### The Categorical Origins of Lebesgue Integration, Revisited

#### Posted by Tom Leinster

I’ve just arXived a new paper: The categorical origins of Lebesgue integration (arXiv:2011.00412). Longtime Café readers may remember that I blogged about this stuff back in 2014, but I’ve only just written it up.

What’s it all about? There are two main theorems, which loosely are as follows:

**Theorem A** The Banach space $L^1[0, 1]$
has a simple universal property. This leads to a unique characterization of
integration.

**Theorem B** The functor $L^1:$ (finite measure spaces) $\to$
(Banach spaces) has a simple universal property. This leads to a unique
characterization of integration on finite measure spaces.

But there’s more! The mist has cleared on some important things since that last post back in 2014. I’ll give you the highlights.

**Universal property of $L^p[0, 1]$**
Let $1 \leq p \lt \infty$. The Banach space $L^p[0, 1]$, with a little bit
of extra structure, has a simple universal property.

To say what that universal property is, I need to put a norm on the direct sum $V \oplus W$ of two Banach spaces, and the one I’ll use is this:

$\| (v, w) \| = \Bigl( \tfrac{1}{2} ( \|v\|^p + \|w\|^p ) \Bigr)^{1/p}.$

In words, it’s the power mean of order $p$ of the norms of $v$ and $w$.

Now consider the category $\mathcal{A}^p$ of triples $(V, v, \delta)$, where $V$ is a Banach space, $v$ is a point in the closed unit ball of $V$, and $\delta$ is a linear contraction $V \oplus_p V \to V$ such that $\delta(v, v) = v$. The maps are the linear contractions that preserve the structure in the obvious sense.

One object of $\mathcal{A}^p$ is the triple $(L^p[0, 1], I, \gamma)$, where $I$ is the constant function $1$ and $\gamma$ takes two functions on $[0, 1]$, juxtaposes them, and scales the domain by a factor of $1/2$:

The theorem is that $(L^p[0, 1], I, \gamma)$ is, in fact, the *initial*
object of $\mathcal{A}^p$. Among other things, this characterizes the Banach
space $L^p[0, 1]$ uniquely up to isometric isomorphism.

When $p = 1$, another object $(V, v, \delta)$ of $\mathcal{A}^1$ is the ground field $\mathbb{F}$ (either $\mathbb{R}$ or $\mathbb{C}$) together with the element $1 \in \mathbb{F}$ and the arithmetic mean operation $m : \mathbb{F} \oplus \mathbb{F} \to \mathbb{F}$. Since $(L^1[0, 1], I, \gamma)$ is initial in $\mathcal{A}^1$, there’s a unique map $(L^1[0, 1], I, \gamma) \to (\mathbb{F}, 1, m)$. It’s nothing but integration.

**Continuous functions on the Cantor set**
The result I just stated characterized $L^p[0, 1]$ for $1 \leq p \lt
\infty$, but excluded $p = \infty$. What about that case?

The category $\mathcal{A}^\infty$ does have an initial object, but it’s not $L^\infty[0, 1]$. In fact, it’s $C(\{0, 1\}^\mathbb{N})$, the continuous functions on the Cantor set.

This isn’t as outlandish as it might seem. First of all, the interval $[0, 1]$ and the Cantor set $\{0, 1\}^\mathbb{N}$ are equivalent as measure spaces. (Here I’m giving the Cantor set the probability measure that corresponds to thinking of its elements as sequences of tosses of a fair coin, and of course I’m giving $[0, 1]$ Lebesgue measure.) So, $L^p[0, 1] \cong L^p(\{0, 1\}^\mathbb{N})$, which means that the previous result could equally well be thought of as a universal characterization of $L^p(\{0, 1\}^\mathbb{N})$.

On that basis, one might expect the initial object of $\mathcal{A}^\infty$
to be $L^\infty(\{0, 1\}^\mathbb{N})$, the essentially bounded functions on
the Cantor set. But it’s not! It’s the *continuous* functions.

**Connecting up with actual function spaces**
It’s all very well characterizing $L^p[0, 1]$ abstractly, but can we relate
its elements to actual *functions* on $[0, 1]$?

The answer is yes, in two different senses.

First, we might want to start with a nice enough function on $[0, 1]$ and produce from it an element of the abstractly-characterized $L^p[0, 1]$. Let’s interpret “nice enough” as “continuous”, so that we’re looking to derive the inclusion map $C[0, 1] \to L^p[0, 1]$ from the universal properties above.

