Exercise 26, § 5

Let $\mu_1, \ldots, \mu_m$ be a finite number of measures on a locally compact space T, which one can write in the form $\mu_k = f_k \cdot \mu$, where $\mu$ is a measure $\geq 0$ and $|f_k| \leq 1$ (No. 9).

a) Let $\Phi$ be a set of functions defined and positively homogeneous on $\mathbf{R}^m$ and such that $u(f_1, \ldots, f_m)$ is locally $\mu$-integrable for every function $u \in \Phi$. Show that if a sequence $(u_n)$ of functions in $\Phi$ converges uniformly on every compact subset of $\mathbf{R}^m$ to a function $u$, then $u(\mu_1, \ldots, \mu_m)$ is the limit of the sequence of measures $u_n(\mu_1, \ldots, \mu_m)$ for the quasi-strong topology on $\mathscr{M}(T)$ (Ch. III, §1, Exer. 8).

b) Suppose that $u \in \Phi$ is continuous on $\mathbf{R}^m$. Show that the mapping

of $(\mathscr{M}(T))^m$ into $\mathscr{M}(T)$ is continuous for the quasi-strong topology (begin by considering the case that $u$ is Lipschitz; then use a) and the Stone–Weierstrass theorem).

c) Suppose that $u$ is continuous on $\mathbf{R}^m$. Let $g$ be any function in $\mathscr{H}(T)$, and K its support. For every $\varepsilon > 0$, show that there exists a finite open covering $(A_i)$ of K formed of relatively compact sets, such that for every finite open covering $(B_j)$ of K, finer than $(A_i)$, and for every family $(h_j)$ of continuous mappings of T into $[0, 1]$ such that $h_j$ has its support in $B_j$ and such that $\sum_j h_j(t) = 1$ on K, one has

(consider first the case that $u$ is Lipschitz and the $f_k$ are continuous, then pass to the case that $u$ is Lipschitz and the $f_k$ are locally integrable, and finally the general case using the Stone–Weierstrass theorem).

d) In $\mathbf{R}^2$, let $u(x_1, x_2)$ be the positively homogeneous function equal to $\sqrt{x_1^2 + x_2^2}$ when $x_1/x_2$ is irrational, and to 0 in the contrary case. Let $\mu$ be Lebesgue measure on $T = [0, 1]$, and let $\nu$ be the measure on T defined by $d\nu(t) = t \, d\mu(t)$. Show that if $g$ is a continuous function $\geq 0$ on T, then $\sum_j u(\int gh_j \, d\mu, \int gh_j \, d\nu)$ does not tend to any limit with respect to the directed set of finite open coverings of the support of $g$.