### The Categorical Origins of Lebesgue Integration

#### Posted by Tom Leinster

*(Updated post here)*

I’ve just come back from the big annual-ish category theory meeting, Category Theory 2014 in Cambridge, also attended by Café hosts Emily and Simon. The talk I gave there was called The categorical origins of Lebesgue integration — click for slides — and I’ll briefly describe it now.

There are two theorems.

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

**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.

The talk’s pretty simple, and I don’t think I can summarize it much better than by repeating the abstract, which went like this:

Lebesgue integration is a basic, essential component of analysis. Yet most definitions of Lebesgue integrability and integration are rather complicated, typically depending on a series of preliminary definitions. For instance, one of the most popular approaches involves the class of functions that can be expressed as an almost everywhere pointwise limit of an increasing sequence of step functions. Another approach constructs the space of Lebesgue-integrable functions as the completion of the normed vector space of continuous functions; but this depends on already having the definition of integration for continuous functions.

So we might wish for a short, direct description of Lebesgue integrability that reflects its fundamental nature. I will present two theorems achieving this.

The first characterizes the space $L^1[0, 1]$ by a simple universal
property, entirely bypassing all the usual preliminary definitions. It
tells us that once we accept two concepts — Banach space and the mean of
two numbers — then the concept of Lebesgue integrability is inevitable.
Moreover, this theorem not only characterizes the Lebesgue
integr*able* functions on $[0, 1]$; it also characterizes Lebesgue
integr*ation* of such functions.

The second theorem characterizes the functor $L^1$ from measure spaces to Banach spaces, again by a simple universal property. Again, the theorem characterizes integration, as well as integrability, of functions on an arbitrary measure space.

## Re: The Categorical Origins of Lebesgue Integration

Very neat! I’m sad that I missed CT this year.

Of course, in practice it’s important to know that an actual (sufficiently nice) function $f:[0,1]\to\mathbb{R}$ can be regarded as an element of $L^1[0,1]$. Is there any way to see this from your characterization, short of going through the proof that it coincides with the usual definition?