What makes an integral an integral?

Question

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:

1. $\int$ is linear.
2. $\int$ 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?

Answer 1

You are given a ground set $X$, a measure $\mu$ on $X$, a subset $B\subset X$, and a function $f:\>B\to{\mathbb R}$. Then you want to know the "total effect" implied by $f$ on $B$, given the measure $\mu$. This "total effect" is called the integral of $f$ over $B$, and is designed by $$\int_B f(x)\>d\mu(x)\ ,$$ or similar. This integral should have the properties $$\int_B \bigl(\alpha f(x)+\beta g(x)\bigr)\>d\mu(x)=\alpha\int_B f(x)\>d\mu(x)+\beta\int_B g(x)\>d\mu(x)\ ,$$ as well as $$\int_B f(x)\>d\mu(x)=\int_{B_1} f(x)\>d\mu(x)+\int_{B_2} f(x)\>d\mu(x)\ ,$$ when $B=B_1\cup B_2$, and $B_1$, $B_2$ are "essentially" disjoint. These ideas lead for a continuous function $f$ to the setup $$\int_B f(x)\>d\mu(x)=\lim_\ldots\>\sum_{k=1}^N f(\xi_k)\>\mu(B_k)\ ,$$ where the $B_k$ are tiny "essentially disjoint" subsets of $B$, $\>\xi_k\in B_k$ $\>(1\leq k\leq N)$, and $B=\bigcup_{k=1}^N B_k$.

These ideas can be realized in a Riemann, Lebesgue, or Henstock-Kurzweil way, all resulting in the same values for the integral in all practical situations, but differing in the collectives of admissible functions and allowed "limit theorems".

All sorts of integrals you meet in differential geometry or in physics are of this kind. The difficulties you encounter with them have nothing to do with Riemann/Lebesgue/Hemstock-Kurzweil, but with the geometrical, linear algebra, or physical background needed to convince you that an interesting ("invariant") quantity is computed in the adopted setup.

Answer 2

I'd like to also append the definition, if it was not clear in the above:

A measure is a function defined on a σ-algebra F over a set X and taking values in the interval $[0,∞[$ such that the following properties are satisfied:

(i) the emptyset has measure zero, $\mu(\emptyset) = 0$;

(ii) countable additivity: if $(E_i)$ is a countable sequence of pairwise disjoint sets in F, then:

$$\mu\left(\bigcup_{i = 1}^{\infty} E_i\right) = \sum_{n=1}^{\infty} \mu(E_i) $$

and (X, F, $\mu$) is called a meaure space.