It can be done! The way it goes is this. The usual map $\{0, 1\}^\mathbb{N} \to [0, 1]$ induces a map $C[0, 1] \to C(\{0, 1\}^\mathbb{N})$. Also, the universal property of $C(\{0, 1\}^\mathbb{N})$ induces a map from it to $L^p[0, 1]$. Composing the two gives a map $C[0, 1] \to L^p[0, 1]$, and I show that in concrete terms, it’s the inclusion.

Second — and in the opposite direction — we might want to start
with an element of the abstractly characterized $L^1[0, 1]$ and produce
from it an actual function on $[0, 1]$. Well: we might *want* to do this,
but it’s not realistic: after all, elements of $L^1[0, 1]$ are *equivalence
classes* of integrable functions, up to equality almost everywhere. So
that’s the best we can hope to get. There can be no hope of evaluating an
element of $L^1[0, 1]$ at an element of $[0, 1]$.

What the universal property of $L^1[0, 1]$ does get us is a canonical map $L^1[0, 1] \to C[0, 1]$. Concretely, this is what generations of undergraduates would write as $f \mapsto F$, where $F(x) = \int_0^x f$. (Thanks to Mark Meckes for pointing this out to me and letting me include it in the paper.)

I’m not saying quite *how* you get this map $f \mapsto F$; it’s
Proposition 2.4 in the paper. But the
point is that this map comes straight from the universal property of
$L^1[0, 1]$, and from $F$ we can extract an actual function representing
$f$. Indeed, it’s a version of the fundamental theorem of calculus that
$F' = f$ almost everywhere. So that’s you extract a genuine function.

**Conjugate pairing** A whole lot of stuff in analysis involves exponents
$p$ and $q$ that are “conjugate”, meaning that $1/p + 1/q = 1$. (And a
whole lot of people have observed that in retrospect, it might have been
better to index the $L^p$ spaces using $1/p$ rather than $p$. Too late!)
Most famously, when $p$ and $q$ are conjugate, $L^p[0, 1]$ and $L^q[0, 1]$
are dual, as long as $p$ and $q$ are not $1$ or $\infty$.

We don’t seem to get this duality directly from the universal property, but
we *do* get the pairing function

$L^p[0, 1] \times L^q[0, 1] \to \mathbb{F}.$

Concretely, this function is $(f, g) \mapsto \int f \cdot g$. In other words, it’s the composite

$L^p[0, 1] \times L^q[0, 1] \to L^1[0, 1] \to \mathbb{F},$

where the first map is multiplication and the second is integration. We already derived the second map from the universal property of $L^1[0, 1]$, and the paper shows how to derive the first map from the universal properties of $L^p$ and $L^q$.

**Sequence spaces** Everything so far has been about spaces of functions on $[0,
1]$. But we can follow a similar strategy to get universal characterizations
of spaces of sequences. All I’ll do here is point to Proposition 2.10 of
the paper, which states a universal
property of the sequence space $\ell^p$. That’s for $p \lt \infty$. When
you put $p = \infty$, what pops out is not $\ell^\infty$ but the space
$c_0$ of sequences converging to $0$.

**The $L^p$ functors** The second half of the paper is about $L^p$ as a
*functor*, rather than $L^p[0, 1]$ or $L^p(\mathbb{N}) = \ell^p$.

The $L^p$ construction takes a measure space as input and spits out a Banach space. There are at least a couple of ways in which it’s functorial:

The one you hear about most is that it’s contravariant with respect to measure-preserving maps, via composition.

But it’s also

*covariant*with respect to embeddings (i.e. inclusions of measure subspaces, up to isomorphism). If you’ve got an $L^p$ function defined on a subspace $Y \subseteq X$, you can simply extend it by $0$ to get an $L^p$ function on all of $X$.Combining these two types of functoriality, $L^p$ is contravariant with respect to measure-preserving

*partial*maps.

I’ll write $\mathbf{Meas}$ for the category of measure spaces and measure-preserving partial maps, so that $L^p$ is a functor

$L^p: \mathbf{Meas}^{op} \to \mathbf{Ban}.$

By “measure space” I always mean a *finite* measure space — not one
whose underlying set is necessarily finite, but where the total measure of
the space is finite.

For each measure space $X$, there’s a special element of $L^p(X)$: the function $I_X$ with constant value $1$. The functor $L^p$, together with the family

$\bigl(I_X \in L^p(X)\bigr)_{X \in \mathbf{Meas}},$

has some elementary properties that we could write down. For example, if $s: X \to Y$ is a measure-preserving map then $L^p(s): L^p(Y) \to L^p(X)$ maps $I_Y$ to $I_X$.

