Lebesgue space - $L^p$ spaces

Question

Let $X=\{a,b\}$,and $\mu(\{a\} )=1$, and $\mu(\{b\} )=\mu(X)=+\infty$ and $\mu(\emptyset)=0$. Is it truth that $L^\infty(\mu)$ is the dual space of $L^1(\mu)$. Whether $L^\infty(\mu)=L^1(\mu)^\ast$? If no, why?

Answer 1

Here we have

$$ \|f\|_{L^1} = \begin{cases}|f(a)| & \text{if } f(b) = 0\\ +\infty & \text{else}\end{cases} $$ so that $L^1(\mu) = \{f \in\mathcal{F}(\{a,b\},\Bbb R) \mid f(b) = 0\}\simeq \Bbb R$ and $L^\infty(\mu) = \mathcal{F}(\{a,b\},\Bbb R)\simeq \Bbb R^2$.

In particular, $\dim L^1(\mu) = 1$ and $\dim L^\infty (\mu) = 2$ ...