What makes an integral an integral?

Solution 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.