Bochner consistency identity for convolution squares ($\hat{p}{\mu }_q=\hat{q}{\mu }_p={\mu }_{p\ast q}$)

For finite sums of compactly supported convolution squares, $\hat{p},\hat{q}\ge 0$. Insert the Bochner theorem representation of $q$ into $p\ast q$ and apply the Fubini theorem: its representing measure is $\hat{p}{\mu }_q$. Reversing $p,q$ and using uniqueness gives the displayed identity. It makes the local quotients ${\mu }_p/\hat{p}$ consistent wherever $\hat{p}>0$.