I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl. Here we use the Bochner integral.
Theorem 1 Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leq p<\infty$, and $(X, |\cdot|)$ be a Banach space. Then $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ where $p^{-1}+q^{-1}=1$, if and only if $X^{*}$ has the Radon-Nikodým property with respect to $\mu$.
I'm trying to extend above result to $\sigma$-finite measure spaces. The idea of the proof is from this comment by @JochenWengenroth.
Could you have a check on my below attempt? Thank you so much for your help!
Proof: Let $\mu$ be $\sigma$-finite. We need the following lemma as a bridge, i.e.,
Lemma Let $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space. There is a finite measure $\nu$ on $(\Omega, \Sigma)$ such that $\mu$ has a density $f:X \to (0, \infty)$ w.r.t. $\nu$.
Let $(f, \nu)$ be given by the Lemma. By Theorem 1, there is an isometric isomorphism $$ \varphi: L_{p}(\nu, X)^* \to L_{q} (\nu, X^*). $$
Notice that $$ g \in L_{p}(\nu, X) \iff \int |g|^p \mathrm d \nu = \int |f^{-1/p}g|^p \mathrm d \mu <\infty \iff f^{-1/p}g \in L_{p}(\mu, X). \quad (\star) $$
We define $$ \psi: L_{p}(\mu, X)^* \to L_{p}(\nu, X)^* $$ by $$ \psi(H) (g) = H (f^{-1/p}g) \quad \forall H \in L_{p}(\mu, X)^*, \forall g \in L_{p}(\nu, X). $$
It follows from $(\star)$ that $\psi$ is an isometric isomorphism. Again, $$ g \in L_{q}(\nu, X^*) \iff \int |g|^q \mathrm d \nu = \int |f^{-1/q}g|^q \mathrm d \mu <\infty \iff f^{-1/q}g \in L_{q}(\mu, X). \quad (\star\star) $$
We define $$ \phi: L_{q} (\nu, X^*) \to L_{q} (\mu, X^*), g \mapsto f^{-1/q}g. $$
It follows from $(\star\star)$ that $\phi$ is an isometric isomorphism. As such, $$ \phi \circ \varphi \circ \psi : L_{p}(\mu, X)^* \to L_{q} (\mu, X^*). $$ is an isometric isomorphism . This completes the proof.
