$(\ell^{1})^{*}$ isometrically isomorphic to $\ell^{\infty}$ as a corollary of Riesz representation Theorem.

Question

I'm following "A Course in Functional Analysis, Conway" and as a corollary of the Riesz theorem (Example 5.9) he states what I have written in the title.
If I consider $\mathbb{N}$ with the discrete topology and the measure $\mu$ that counts the points, I have a locally compact space so I can apply the Riesz Theorem obtaining $C_{0}(\mathbb{N})^*$ isometrically isomorphic to $M(\mathbb{N})$. I agree that $C_{0}(\mathbb{N})^*=\ell^{1}$ but I can't see why $M(\mathbb{N})^*= \ell^{\infty}$. I also don't understand the example 5.10 but I think that solving 5.9 would help me understand better 5.10.