Proposition 1

— Let G be a locally compact group, $\lambda, \mu, \nu$ measures $\neq 0$ on G.

(i) If $\lambda, \mu, \nu$ are convolvable, then so are $\lambda$ and $\mu$, $|\lambda| * |\mu|$ and $\nu$, $\mu$ and $\nu$, $\lambda$ and $|\mu| * |\nu|$, and one has

(ii) If $\lambda$ and $\mu$ are convolvable, as well as $|\lambda| * |\mu|$ and $\nu$, then $\lambda, \mu, \nu$ are convolvable. Similarly if $\mu$ and $\nu$ are convolvable, as well as $\lambda$ and $|\mu| * |\nu|$.

This follows from Prop. 1 of §1, No. 2.

There can exist measures $\lambda, \mu, \nu$ on G such that the convolution products $\lambda * \mu$, $(\lambda * \mu) * \nu$, $\mu * \nu$, $\lambda * (\mu * \nu)$ are all defined, and yet $(\lambda * \mu) * \nu \neq \lambda * (\mu * \nu)$ (cf. Exer. 4).

Let $\rho$ be a lower semi-continuous finite function $> 0$ on $G$ such that $\rho(st) \leq \rho(s)\rho(t)$ for all $s, t$ in $G$. We denote by $\mathscr{M}^\rho(G)$ the vector space of measures $\lambda$ on $G$ such that $\rho$ is $\lambda$-integrable, and by $\| \lambda \|_\rho$ (or simply $\| \lambda \|$) the norm $\int_G \rho(s) d|\lambda|(s)$ on this space. When $\rho = 1$, one recovers the set $\mathscr{M}^1(G)$ of bounded measures on $G$.