91Ó°ÊÓ

(i) Show that \(C([0,1])\) has a separable dense subset. (ii) Show that the space \(\left(C_{b}([0, \infty)),\|\cdot\|_{\infty}\right)\) of bounded continuous functions, equipped with the supremum norm, is not separable. (iii) Show that the space \(C_{c}([0, \infty))\) of continuous functions with compact support, equipped with the supremum norm, is separable.

Let \(\mu\) be a locally finite measure. Show that \(\mu(K)<\infty\) for any compact set \(K\).

Let \(E=\mathbb{R}\) and \(\mu_{n}=\delta_{n}\) for \(n \in \mathbb{N}\). Show that \(\mathrm{v}-\lim _{n \rightarrow \infty} \mu_{n}=0\) but that \(\left(\mu_{n}\right)_{n \in \mathbb{N}}\) does not converge weakly.