Exercise 2, § 4

Let X and $\nu$ be the locally compact space and the measure defined in Exer. 5 of Ch. IV, §1; let $f$ be the characteristic function of the set D of points $(0,y)$ in X. Show that, in the following two cases, $\nu = \int g(t) \varepsilon_{\pi(t)} d\mu(t)$ and

a) Take $T = X$, and for $\mu$ the measure defined by the mass $\log n / n^3$ at each of the points $(1/n, k/n^2)$. Take for $\pi$ the identity mapping, and $g$ defined by the following conditions: $g$ is continuous, but $g(t) > 0$ does not hold for every $t \in T$.

b) Take for T the set X equipped with the discrete topology, and for $\mu$ the measure defined by the same masses as $\nu$. Take $g(t) = 1$ for all $t$, and for $\pi$ the identity mapping: the latter is continuous, but not proper.