Exercise 3, § 3

For $x = (x_1, \ldots, x_n) \in \mathbf{R}^n$ one sets $|x|^2 = x_1^2 + \cdots + x_n^2$. Let $\mathscr{M}_1$ be the set of measures $\mu$ on $\mathbf{R}^n$ for which there exists a real number $k$ such that the function $(1 + |x|^2)^k$ is $\mu$-integrable. Let $\mathscr{M}_2$ be the set of measures $\nu$ on $\mathbf{R}^n$ for which the function $(1 + |x|^2)^k$ is $\nu$-integrable for every $k$. Show that if $\mu \in \mathscr{M}_1$ and $\nu \in \mathscr{M}_2$, then $\mu$ and $\nu$ are convolvable, and $\mu * \nu \in \mathscr{M}_1$; if $\mu \in \mathscr{M}_2$ and $\nu \in \mathscr{M}_2$, then $\mu * \nu \in \mathscr{M}_2$. (One first shows that if $u \geq 0,\ v \geq 0$, then

from this, one deduces that for any $x, y$ in $\mathbf{R}^n$,

Then let $\mu \in \mathscr{M}_1,\ \nu \in \mathscr{M}_2$ with $\mu \geq 0,\ \nu \geq 0$. Let $f$ be a continuous function $\geq 0$ on $\mathbf{R}^n$. There exists a $k$ such that $\mu = (1 + |x|^2)^k \cdot \mu_1$ with $\mu_1$ bounded; let $\nu_1 = (1 + |x|^2)^k \cdot \nu$, which is bounded. Then

Whence the convolvability of $\mu$ and $\nu$ and the fact that $\mu * \nu \in \mathscr{M}_1$. Argue in an analogous manner for $\mu, \nu$ in $\mathscr{M}_2$.