Exercise 3, § 2

Let T be the compact interval $[0, 1]$ of $\mathbf{R}$; for every integer $n > 0$, let $\lambda_n$ be the measure

on T. Show that, in the space $\mathscr{M}(T)$ equipped with the vague topology (or with the quasi-strong topology (Ch. III, §1, Exer. 8)), the sequence of measures $(\lambda_n)$ is summable, but the sequence $(|\lambda_n|)$ is not. (Observe that if $\alpha_n = n \cdot \lambda_n(f)$ then $\sum_n \alpha_n^2 < +\infty$, by making use of Bessel’s inequality, and deduce from this that $\sum_n |\lambda_n(f)| < +\infty$ by the Cauchy–Schwarz inequality.)