### Universal Measures

#### Posted by Tom Leinster

There is much that is odd about motivic measure if it is judged by measure theory in the sense of twentieth century analysis […] The first peculiarity is that the measure is not real-valued.

—Thomas Hales, What is motivic measure?,
*Bulletin of the AMS* 42 (2005), 119–135.

This post isn’t about motivic measure, though you should definitely take a look
at Hales’s excellent article (especially to find out what the *second*
peculiarity is). This post will, however, share something of the spirit of motivic measure, including a flexible attitude towards where measure takes its values.

Suppose that we have some set, and a collection of “subsets” that we want to be able to “measure”. I’ll
keep
this very vague and general for the moment, though when I make it precise it
genuinely *will* be quite general. The word “measure” isn’t used here with its standard meaning: we’re just assigning a “quantity”
to each of our sets in some plausible way.

The crucial point is that whatever “quantity” means, it needn’t mean “real number”. And all I ask of a “measure” $\phi$ is that it satisfies the inclusion-exclusion principle: $\phi(A_1 \cup \cdots \cup A_n) = \sum_i \phi(A_i) - \sum_{i \lt j} \phi(A_i \cap A_j) + \cdots$ whenever $n \geq 0$ and $A_1, \ldots, A_n$ are sets for which this makes sense.

What is the universal way to measure a collection of sets?

I’ll answer that question, but first it needs making precise.

What kind of thing are the “quantities”? The inclusion-exclusion equation is an equation between two quantities, and involves only addition and subtraction, so all we need to ask of quantities is that they form an abelian group.

What kind of thing are the “sets”? The inclusion-exclusion equation involves unions and intersections of the sets $A_i$ concerned. For that to make any kind of sense, these “sets” had better all be subsets of one big set $S$, say. So we’ve got a distinguished collection $M$ of subsets of $S$, and each $A_i$ is supposed to be an element of $M$. Here $M$ should make you think of “measurable”.

More
exactly, the equation involves *one* union and *lots* of
intersections. It’s going to be a real
pain if the collection $M$ of measurable sets isn’t closed
under intersection, so let’s assume that it is. But we won’t assume that it’s closed
under union: we just ask that the equation holds *if* $A_1 \cup \cdots
\cup A_n \in M$ (and, of course, $A_i \in M$ for each $i$).

A good example to keep in mind is this: $M$ is the set of compact convex subsets of $S = \mathbb{R}^N$ (counting $\emptyset$ as convex). This is closed under binary intersection, but not union.

In other examples, $M$ might be closed under finite union. Then the inclusion-exclusion principle reduces to the cases $n = 0$ and $n = 2$. The usual terminology then is that $\phi$ is a valuation.

AsideYou could formulate it more generally, taking $M$ to be an abstract poset — not necessarily one represented as a sub-poset of $(\mathcal{P}(S), \subseteq)$. In that generality, I don’t know what the universal measure is.

Fix a set $S$ and a collection $M$ of subsets of $S$, closed under binary intersection.

With apologies for abusing an already-abused word, I’ll make the following definition.

DefinitionAmeasureon $M$ is a pair $(X, \phi)$ where $X$ is an abelian group (of “quantities”) and $\phi: M \to X$ is a function satisfying the inclusion-exclusion principle:$\phi(A_1 \cup \cdots \cup A_n) = \sum_i \phi(A_i) - \sum_{i \lt j} \phi(A_i \cap A_j) + \sum_{i \lt j \lt k} \phi(A_i \cap A_j \cap A_k) - \cdots$

whenever $n \geq 0$ and $A_1, \ldots, A_n \in M$ with $A_1 \cup \cdots \cup A_n \in M$.

For example, if $S = \mathbb{R}^2$ and $M$ is the set of compact convex subsets, then there is a measure $(\mathbb{R}, perimeter)$ on $M$.

There’s an obvious notion of **map of measures**: a map $(Y, \psi) \to
(X, \phi)$ is a homomorphism $\alpha: Y \to X$ such that $\alpha \circ \psi =
\phi$. So there’s a **category of measures** on $M$. Let’s call it
$Meas(M)$.

Here’s the formal version of the question I originally posed, “what is the universal measure?”:

What is the initial object of $Meas(M)$?

