# Category Theory and Lebesgue Integration.

I'm wondering if there's any Category Theory floating around in the theory of Lebesgue Integration. To avoid things becoming too broad, let's keep this focused on the basics. Here's how I see the general set up (missing a few details to keep things brief!).

Definition 1: A function from $$\mathbb{R}^k$$ to $$\mathbb{R}$$ is a step function if there exists a partition $$P$$ of $$\mathbb{R}^k$$ such that $$f$$ is constant for each interval (of $$\mathbb{R}^k$$) associated with $$P$$ and zero on the unbounded region associated with $$P$$.

Theorem 1: Step funtions form a vector space over $$\mathbb{R}$$ and $$\int$$ (defined for step functions) is a linear transformation from this space to $$\mathbb{R}$$.

Theorem 2 (Lattice Properties): If $$f, g$$ are step functions on $$\mathbb{R}^k$$, then so are $$\max (f, g)$$, $$\min (f, g)$$, the positive & negative parts of $$f$$, and $$\lvert f\rvert$$.

Definition 2: A function $$f:\mathbb{R}^k\to\mathbb{R}$$ is an upper function if there is an increasing sequence of step functions $$(f_n)_{n\in\mathbb{N}}$$ such that $$\int f_n$$ converges and $$f_n\to f$$ a.e. as $$n\to\infty$$. The set (or whatever) of such functions is denoted $$\mathscr{L}^{\text{inc}}(\mathbb{R}^k)$$. We define the integral of an upper function as $$\int f=\lim_{n\to\infty}\underbrace{\int f_n.}_{\text{These are integrals of step functions.}}$$

Theorem 3: Upper functions don't form a vector space over $$\mathbb{R}$$.

Definition 3: A function $$f:\mathbb{R}^k\to\mathbb{R}$$ is Lebesgue integrable on $$\mathbb{R}^k$$ if there exist upper functions $$g, h$$ on $$\mathbb{R}^k$$ with $$f=g-h$$. We define $$\int f=\int g -\int h$$. The set (or whatever) of such functions is denoted $$\mathscr{L}^1(\mathbb{R}^k)$$.

Theorem 4: $$\mathscr{L}^1(\mathbb{R}^k)$$ is an $$\mathbb{R}$$-vector space and $$\int$$ is a linear map.

Theorem 5: Functions in $$\mathscr{L}^1(\mathbb{R}^k)$$ satisfy the same "Lattice Properties" as in Theorem 2.

Do step functions, upper functions, and Lebesgue integrable functions form categories? Is there a way to describe the "Lattice Properties" above of the respective functions using Category Theory? What's the "significance" of some of these functions but not others forming vector spaces from a categorical viewpoint (if there be such)?

I'm very sorry if this is too broad. It just seems like the sort of thing someone would've investigated . . .

• Curious question. +1 :) Jan 28 '14 at 11:13
• Excellent question +1 Jan 28 '14 at 13:24
• Which question? It isn't really focussed at all. It may be better to narrow it. It happens quite often that people ask here "Can X be seen from a categorical perspective", without giving any motivation or explanation why this should be interesting or possible. "What does $\sqrt{2}$ mean from a categorical perspective" ... Jan 28 '14 at 15:00
• I'm very sorry, @MartinBrandenburg. The questions are highlighted and I don't think adding to them would help. I believe it to be an interesting set of questions in its own right and my motivation was simply ignorance & curiosity. How else am I supposed to find out about this sort of thing if I don't ask? For all I know/knew, this could've had a straightforward, readily available answer. I'd be happy if these functions turn out to be either objects or morphisms; I had the former in mind but why should I exclude the latter? Jan 28 '14 at 16:09
• The above construction is known as the Daniell Scheme, and can be performed over any space $X$ with a collection of "basic functions" $B$ that form a lattice vector space say over $\Bbb R$ and where a positive continuous (in a certain sense) functional $B\to \Bbb R$ is defined. Then one can construct a functional over the whole space which we call an integral. Mar 14 '14 at 21:46

It's not quite what you were asking, but I feel compelled to link to my own doctoral dissertation here.

