Easy example of $f\in L_1^*\backslash L_\infty$?

Question

If I'm not mistaken the dual of $L_1$ is $L_\infty$ whenever the measured space is $\sigma$-finite. So I know where not to look for an easy example of $f\in L_1^*\backslash L_\infty$. Does anyone know where to do look ?

Answer 1

Let $X=[0,1]$. Let $\mu$ be counting measure on $X$, except restricted to the $\sigma$-algebra of sets $E$ such that $E$ or $X\setminus E$ is countable. Define $\Lambda\in(L^1(\mu))^*$ by $$\Lambda f=\sum_{x\in[0,1/2]}f(x).$$

Then $\Lambda$ cannot be represented as integration against any $L^\infty$ function; loosely speaking, it's clear that that function would have to be $\chi_{[0,1/2]}$, which is not measurable.