$?

Edit: for clarity, I define $\ell^\infty(\mathbb{N};X)$ to be the space of *all* bounded functions $\mathbb{N} \to X$. When $X$ is infinite dimensional, the set of simple functions (i.e. functions with finite range) is not dense in $\ell^\infty(\mathbb{N};X)$, because bounded sets in $X$ are not totally bounded. I only know the answer to this question in the `trivial' case where $X = L^\infty(\Omega)$ for some $\Omega$.

Vector Measuresby Dinculeanu. I suspect the answer is going to be "finitely additive, bounded variation, absolutely continuous, $X^*$-valued measures on $\Omega$". There is a general theory of measures taking values in a Banach space, and it generally works like you think it does. $\endgroup$ – Nate Eldredge Sep 25 '16 at 14:12