I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl.

THEOREM 1. Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leq p<\infty$, and $X$ 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 $\mu$.

Now I would like to extend it to $\sigma$-finite measure space. However, I'm stuck at proving that $\varphi$ is an isometry. Could you elaborate on how to prove it?

Thank you so much!

My attempt: Let $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space. There is a countable measurable partition $(\Omega_n)$ of $\Omega$ such that $\mu(\Omega_n) < \infty$. Let $\mu_n(A) := \mu(A \cap \Omega_n)$ for all $n$. Then $\mu_n$ is concentrated on $\Omega_n$. Also, $\mu = \sum \mu_n$ and $(\Omega, \Sigma, \mu_n)$ is a finite measure space. By Theorem 1, for each $n$ there is an isometric isomorphism $$ \varphi_n : L_{p}(\mu_n, X)^* \to L_{q} (\mu_n, X^*). $$

For $L \in L_{p}(\mu, X)^*$, we define $L_n \in L_{p}(\mu_n, X)^*$ by $$ L_n (f) := L(f 1_{\Omega_n}) \quad \forall f \in L_{p}(\mu_n, X). $$

Then we define $$ \varphi : L_{p}(\mu, X)^* \to L_{q} (\mu, X^*), L \mapsto \sum_n \varphi_n (L_n) $$

It's easy to see that $\varphi$ is an isomorphism.