2 · Compare two analytic measure rules

The p-adic rule and the solid rule over a discrete ring both pass the analytic test. They support different infinite operations because their coefficient rings and measure modules differ.

Try this. Add terms in both examples. The p-adic side follows a convergent series. The discrete-ring side shows a measure with an unrestricted family of coordinates.

Proposition 7.8 proves that both rules are analytic. In the discrete example, the ring itself has no topology; the completion is carried by the chosen category of modules.