I'm reading *Theorem 1* at page 98 of *Vector Measures* by Joseph Diestel, John Jerry Uhl. Here we use the [Bochner integral](https://math.stackexchange.com/questions/4298588/dominated-convergence-theorem-for-banach-space).



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> **Theorem 1** Let $(\Omega, \Sigma, \mu)$ be a $\sigma$-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](https://math.stackexchange.com/questions/4546633/construct-conditional-expectation-on-banach-spaces-with-radon-nikod%c3%bdm-property) with respect to $\mu$.
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The isometric isomorphism given in the book is $\varphi: L\_{p}(\mu, X)^\* \to L\_{q} (\mu, X^\*)$ such that
$$
H (f) = \int\_\Omega \langle \varphi(H), f \rangle \mathrm d \mu \quad \forall H \in L\_{p}(\mu, X)^\*, \forall f \in L\_{p}(\mu, X).
$$


I feel that below result holds, i.e.,



> 
> **Corollary:** Let $N \in \Sigma$ and $K \in L\_{p}(\mu, X)^\*$ such that $K(f1\_N)=0 \quad \forall f \in L\_{p}(\mu, X)$. Then $\varphi (K)1\_N = 0$, i.e., $\varphi(K)=0$ on $N$.
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Could you provide me with some hints? Does it hold in case of other isometric isomorphisms?


Thank you so much for your elaboration.




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**My attempt:** We need to show that
$$
\int\_N |\varphi(K)|\_{X^\*} \mathrm d \mu =0.
$$


We have
$$
K(f1\_N) = \int \langle \varphi(K)1\_N, f \rangle \mathrm d \mu = 0 \quad \forall f \in L\_{p}(\mu, X).
$$


It suffices to prove that if $T \in L\_{q} (\mu, X^\*)$ such that
$$
\int \langle T, f \rangle \mathrm d \mu = 0 \quad \forall f \in L\_{p}(\mu, X),
$$
then $T=0$ $\mu$-a.e.




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I posted [this question](https://math.stackexchange.com/questions/4578140/if-h-in-l-q-mu-x-such-that-int-langle-h-f-rangle-mathrm-d-mu) on MSE, but have not received any answer so far. So I post it here.



If $H \in L_{q} (\mu, X^*)$ such that $\int \langle H, f \rangle \mathrm d \mu = 0$ for all $f \in L_{p}(\mu, X)$, then $H=0$ $\mu$-a.e
It is always true that $L\_q(\mu,X^\*)\hookrightarrow L\_p(\mu,X)^\*$ isometrically (without the assumption that $X$ has the Radon-Nikodym property). We need the Radon-Nikodym property to guarantee that this isometric embedding is a surjection.


From this it follows that for any $T\in L\_q(\mu,X^\*)$, $$\|T\|=\sup\Bigl\{\Bigl|\int \langle T,f\rangle d\mu\Bigr| : f\in B\_{L\_p(\mu,X)}\Bigr\}.$$ If $\int \langle T,f\rangle d\mu=0$ for all $f\in L\_p(\mu,X)$, then it follows that $\|T\|=0$, and $T$ is the zero vector. From this it follows that $T=0$ $\mu$ a.e.