The answer can be found in a result called *Groemer’s
integral theorem*. Actually, I don’t know how widely it’s called that. It’s the
name used by Klain and Rota in their
book
(page 9), but I suspect they coined it. The original result of Helmut Groemer —

— was about convex sets. Klain and Rota gave a more general lattice-theoretic formulation. The current formulation as a universal property is new, as far as I’m aware.

In a slogan, the answer is:

The universal measure is indication.

And while I’m sloganeering:

The universal property asserts that simple functions can be integrated.

Let me explain those cryptic utterances.

Every subset $A$ of our big set $S$ has an
**indicator function**, or characteristic function, $I_A: S \to
\mathbb{Z}$. It takes value $1$ inside $A$ and $0$ outside $A$. A function $f: S
\to \mathbb{Z}$ is **simple** if it’s a finite $\mathbb{Z}$-linear
combination of indicator functions of measurable sets:

$f = \sum_{i = 1}^n \alpha_i I_{A_i}$

($\alpha_i \in \mathbb{Z}$, $A_i \in M$). The simple functions form an abelian group $Simp(M)$ under addition.

Every measurable set $A$ has an associated indicator function $I_A$, and this process — which I facetiously called “indication” — defines a map

$I_\bullet: M \to Simp(M).$

It’s easy to see that $I_\bullet$ satisfies the inclusion-exclusion principle. For example, the case $n = 2$ is that

$I_{A \cup B} = I_A + I_B - I_{A \cap B}.$

So $(Simp(M), I_\bullet)$ is a measure on $M$.

Theorem (Groemer)The initial object of $Meas(M)$ is $(Simp(M), I_\bullet)$.

Let’s unwind this universal property and see what it says. By definition of the category $Meas(M)$, it says that whenever $X$ is an abelian group and $\phi: M \to X$ satisfies the inclusion-exclusion principle, there is a unique homomorphism $\overline{\phi}: Simp(M) \to X$ such that $\overline{\phi} \circ I_\bullet = \phi$:

$\begin{matrix} M &\stackrel{I_\bullet}{\to} &Simp(M) \\ &\phi\searrow &\downarrow \exists!\overline{\phi}\\ & &X. \end{matrix}$

But the commutativity of this diagram just says that for any measurable set $A$, we have

$\overline{\phi}(I_A) = \phi(A).$

What usually turns the indicator function of a set into the measure of that set? Integration! So we should write $\overline{\phi}$ as $\int - d\phi$. Then the diagram asserts that

$\int I_A d\phi = \phi(A)$

for all $A \in M$.

Since every simple function is a $\mathbb{Z}$-linear
combination of indicator functions, and since $\int - d\phi$ is supposed to be a
$\mathbb{Z}$-linear map, the *uniqueness* part of the universal property
is not in question. The substantial part is existence. What we have to prove
here is that the integral of a simple function can be defined
*consistently* by

$\int \Bigl(\sum_i \alpha_i I_{A_i}\Bigr) d\phi = \sum_i \alpha_i \phi(A_i).$

In other words, if we represent a simple function as a linear combination of indicator functions in two different ways, we get the same answer for the integral. Proving that is going to depend on $\phi$ having some nice property; and that nice property turns out to be exactly the inclusion-exclusion principle.

There’s a more general result, too. All of the above was in a stripped-down world where quantities are only assumed to form an abelian group. In other words, we worked over $\mathbb{Z}$. But everything works over an arbitrary ring $k$. Then an object of $Meas(M)$ is a pair $(X, \phi)$ where $X$ is a $k$-module and $\phi$ is exactly as before; a simple function is a $k$-linear combination of indicator functions; and so on. It all works.

The traditional choice is $k = \mathbb{R}$, so that we’re integrating real-valued functions. But the proof of Groemer’s theorem uses nothing about $\mathbb{R}$ beyond that it’s a ring.

## Re: Universal Measures

Very nice!

I don’t suppose it helps that by the Yoneda lemma, every poset $M$ is isomorphic to a sub-poset of $(\mathcal{P}(M),\subseteq)$ — namely the sub-poset consisting of the “principal ideals” (= representable functors) $\downarrow(a) = \{ b\in M | b \le a \}$. I guess the problem is that the Yoneda embedding doesn’t preserve unions. Makes me feel like there’s some sheafy sort of thing that wants to be going on.