p-adic measure ($\mu :C(G,I)\to I$)

For a profinite group $G$ and a complete discrete valuation ring $I$ with a continuous ${\mathbb{Z}}_p$-algebra structure, an $I$-valued measure is a bounded $I$-linear functional on continuous $I$-valued functions, with the uniform norm. Equivalently it assigns finitely additive values in $I$ to clopen sets. Locally constant functions are uniformly dense, so integration extends uniquely from finite sums. Countable additivity in the usual real-measure sense is not imposed. The coefficient ring may have infinite residue field and need not be compact.

Let $I$ be a complete discrete valuation ring with a continuous ${\mathbb{Z}}_p$-algebra structure. Every continuous function on ${\mathbb{Z}}_p$ with values in $I$ has a unique uniformly convergent expansion in the binomial polynomials, with $a_n={\Delta }^{n}f(0)$. This is the coefficient-ring version of the Mahler theorem. To see that $a_n\to 0$, approximate the function modulo any power of the maximal ideal by a function of period $p^k$. In the expansion of $(E-1{)}^{p^k}$, all intermediate binomial coefficients are divisible by $p$, while $(E^{p^k}-1)$ kills the periodic function. Iterating makes its differences vanish modulo the prescribed ideal power. Finite differences do not increase the uniform norm, so the same conclusion holds for the original function. Binomial inversion gives agreement at the nonnegative integers, and their density in the p-adic integers gives the expansion everywhere. This gives the dual description of p-adic measures by arbitrary bounded sequences of binomial moments, and hence the Amice transform.

The coefficient of $T^n$ is $\int \left(\genfrac{}{}{0ex}{}{x}{n}\right)d\mu (x)$. The Mahler expansion of continuous p-adic functions identifies $I$-valued p-adic measures on ${\mathbb{Z}}_p$ with $I[[T]]$. If $D=(1+T)d/dT$, then $D^k{\mathcal{A}}_{\mu }(0)=\int x^{k}d\mu (x)$. A measure supported on ${\mathbb{Z}}_p^{\times }$ has ${\sum }_{{\zeta }^p=1}{\mathcal{A}}_{\mu }(\zeta (1+T)-1)=0$, because the sum of ${\zeta }^x$ is zero for $p\nmid x$.

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.