I am trying to answer this question (#8 on page.404 in Royden and Fitzpatrick "Real Analysis") it is for general measure space and for Riesz representation theorem for the dual of $L^q$:
Give an example of a measure space $(X, \mathfrak{M}, \mu)$ for which the Riesz Representation Theorem does extend to the case $p=\infty.$
But, I have found this question here:
riesz representation theorem for the case when p is infinity for the case of lebesgue measure
Does this means that my question is wrong? If not, is there any hint for solving it?
Read the exercise, and the linked question, more carefully.
The linked question shows that the statement is not true in general; that there exist measure spaces for which it fails. That does not mean that it fails for every measure space; there could still be other measure spaces $(X,\mu)$ for which the dual of $L^\infty(X,\mu)$ is indeed $L^1(X,\mu)$. All we know from the linked question is that $X=\mathbb{N}$ (with $\mu$ being counting measure) won't work.
There is nothing wrong with the question in Royden. Such spaces do exist, and in fact some extremely simple examples will work...