The second main theorem is that the pair $(L^p, I)$ is *initial* with these
simple properties.

That is, I define a category $\mathcal{B}^p$ of pairs $(F, v)$ where $F: \mathbf{Meas}^{op} \to \mathbf{Ban}$ and $v$ is a family $(v_X \in F(X))_{X \in \mathbf{Meas}}$ satisfying those elementary properties, and the theorem is that $(L^p, I)$ is the initial object of $\mathcal{B}^p$.

**Baby versions** Actually, this theorem is naturally the third of a trilogy
of results. In the paper, I used the first two as stepping stones to get to the third.

The first is about a similar category of pairs $(F, v)$, where $F$ is
now a functor from $\mathbf{Meas}^{op}$ to the category of mere *vector
spaces* (not Banach spaces). The initial object is the pair $(\mathcal{S},
I)$, where $\mathcal{S}(X)$ is the vector space of *simple functions* on $X$.

The second is about a category of pairs $(F, v)$, where $F$ is a functor
from $\mathbf{Meas}^{op}$ to the category of *normed vector spaces*. The
initial object is the pair $(S^p, I)$, where $S^p(X)$ is the normed vector
spaces of *simple functions up to equality almost everywhere*, with the
$p$-norm.

So, informally:

The universal vector space on a measure space consists of the simple functions.

The universal normed vector space consists of the a.e. equivalence classes of simple functions.

The universal Banach space consists of the a.e. equivalence classes of integrable functions.

**Integration on an arbitrary measure space** The universal
characterization of the functor $L^1$ gives a unique characterization of
integration.

Just as for $[0, 1]$, this comes about by choosing a suitable object of $\mathcal{B}^1$ and applying the fact that $(L^1, I)$ is initial in $\mathcal{B}^1$. That “suitable object” is the constant functor $\mathbb{F}$ (the ground field) together with, for each measure space $X = (X, \mu_X)$, the total measure $\mu_X(X) \in \mathbb{F}$. The initiality of $(L^1, I)$ gives us a map $L^1(X) \to \mathbb{F}$ for each measure space $X$, and it won’t surprise you that it’s our old friend $\int_X$.

**The action of functions on measures** Whenever you have a measure space
$(X, \mu)$ and an integrable function $f$ on it, you get a new measure on
$X$ that’s traditionally written as $f\,d\mu$ but which I prefer to write
as $f\mu$. (It seems to me that the $d$s just make life messier.) It’s
defined on measurable sets $A \subseteq X$ by

$(f \mu)(A) = \int_A f \, d\mu.$

This construction, too, can be derived from the universal property of
$L^1$. This time I won’t say how! It’s not at all hard, but I’m running out of steam, so I’ll just refer you to Proposition 3.10 of the
paper. And I’ll note that being able to integrate $f$ on arbitrary, “small”, regions of $X$ is as close as we can truthfully come to realizing $f \in L^p(X)$ as an actual function, given that in reality $f$ is only an *equivalence class* of functions.

**Universal characterization of $L^2$** We all know that there’s something
special about $p = 2$, and there’s a theorem that does a little bit to
express that special nature. It says that $(L^2, I)$ is the initial object
of a category of pairs $(F, v)$, where $F$ is now a functor
from $\mathbf{Meas}^{op}$ to *Hilbert* spaces. And the axioms that these
pairs are required to satisfy are slightly simpler and more intuitive than
for general $p$.

**What’s the point?** I’ve written about as much as I want to for one post,
but I didn’t want to finish without mentioning this general question.

A certain kind of no-nonsense mathematician would find all these results very odd. What do they tell us that we didn’t already know? I did a lot of thinking about how to respond, and I gave five answers in the introduction (page 2). I think my favourite one is this:

Fourth, the second main theorem, characterizing the $L^p$ functors, provides a guide for the discovery of new theories of integration. A researcher seeking the right notion of integration in some new context (perhaps some new kind of function on a new kind of space) could follow the same template: decide what kind of spaces the integrable functions should form (perhaps not Banach spaces) and what kind of functoriality should hold, formulate a universal property analogous to the one used below, and find the functor satisfying it.

I leave you to discuss this in the comments!

## Re: The Categorical Origins of Lebesgue Integration, Revisited

Looks like a good read, Tom!

We had a small burst of functional analysis meets category theory earlier this year with talk of Smith spaces (aka Waelbrock dual spaces) forming $Ban^{op}$.

Presumably, your universality results could be dualized to Smith spaces.