Several of the most basic results of Lebesgue integration are stated for nonnegative (Lebesgue measurable) maps $X \to [0,+\infty]$ and may involve the order structure $\leq$ of $[0,+\infty]$:
- Fatou's lemma: for measurable maps $f_n:X \to [0,+\infty]$ one has $$\int_X \liminf_n f_n \, d\mu \leq \liminf_n \int_X f_n \, d\mu$$
- Monotone convergence theorem: for a nondecreasing sequence $0\leq f_0 \leq f_1 \leq \cdots$ of measurable maps $f_n:X \to [0,+\infty]$ one has $$\lim_n \int_X f_n \, d\mu = \int_X \lim_n f_n \, d\mu$$
- Fubini-Tonelli theorem: given two $\sigma$-finite measure spaces $(X,\mathcal{A},\mu)$ and $(Y,\mathcal{B},\nu)$ and a nonnegative measurable map $X \times Y \to [0,+\infty]$ one has $$\int_{X\times Y} f d\mu\otimes\nu = \int_{X} \left(\int_{Y} f_y \, d\nu \right) \mu = \int_{Y} \left(\int_{X} f_y \, d\mu \right) \nu$$ where we write $f_x \equiv \{ y \mapsto f(x,y) \}$ etc ...
On a personal level I am very fond of these results on account of the simplicity of their assumptions: nonnegativity is all you need. I am curious to what extent the above can be generalized. In particular: what would replace $[0,+\infty]$ ?
Lebesgue integration extends to Banach space valued maps (Bochner integral). There is also a notion of Banach lattices and more generally of Riesz spaces $(E, \leq)$, which have a positive cone $P = \{ x \in E \mid x \geq 0 \}$, and for which one can define various notions of completeness and order topologies.
Question: Is there a theory of integration of measurable functions with values in some type of abstract complete positive cone ?
By complete I mean in particular that there should be a top element (so $[0,+\infty)$ wouldn't qualify but $[0,+\infty]$ should, and similarly the positive cone $P$ of a Riesz space woudln't directly qualify but some completion thereof might). By abstract I mean that there need not exist an ambient ordered vector space of which it is the positive cone.
The examples I'm interested in are
- integration of functions with values in the space of measures $\mu:\mathcal{A} \to [0,+\infty]$
- integration of functions with values in the space of nonnegative measurable functions
- integration of functions with values in the space of e.g. compact (?) convex sets with Minkovski sum as addition, scalar multiplication as one might imagine and order structure given by inclusion
(Wrt the last example I imagine that one might have to take some type of compactification of the space of compact convex sets to have a top element.)
Neither of these examples are cones in ordered vector spaces. This is due to allowing infinite values in examples 1 and 2 (and thus running into difficulties when subtracting two such objects) compact convex sets not having opposites wrt Minkowski sum.
