Exercise 9, § 1

Let $X$ be the compact interval $[0,1]$ of $\mathbf{R}$.

a) Let $\mu_n$ be the measure defined by the mass +1 at the point 0 and the mass -1 at the point $1/n$. Show that the sequence $(\mu_n)$ tends vaguely to 0 in $\mathscr{M}(X; \mathbf{C})$, but that $|\mu_n|$ does not tend vaguely to 0.

b) Let $\mu$ be Lebesgue measure on $X$ and let $g_n(x) = \sin nx$. Show that the measure $(1 - g_n) \cdot \mu$ is positive and tends vaguely to $\mu$ as $n$ tends to infinity, but does not tend strongly to $\mu$. From this, deduce that the topologies induced on $\mathscr{M}_+(X)$ by the vague topology and the strong topology (which is identical here to the quasi-strong topology) are distinct (cf. Exer. 10).