I recently found an entire category of Wikipedia pages discussing different definitions of the integral. It includes the familiar definitions of Riemann and Lebesgue, the less familiar but still well known Stieltjes integrals, and quite unfamiliar (but cool) definitions like the Henstock–Kurzweil integral and the Khinchin integral.
Now my background is in physics, where we have another kind of integration called path integral (or functional integration), which remains generally ill-defined despite many decades of fruitful use in theoretical physics. Naturally, one of my first thoughts upon discovering the exotic integrals in the aforementioned Wikipedia pages (like Henstock–Kurzweil) was whether they could help give rigorous definition to physicists' path integrals, but unfortunately it doesn't look like that's the case.
Thinking about all these different sorts of integrals got me to wondering:
What are the common features of all these different sorts of integrals that make them "integrals"?
In other words, what are the minimal requirements for some mathematical definition to be an integral? If I had to guess, I would say the following is a plausible, though imprecise, start:
Given a vector space of functions $V$, a definite integral $\int$ on this space is a function from a subspace $I\subseteq V$ of "integrable functions" to a number field $F$, such that:
- $\int$ is linear.
- $\int$ agrees with our intuition for certain simple functions. In finite dimensions, this could be e.g. that $\int$ applied to the indicator function of a cube is the volume of the cube. In infinite dimensions, you might prefer to work with Gaussians instead of indicator functions of cubes.
I would be tempted to add some condition about continuity, but I'm not sure that would be appropriate in infinite dimensions (i.e. $\int$ might be unbounded?). I wonder if anyone has tried to define integrals in the abstract along these lines, or if these conditions are roughly the minimal "common features" of all integrals?