PomeDE6440832625Corollary 10.31.51.DE6440832625 For fixed $\lambda >0$, write $u\star _\lambda v$ for the solution $a$ in the theorem. This operation on $\mathcal E_\varphi $ is commutative and associative, and, for $c\in \mathbb R$,
Proof. Put $F_\lambda (u)=\mathrm{e}^{-\lambda u}\mu _ u$, with zero mass on the poles. The theorem says that $F_\lambda (u\star _\lambda v)=F_\lambda (u)+F_\lambda (v)$. By corollary 13.19.3, the map $F_\lambda $ is injective on $\mathcal E_\varphi $. Commutativity and associativity therefore follow from those of addition of positive measures, including measures of infinite total mass. Finally, $F_\lambda (u+c)=\mathrm{e}^{-\lambda c}F_\lambda (u)$, which together with injectivity gives both identities. $\square$
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