http://www.andrew.cmu.edu/user/awodey/students/jackson.pdf

I started from the well known facts that if $\langle \Omega,\mathcal{F}\rangle$ is a measurable space, and $j$ is the countable join topology on $\mathcal{F}$, then:

1. The object of measurable real valued functions is the Dedekind real numbers object $\mathbb{R}$ in $\textrm{Sh}_{j}(\mathcal{F})$.
2. The presheaf of measures $\mathbb{M}$ is also a sheaf.
3. (Lebesgue) integration is definable as a natural transformation $$\textstyle\int\displaystyle:\mathbb{R}^{+}\times\mathbb{M}\to\mathbb{M}.$$

(3) wasn't "well known" as such, but is obvious when you think about it.

The work I did was to find definitions for $\mathbb{M}$ and $\int$ using the internal language of $\textrm{Sh}_{j}(\mathcal{F})$, and to provide categorical proofs of the Monotone Convergence Theorem and the Radon-Nikodym Theorem (spoiler alert: derivatives exist with respect to $\mu$ when the topology of $\mu$-almost everywhere equivalence induces a Boolean topos). These definitions and theorems can then be applied in any topos with a natural numbers object, and not just in the traditional setting of sheaves on a $\sigma$-field.

• Wow, that sounds awesome! Thank you so much! :) Mar 14 '14 at 21:52
• It was! Actually, I should add that that my construction of the Lebesgue integral followed the usual process (more or less). Instead of simple functions, I used measurable rational-valued functions (which can obviously only take countably many values): these functions comprise the rational numbers object inside $\textrm{Sh}_{j}(\mathcal{F})$. A Dedekind real is expressed as a lower and upper cut, so the integral of such a thing was just the supremum of the integrals of the rationals in the lower cut. Mar 14 '14 at 22:09
• I feel like all the questions with the category theory tag should come with a link to either Vakil's Foundations of Algebraic Geometry (with chapter 1 being about 100 pages of category theory) or Dr. Awodey's notes (and possibly a link to the YouTube videos from his talks in Oregon) or both/all three... Mar 13 '18 at 19:46

I don't really understand the question. You give the standard definitions of Lebesgue integration and finally ask if Lebesgue measurable functions form a category? Do you mean if they are the morphisms of a category? Anyway, here is something which might interest you:

One can show that $X=(L^1[0,1],1,\xi)$ is the initial pointed Banach space equipped with a pointed map $\xi : X \oplus X \to X$, see here. Actually we can construct $L^1[0,1]$ this way using abstract nonsense. Applying this to the pointed Banach space $(\mathbb{R},1,m)$ with the mean $m(a,b)=\frac{a+b}{2}$, we obtain a unique map of Banach spaces $\int : L^1[0,1] \to \mathbb{R}$, $f \mapsto \int f(x) \, dx$ such that $\int 1 \, dx =1$ and $$2 \cdot \int f(x) \, dx = \int f\bigl(\tfrac{x}{2}\bigr) \, dx + \int f\bigl(\tfrac{x+1}{2}\bigr) \, dx.$$ I've learned this from a note by Tom Leinster.

• Not a bad answer from someone who did not really understand the question :-) Jan 28 '14 at 18:19
• It does interest me. Thank you :) Jan 29 '14 at 19:50
• I gave a copy of that note to my lecturer on Lebesgue Integration. He hasn't done Category Theory for over a decade but he mentioned something (off the top of his head) about the category of measure spaces and the "Daniell Process". This doesn't mean much to me but maybe someone here (might be interested and) could guess at what he meant $\ddot\smile$ Jan 30 '14 at 18:55
• This does seem to be what I was after actually, although it's "only" for the interval $[0, 1]$ (and it describes the space of integrable functions without suggesting how to integrate functions within that space). I guess I was hoping for some functors or something from, say, $\mathscr{L}^1(\Bbb{R}^k)$ to $\Bbb{R}$-vector spaces (or maybe lattices), but in such a way that they explain why $\mathscr{L}^{\text{inc}}(\Bbb{R}^k)$ is not a vector space or why not all the lattice properties hold there. Do you see what I mean? Jan 31 '14 at 12:51
• Well, $L^1$ is just not a category! Jan 31 '14 at 14:03