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Measures on $\mathbb Q_p$

The $p$-adics and Related Concepts

Because $\mathbb Q_p$ is a locally compact abelian group it has a natural translation invariant measure, Haar measure, which allows us to formulate a natural theory of integration of (certain) real valued functions over subsets of the $p$-adics. Here we talk about this Haar measure and it’s basic properties.

A $\sigma$-algebra is a collection of sets (called measurable) that play a similar role in measure theory that topologies of open sets play in topology.

Definition: A collection of subsets $\mathcal F$ of $\mathbb Q_p$ is a $\sigma$-algebra if

  • $\mathbb Q_p \in \mathcal F$;
  • $E \in \mathcal F$ implies $E^c \in \mathcal F$;
  • $\mathcal F$ is closed under countable unions.

Like for topologies, we define the $\sigma$-algebra generated by a collection of sets $\mathcal C$ to be the intersection of all $\sigma$-algebras containing $\mathcal C$.

Definition: The $\sigma$-algebra $\mathcal B$ generated by the Borel topology $\mathcal T$ is called the Borel $\sigma$-algebra.

In general $\sigma$-algebras are big complicated things, but the topology of $\mathbb Q_p$, and in particular the fact that it is totally disconnected means that the Borel $\sigma$-algebra on $\mathbb Q_p$ is not so bad. The following theorem characterizes Borel sets and the Borel $\sigma$-algebra.

Theorem: A Borel set $A \in \mathcal B$ can be given as the disjoint union of a (finite or) countable collection of balls.

A measure is a device to ascribe some notion of size to elements of a $\sigma$-algebra.

Definition: A measure on $\mathcal B$ is a function $\mu: \mathcal B \rightarrow [0, \infty)$ such that

  • $\mu(\emptyset) = 0;$
  • If $A_1, A_2, \ldots$ is a collection of disjoint sets $$\mu\left(\bigcup A_n \right) = \sum \mu(A_n).$$

The second condition is convenient in light of the previous theorem. In particular, it suffices to know the measure of balls to determine the measure of all sets in $\mathcal B$.

Haar Measure on $\mathbb Q_p$

$\mathbb Q_p$ is not just a set, it is a compact abelian group (in fact it is a topological field, but for now we only need the operation addition). A measure $\mu$ is said to be translation invariant if for all $x \in \mathbb Q_p$ and $A \in \mathcal B$, $\mu(x + A) = \mu(x)$. Translation invariant measures behave naturally with respect to the group operation and it turns out are almost unique.

Theorem: There is a unique translation invariant measure $\lambda$ on $\mathcal B$ such that $\lambda(\mathbb Z_p) = 1$. This measure is called the normalized Haar measure on $\mathbb Q_p$.

We can always restrict measures and $\sigma$-algebras to subsets of $\mathbb Q_p$, and in particular $\lambda$ forms a measure on the Borel subsets of $\mathbb Z_p$. Because $\lambda(\mathbb Z_p) = 1$ we say $\lambda$ is a probability measure on $\mathbb Z_p$.

Let’s investigate some properties of Haar measure. First note that $\mathbb Z_p$ is the disjoint union of the cosets $p \mathbb Z_p, 1 + p \mathbb Z_p, \ldots, p – 1 + \mathbb Z_p$. Because there are a total of $p$ cosets and because each is the translation of $p \mathbb Z_p$ we conclude that $$\lambda(r + p \mathbb Z_p) = \frac{1}{p}; \qquad r=0,1,\ldots,p-1.$$ This is a completely general phenomenon.

Theorem: The Haar measure of the ball/coset $B = r + p^n \mathbb Z_p$ is $\lambda(B) = p^{-n}$. That is balls of radius $p^{-n}$ have Haar measure $p^{-n}$.

Though not explicitly designed to, $\lambda$ behaves nicely with respect to multiplication in $\mathbb Q_p$ as well.

Theorem: Let $B$ be a Borel set and let $x \in \mathbb Q_p$ then the Haar measure of the multiplicative translate $x B = \{ x y : y \in B \}$ satisfies $$\lambda(x B) = |x|_p \lambda(B).$$

Haar measure on $\mathbb Q_p^{\times}$

$\mathbb Q_p^\times := \mathbb Q_p \setminus \{0\}$ is itself a locally compact abelian group under multiplication. It likewise has a (multiplicative) translation invariant measure, which is unique once we specify the measure of a specified reference set. For Haar measure on $\mathbb Q_p$ under addition we used $\mathbb Z_p$ for the reference set (which we normalized to have Haar measure 1). Here it makes sense to use a different reference set.

Definition: The set $U_p = \{ x \in \mathbb Q_p : |x|_p = 1\}$ is a group under multiplication called the group of units in $\mathbb Q_p$ (or in $\mathbb Z_p$).

Exercise: Show that ${\displaystyle U_p = \bigcup_{r=1}^{p-1} (r + p\mathbb Z_p).}$ Hence $U_p$ is compact.

Definition: We denote by $\lambda^{\times}$ the unique multiplicative translation invariant (Haar) measure on $\mathbb Q_p^{\times}$ satisfying $\lambda^{\times}(U_p) = 1$. This measure restricted to $U_p$ is a probability measure.

It is worth remarking that some people (in particular John Tate, who was usually right about things) choose a different normalization for Haar measure on $\mathbb Q_p^\times$. It is sometimes convenient to have the measure of $U_p$ be $(p-1)/p$ as this is the Haar measure of that set in the additive group. This can be useful when you have need to integrate both with respect to $\lambda$ and $\lambda^{\times}$ (which Tate did) as both measures give the same value on $U_p$. Since we’ll not be working with $\lambda$ and $\lambda^{\times}$ simultaneously, I choose instead to normalize $\lambda^{\times}$ so that it is a probability measure when restricted to $U_p$.

Exercise: Show that $\lambda^{\times}(\mathbb Z_p \setminus \{0\}) = \infty.$

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