Convolution of p-adic measures ($(\mu \ast \nu )(h)=\int \int h(xy)d\mu (x)d\nu (y)$)

On a profinite group, convolution is first defined for functions on a finite quotient by summing products of the two measures' coset coefficients, then extended by uniform approximation. Finite-quotient associativity proves associativity. For an abelian group it is commutative; the Dirac measure at the identity is the identity element. Under the measure realization of an Iwasawa algebra, it becomes group-algebra multiplication.