Exercise 9, § 1

Let $G$ be a locally compact group (not necessarily commutative) endowed with a left Haar measure.

a) If $f\in L^1(G)$, there exist $f_1$ and $f_2$ in $L^1(G)$ such that $f_1*f_2=f$. (Use exercise 30 of I, p. 185.)

b) Let $f_1, f_2\in L^1(G)$ with $f_1\geqslant 0,f_2\geqslant 0$. Now prove that $f_1*f_2$ is almost everywhere equal to a lower semi-continuous function. (Consider $f_1$ as the simple limit of an increasing sequence of functions $\geqslant 0$ of $L^{\infty}(G)$.)

c) Take $G =\mathbf{R}$. For every open interval $I = ]a, b[$, put

Define intervals $I_n$ for $n\geqslant 1$ and $I'_n$ for $n\geqslant 0$ by $I'_0= ]0,1[$ and

for $n\geqslant 1$. Let F be the union of the intervals $I_n$ for $n\geqslant 1$. There does not exist any pair $(F_1,F_2)$ of subsets of $\mathbf{R}$ such that $F_1+ F_2= F$. d) If $f$ is a continuous function $\geqslant 0$ on $\mathbf{R}$ such that $F =\{x\in \mathbf{R}|f(x)>0\}$, then there does not exist any pair of functions $(f_1, f_2)$ in $L^1(\mathbf{R})$ such that $f_1\geqslant 0,f_2\geqslant 0$ and $f=f_1*f_2$. (Suppose such an equality. Let $E_i$ be the set of the $x\in \mathbf{R}$ such that $f_i(x)>0$. Let $F_i$ be the set of points of density of $E_i$ (INT, V, §6, exerc. 15). Prove that $F = F_1+ F_2.$)