Exercise 25, § 1

a) Let $f$ be a positive numerical function defined on $\mathbf{R}$, integrable (for the Lebesgue measure $\mu$ on $\mathbf{R}$), bounded, and with compact support. Let $\gamma = \sup_{t \in \mathbf{R}} f(t)$. For every $w \in \mathbf{R}$, denote by $U_f(w)$ the set of $t \in \mathbf{R}$ such that $f(t) \geq w$; set $\nu_f(w) = \mu^*(U_f(w))$. Show that for every $\alpha > 1$,

b) Let $g$ be a second numerical function satisfying the same conditions as $f$, and set $\delta = \sup_{t \in \mathbf{R}} g(t)$. Let $h$ be the function defined on $\mathbf{R}^2$ by $h(u, v) = f(u) + g(v)$ if $f(u)g(v) \neq 0$, and $h(u, v) = 0$ otherwise; finally, set $k(t) = \sup_{u+v=t} h(u, v)$, so that $k$ is positive, integrable, bounded, and has compact support. Show that for every $\alpha > 1$,

(Observe that for $0 < w < 1$, one has $U_k(\gamma w + \delta w) \supset U_f(\gamma w) + U_g(\delta w)$, and use a) and Exer. 23 d).)

c) Let $\mu_n$ be Lebesgue measure on $\mathbf{R}^n$, A and B two integrable subsets of $\mathbf{R}^n$. Show that (Brunn–Minkowski inequality)

(Reduce to the case that A and B are compact. Then argue by induction on $n$ using Exer. 23 d), the Lebesgue–Fubini theorem, the inequality proved in b), as well as Hölder’s inequality.)