Suppose $X$ is a topological space, and $\mu$ is a Borel measure on $X$. Also suppose we have an $n$-dimensional vector bundle $E \to X$, with an inner product $\langle \cdot,\cdot \rangle_x$ on the fibre $E_x$ for all $x \in X$, in such a way that each $E_x$ is complete and such that there exists a vector bundle trivialisation which is compatible with the fibrewise inner products. The inner product $\langle \cdot, \cdot \rangle_x$ determines a norm $||\cdot||_x$ on $E_x$.

Say that a (not necessarily continuous) section $\sigma \colon X \to E$ is *measurable* if its restriction to each trivialising open set $U \subset X$ is given by a measurable function $U \to \mathbb{F}^n$ (here $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$, depending on how you feel). Denote the set of measurable sections (understood as being defined up to measure zero) by $\Gamma(E)$.

Given a section $\sigma \in \Gamma(E)$ and a number $p \in (0,\infty)$, we can define $$||\sigma||_p := \left( \int_X ||\sigma(x)||_x^p \; d\mu(x) \right)^{1/p},$$ and we can then define the corresponding $E$-valued Lebesgue space $L^p(X;E)$ in the obvious way.

Question:do we have the usual duality relations for Lebesgue spaces, i.e. $(L^p(X;E))^* \cong L^{p^\prime}(X;E)$ where $1 = \frac{1}{p} + \frac{1}{p^\prime}$?

As one would expect, there is a kind of Hölder inequality: if $\sigma \in L^p(X;E)$ and $\tau \in L^{p^\prime}(X;E)$, then the function $\langle \sigma,\tau \rangle$ on $X$, given by $$\langle \sigma,\tau \rangle(x) := \int_X \langle \sigma(x), \tau(x) \rangle_x \; d\mu(x),$$ satisfies $|| \langle \sigma,\tau \rangle||_{L^1(X)} \leq ||\sigma||_p ||\tau||_{p^\prime}$. It follows that the pairing $\langle \cdot,\cdot \rangle$ can be used to isometrically embed $L^{p^\prime}(X;E)$ into $(L^p(X;E))^*$.

However, I haven't been able to prove the reverse containment - that each functional $\phi \in (L^p(X;E))^*$ is given by pairing with an element of $L^{p^\prime}(X;E)$ - without additional assumptions, such as the existence of a finite trivialising cover for $E$ with uniform norm control (for example, when $X$ is compact). In this case, one can recover the result from the corresponding result for trivial bundles - which is essentially the case of vector-valued Lebesgue spaces - but constants appear which depend on the cardinality of a trivialising cover, which is somewhat unexpected.

Has this been explicitly proven anywhere? Is it even true in general? (I'll be very surprised if it isn't)