3. Measures and the Iwasawa algebra
This chapter develops the p-adic analysis underlying the whole story: p-adic
measures on a profinite abelian group G, their identification with the Iwasawa
algebra \Lam(G), and — in the key case G = \Zp — a fourth description as
power series via the Mahler transform. We then build a toolbox of operations on
measures, introduce pseudo-measures (to accommodate the simple pole of the zeta
function), and close with locally analytic distributions. The treatment follows
Jacinto and Williams (2023) §3.
Throughout we fix a finite extension L of \Qp, with p-adic valuation
\vp normalised by \vp(p) = 1; this is the coefficient field, and \OL
denotes its ring of integers.