I want to carry the notion of Riemann integration to a more general setting. I have already given the following axioms on area function defined on an arbitrary Cartesian product: Let $X$ and $Y$ be two sets. Then by area we mean a function from some subset of the power set of their cross product to extended non-negative reals defined as follows:
- Area of countable union of disjoint subsets of the domain equals the sum of areas of the subsets.
- Area of the graph of any function from any subset of $X$ to $Y$ is zero.
- Area of the graph of any function from any subset of $Y$ to $X$ is zero.
Using the three axioms I have been able to show that areas of points,curves,etc. on Cartesian plane is zero, and also that area is monotonic. I want to extend this model to include the concept of integration. Any ideas???
I think some clarifications have to be made here
First: you first axiom is dangerous. If the subsets overlap or even coincide, you end up with something too large or even infinite. Not that that is wrong per se, but it is either not the concept of area one has in mind or a useless measure.
To give an example: if $X=[0,1]$, $Y=[0,1]$, let $S_n=[0,1]\times[0,1], n\in \mathbb{N}$, then $X\times Y=\bigcup\limits_{n=1}^{\infty}S_n$. Unless your measure of $[0,1]\times [0,1]$ is zero or infinite, you get a contradiction.
In order for your axiom to make sense, you must ask that these subsets are disjoint.
Second: you never specified what is the building block, that is, what are the sets for which you know how to compute the area/length. Usually these are intervals (or cartesian product of intervals) $I=[a,b]$, since there our intuition says their length must be $\ell(I)=|b-a|$. Then, you work your way towards more general sets, using these building blocks to define their measure.
Third: notice that your measure should not be defined as a function from $X\times Y$ to the extended real. Instead, it should be defined on some subset of the power set of $X\times Y$, since you want to measure subsets of $X\times Y$. Now, I don't know what $X$ and $Y$ are (if they're nasty sets, it could be really hard to define a measure), but assuming they are intervals, then you cannot build a reasonable measure on the power set of $X\times Y$ that is not the trivial one (Ulam's theorem).
Fourth: you never specified what condition your sets must satisfy in order to be measurable. For what I said in the previous point, it is clear you must restrict your focus to something smaller than the power set $\mathcal{P}(X\times Y)$, call it $\Sigma$. This subset $\Sigma$ (notice, subset of the power set of $X\times Y$, not a subset of $X\times Y$ itself) must satisfy some properties. For instance, you should know how to measure the whole $X\times Y$ or the empty set. Or, if you know how to measure a subset $Z$, you should be able to measure $(X\times Y) \setminus Z$. These, plus the closure under countable union, are the properties that characterize a $\sigma$-algebra. However, there are many $\sigma$-algebras. The one that suites your interest would have to satisfy some characterization property, which would tell us immediately if a given set belongs to it or not.