Exercise 25, § 1

a) Let $f$ be a positive numerical function defined in $\mathbf{R}$, 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 the $t \in \mathbf{R}$ such that $f(t) \geq w$; put $v_f(w) = \mu^*(U_f(w))$. Now prove that for every $\alpha > 1$, one has

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

(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 Exercise 23 d)).

c) Let $\mu_n$ be the Lebesgue measure in $\mathbf{R}^n$, A, B two integrable subsets of $\mathbf{R}^n$. Now prove that one has (“Brunn-Minkowski inequality”)

(Reduce to the case where A and B are compact. Reason then by induction on $n$ using Exercise 23 d), the Lebesgue-Fubini Theorem, the inequality proved in b), as well as Hölder's inequality).