= Solution
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 \b[$I$ 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 functional
$$
\mu:C(G,I)\longrightarrow I,
$$
on 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 by
$$
\int_G\sum_j a_j\mathbf1_{C_j}\,d\mu=\sum_j a_j\mu(C_j).
$$
Refining finite clopen partitions shows that this is independent of the expression. Since every $\mu(C_j)\in I$, the nonarchimedean triangle inequality gives $|\int f\,d\mu|_p\leq\|f\|_\infty$.
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 $p^nI$; choose one value of $f$ on each part. These approximations converge uniformly, and completeness of $I$ extends integration uniquely. Conversely, a bounded functional gives $\mu(C)=\mu(\mathbf1_C)$ 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> by
$$
I[[G]]=\varprojlim_{U}I[G/U]=\varprojlim_{U,n}(I/p^nI)[G/U],
$$
where $U$ 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 by
$$
\mu_U=\sum_{gU\in G/U}\mu(gU)[gU].
$$
Finite 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 are
$$
(\mu*\nu)(gU)=\sum_{aU\in G/U}\mu(aU)\nu(a^{-1}gU).
$$
This is exactly the coefficient of $[gU]$ in $\mu_U\nu_U$. Equivalently,
$$
\int_G h\,d(\mu*\nu)=\int_G\!\int_Gh(xy)\,d\mu(x)\,d\nu(y),
$$
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:
$$
\boxed{I[[G]]\simeq\operatorname{Meas}(G,I).}
$$
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