# Measures on formal power series over a finite field

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.

• It seems that you're just getting the probability measure where the coefficients are iid uniformly distributed on $F_p$, which does seem completely natural. In other words, the infinite product of uniform measure with itself. It lives on the product $\sigma$-algebra and is countably additive. This construction will be discussed in any probability theory book. It's the same construction you would use to talk about an infinite sequence of dice rolls. – Nate Eldredge Mar 6 at 14:52
• OK, that gives me some keywords to look for. Any particular book(s) you might recommend (I'll see what we have in our library, though) – Max Horn Mar 6 at 14:56
• Also: $F_p[[t]]$ is a compact group (in the product topology) and the measure described by Nate is the Haar measure. This means it has nice invariance properties. And maybe your "small" condition could be done with Baire category instead of measure. – Gerald Edgar Mar 6 at 15:08
• @Lubin could you elaborate? – Max Horn Mar 6 at 16:21
• @MaxHorn: Great, glad you found it. Another thing to look at is the Kolmogorov zero-one law; this is the main tool for showing that a set has measure zero with respect to an infinite product measure. – Nate Eldredge Mar 6 at 17:42