$L^\infty([0,1])$ is dual space of space of Radon measure

Question

In the lecture here (at 16:14s), he claims that the (continuous) dual space of space of Radon measures on $\Omega=[0,1]$ is $L^\infty(\Omega)$. So, I am wondering

1. Where I can cite such results?
2. Do we have similar result for (locally) compact set $\Omega$?

Answer 1

I think the instructor made a mistake there. It is not even true that $L^\infty(\Omega)$ is contained in the dual of the space $X$ of Radon measures (at least not in a natural way): for instance, it is impossible to define $$ \int_\Omega f\,d\mu, $$ for $f\in L^\infty(\Omega)$, in case, say, $\mu=\delta_x$ is the Dirac measure supported on $x$, since $f(x)$ is not well defined.

If you replace $L^\infty(\Omega)$ by the space $B(\Omega)$ of bounded measurable functions on $\Omega$ (not moding out when functions agree a.e.) then at least you may embedd $$B(\Omega) \hookrightarrow X'$$ and I think this is what the instructor had in mind.