Let $(X,M, μ)$ be the measure space where $X = \{a, b\}, M= P(X),$ and $μ(\{a\}) = 1, μ(\{b\}) = ∞.$ Discuss the duality relation between $L^p(X)$ and $L^q(X), 1/p + 1/q = 1, $ for $1 ≤ p < ∞.$
My Work:
Here, $\mu$ is not $\sigma$- finite. For $1<p<\infty$ $\sigma$- finiteness of $\mu$ is not needed. Here $E=\{a\}$ is a $\sigma$- finite subset of $X$. So, given a bounded linear functional $L$ on $L^p(X)$, there is a unique $g\in L^q(E)$ such that $g$ vanishes off $E$ and $Lf=\int_X fg \;d\mu$ for all $f\in L^p(X)$.
My problems are,
i) Is my argument correct upto this point?
ii) What else I can discuss for the case where $1<p<\infty$?
iii) When $p=1$, $\sigma$- finiteness of $\mu$ is needed. So, what can I mention for this case here?
Any help would be appreciated.