Category Theory and Lebesgue Integration.

Question

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 . . .

Answer 1

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.

Answer 2

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.