Exercise 2, § 3

The spaces X and E and the mapping f being defined as in Exer. 1, show that the mapping $\lambda \mapsto \lambda(f)$ of $\mathscr{M}(X; \mathbf{C})$ into E is not continuous for the vague topology. (If $g_k$ ($1 \leqslant k \leqslant n$) are n continuous scalar functions on X, show that there exists a discrete measure $\lambda$ on X such that $\int g_k\ d\lambda = 0$ for $1 \leqslant k \leqslant n$ but $\| \int f\ d\lambda \|$ is arbitrarily large.)