= Measure realization of an Iwasawa algebra
{title2=$I[[G]]\simeq\operatorname{Meas}(G,I)$}
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.
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