Let $X$ be a Banach lattice and let $K=\{x^{*}\in B_{X^{*}}:x^{*}\geq 0\}$. Then $K$ is a compact Hausdorff space under the $weak^{*}$-topology. Let $\mu$ be a regular Borel probability measure on $K$. We denote by $i_{X}$ the embedding $X\rightarrow C(K)$ given by$\langle i_{X}(x),x^{*}\rangle=\langle x^{*},x\rangle, x\in X, x^{*}\in K.$ It is easy to see that $i_{X}$ is an isomorphic embedding.
Let $1\leq p<\infty$. Let us define a seminorm $|||\cdot|||$ on vector space $i_{X}(X)$ by $$|||f|||=[\int_{K}\langle x^{*},|x|\rangle^{p}d\mu(x^{*})]^{\frac{1}{p}}, f=i_{X}(x)\in i_{X}(X).$$ We set $$R=\{f\in i_{X}(X):|||f|||=0\}.$$ Then $i_{X}(X)/R$ becomes a normed space under the norm $$\|[f]\|=|||f|||,f\in i_{X}(X).$$ Let $L^{p}_{0}(\mu)$ be the completion of the normed space $(i_{X}(X)/R,|||\cdot|||)$.
Question 1. what's the dual of $L^{p}_{0}(\mu)$?
Question 2. Is $L^{p}_{0}(\mu)$ reflexive for $1<p<\infty$?
Question 3. Is $L^{p}_{0}(\mu)$ an $\mathcal{L}_{p}$-space?
Thank you!
