For various reasons not important for this question, I'd like to show that certain subsets of $F_p[[t]]$, the ring of formal powers over the finite (prime) field $F_p$ in the variable `t`

, are "small", i.e., that their elements are "rare" and if you pick a "random" one, you are unlikely to hit one of these. To make that precise, I feel that I should have a measure on this ring.

**First question: Are any such measures "well-known", e.g. have been discussed in books or at least papers? Any references?**

My naive idea was to attempt to define a "measure" as follows: for $d\in\mathbb{N}$, let $$ \pi_d:F_p[[t]] \to F_p[t],\ \sum_{n=0}^{\infty} a_n t^n \mapsto \sum_{n=0}^{d-1} a_n t^n, $$ i.e., we keep the first $d$ terms and "cut off" the rest. Then the image of $\pi_d$ has size $p^d$. Now we can define a map $\mu$ on subsets of $F_p[[t]]$ as follows: for $X\subset F_p[[t]]$, let $$ \mu(X) := \lim_{d\to\infty} \frac{|\pi_d(X)|}{p^d}. $$

This should be well-defined, as $|\pi_{d+1}(X)|\leq p\cdot |\pi_d(X)|$. Also $0\leq \mu(X)\leq 1$ (which fits, as our ring is compact). But is it countably additive (with respect to which $\sigma$-algebra)? Unfortunately it's been a long time I learned about measures...

**Second question: Is this a measure? (And if so, is it a "sensible" resp. "standard" one?) Which approaches could one use to prove this?**

I've been trying to read up on this a bit, and am planing to luck closer at the theory of Haar measures and Radon measures, as that sounds as if it might provide tools to either study this candidate for a "measure", or else define a different measure (which then hopefully I can use for my purposes). But I am hoping that perhaps some experts can help me get on the right track here.