Exercise 20, § 6

a) Let Y be a metric space; for every $y \in Y$ and every $r > 0$, denote by $B(y; r)$ the open ball in Y with center $y$ and radius $r$. Let $\mathscr{G}$ be a set of open balls in Y whose diameters form a bounded set in $\mathbf{R}$, and which is such that, for every sequence $(B(y_n; r_n))$ of pairwise disjoint balls belonging to $\mathscr{G}$, one has $\lim_{n \to \infty} r_n = 0$. Show that if M is the union of the balls $B \in \mathscr{G}$, there exists a sequence of pairwise disjoint balls $B(y_n; r_n) \in \mathscr{G}$ such that the balls $B(y_n; 4r_n)$ form a covering of M. (If $k > 0$ is an upper bound for the set of radii of the balls $B \in \mathscr{G}$, define by induction on $h$ a sequence of families $(\mathfrak{F}_h)$ of balls $B(y_{hj}; r_{hj}) \in \mathscr{G}$ such that $\mathfrak{F}_h$ is maximal among the (finite) families of balls belonging to $\mathscr{G}$, pairwise disjoint and disjoint from the balls belonging to the families $\mathfrak{F}_i$ for $i < h$, and with radii between $(2/3)^{h+1}k$ and $(2/3)^hk$.)

b) Let X be a metrizable locally compact space, $d$ a metric on X compatible with its topology; again denote by $B(x; r)$ the open ball with center $x$ and radius $r > 0$ for this metric, and by $\delta(A)$ the diameter of a subset A of X for the metric $d$. Let $\mu$ be a positive measure on X satisfying the following conditions: 1° every open ball is $\mu$-integrable; 2° $\mu(B(x; 4r)) \leq K \cdot \mu(B(x; r))$, where K is a constant $> 1$ independent of $x$ and $r$; 3° if a sequence $(B_n)$ of open balls is such that $\lim_{n \to \infty} \mu(B_n) = 0$, then $\lim_{n \to \infty} \delta(B_n) = 0$; 4° if a sequence $(B_n)$ of open balls is such that $\lim_{n \to \infty} \delta(B_n) = +\infty$, then $\lim_{n \to \infty} \mu(B_n) = +\infty$.

Let $f \in \mathscr{L}^p(X; \mu)$ be a function with values $\geq 0$ ($1 < p < +\infty$); for every $x \in X$, set

where B runs over the set of open balls with center $x$. Show that $\overline{f}$ is finite and lower semi-continuous on X.

c) Let $f^*$ be the decreasing rearrangement of $f$ (§5, Exer. 29); for $t > 0$, set

which is a decreasing continuous function such that $f^* \leq \beta_f$; one has $\beta_f(t) = o(t^{-1/p})$ as $t$ tends to 0 or to $+\infty$ (Exer. 19 a)). Finally, denote by $\gamma_f$ the inverse function of $\beta_f$, defined on the interval $]0, \beta_f(0+)[$, and extended by 0 to the exterior of this interval if $\beta_f(0+)$ is finite.

For every $t > 0$, denote by $M_t$ the set of $x \in X$ such that $\overline{f}(x) > t$. Show that $\mu(M_t) \leq K \cdot \gamma_f(t)$. (For every $x \in M_t$, let $B_x$ be an open ball with center $x$ such that $\int_{B_x} f(y) d\mu(y) \geq t \cdot \mu(B_x)$; apply a) to the family of balls $B_x$.)

d) Deduce from c) that

(Note that the first member may also be written $\int_0^{+\infty} p t^{p-1} (\mu(M_t)) dt$, and make use of Hardy’s inequality (Exer. 19).)