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A p-adic measure gives the compatible finite-quotient elements ${\sum }_{gU}\mu (gU)[gU]\in I[G/U]$. The transition maps sum coefficients over fibers. Conversely compatible coefficients define values on cosets, then all clopen sets, and hence integration. Thus the inverse limit Iwasawa algebra is the ring of measures, with multiplication given by convolution of p-adic measures. Completeness of $I$ suffices even when its residue field is infinite.

For a profinite abelian group $G$, its Iwasawa algebra is the completed group algebrawhere $U$ runs over open subgroups. The transition map sums the coefficients over the fibers of $G/V\to G/U$ when $V\subset U$. Consequently an element gives compatible values $\mu (gU)\in {\mathbb{Z}}_p$ on cosets, equivalently a finitely additive ${\mathbb{Z}}_p$-valued p-adic measure on the clopen sets of $G$. Conversely these values specify every finite-quotient coefficient. Integration of a continuous function with values in ${\mathbb{Z}}_p$ is obtained by uniform approximation by locally constant functions; the integrals converge because all measure values have $p$-adic absolute value at most $1$. Multiplication in the Iwasawa algebra becomes convolution of p-adic measures. This is its measure realization of an Iwasawa algebra.

The Mahler theorem states that each continuous function $g:{\mathbb{Z}}_p\to {\mathbb{Z}}_p$ has the unique expansion converging in the sense of uniform convergenceIts Mahler coefficients areand $\Vert g{\Vert }_{\infty }={\sup }_n|b_n{|}_p$. The same assertion holds for continuous functions with values in ${\mathbb{Q}}_p$ with coefficients in ${\mathbb{Q}}_p$.

Given $f(T)={\sum }_{n\ge 0}{a}_n{T}^n\in R$, define the bounded linear functionalThe sum converges because $a_n\in {\mathbb{Z}}_p$ and $b_n\to 0$. Its values on indicator functions of clopen sets give a p-adic measure, denoted $\lambda (f)$, withConversely, a p-adic measure determines these coefficients $a_n\in {\mathbb{Z}}_p$, and continuity allows termwise integration of every Mahler expansion. Thus the inverse iswhich is the Amice transform. These two constructions are inverse, proving the canonical bijection $\lambda :R\simeq \Lambda ({\mathbb{Z}}_p)$. In fact it is an isomorphism of rings: the identity $\left(\genfrac{}{}{0ex}{}{x+y}{n}\right)={\sum }_{i+j=n}\left(\genfrac{}{}{0ex}{}{x}{i}\right)\left(\genfrac{}{}{0ex}{}{y}{j}\right)$ identifies convolution of p-adic measures with multiplication of formal power series. For example, $\lambda (1)={\delta }_0$ and $\lambda (1+T)={\delta }_1$, where ${\delta }_a$ is the Dirac measure at $a$.

For the moment identity, let $\left\{\genfrac{}{}{0ex}{}{k}{n}\right\}$ denote a Stirling number of the second kind. The finite polynomial identitygives $\int x^{k}d\lambda (f)={\sum }_{n=0}^k{a}_{n}n!\left\{\genfrac{}{}{0ex}{}{k}{n}\right\}$. On the other hand, $D=(1+T)d/dT$ satisfies $D^k(1+T{)}^x=x^k(1+T{)}^x$ for every nonnegative integer $x$. Expanding $(1+T{)}^x$ and evaluating at $T=0$ yieldsComparison in the basis of binomial polynomials gives $(D^k{T}^n)(0)=n!\left\{\genfrac{}{}{0ex}{}{k}{n}\right\}$. Terms with $n>k$ have zero constant term after at most $k$ differentiations, so no infinite-series interchange is needed. ThereforeThe case $k=0$ says that the total mass is $f(0)$.

The coefficient ring $I$ is complete for its $p$-adic norm, with uniformizer $p$ and residue field ${\overline{\mathbb{F}}}_p$. Its residue field is infinite, so $I$ is not compact. The measure construction uses completeness, not compactness of the coefficient ring.

An $I$-valued p-adic measure on $G$ is a bounded $I$-linear functionalon the continuous functions with the uniform norm. Equivalently it assigns a value $\mu (C)\in I$ to each clopen subset, finitely additively on disjoint unions. Integration of a locally constant function is defined byRefining finite clopen partitions shows that this is independent of the expression. Since every $\mu (C_j)\in I$, the nonarchimedean triangle inequality gives $|\int fd\mu {|}_p\le \Vert f{\Vert }_{\infty }$.

Locally constant functions are uniformly dense in $C(G,I)$. Indeed, for every $n$, continuity and compactness of $G$ supply a finite clopen partition on which $f$ is constant modulo $p^{n}I$; choose one value of $f$ on each part. These approximations converge uniformly, and completeness of $I$ extends integration uniquely. Conversely, a bounded functional gives $\mu (C)=\mu (1_C)$ and finite additivity. This proves the equivalence of the two definitions. Ordinary countable additivity of real measures is not part of the definition.

Define the Iwasawa algebra bywhere $U$ runs over open subgroups. Since $G$ is an abelian profinite group, every such quotient is a finite abelian group. The transition map for $V\subseteq U$ sends a basis element $[gV]$ to $[gU]$, so sums coefficients over the fibers.

A p-adic measure defines an element of this inverse limit byFinite additivity is precisely compatibility under the transition maps. Conversely, compatible coefficients define values on every coset and hence on every clopen subset: any clopen subset is a union of cosets of one open subgroup, by compactness and a finite common refinement. Compatibility makes the resulting value independent of that subgroup. The preceding integration argument then constructs the p-adic measure.

Finally, multiplication is convolution of p-adic measures. On a finite quotient its coefficients areThis is exactly the coefficient of $[gU]$ in ${\mu }_U{\nu }_U$. Equivalently,first for locally constant $h$, then by uniform approximation. The Dirac measure at the identity corresponds to the identity element of the group algebra. We have proved a bijection respecting addition, scalar multiplication and multiplication: