In what sense is Lebesgue integral the "most general"?

Solution 1:

There are other notions of integral that possess the basic properties you'd expect of an integral but which can be more general than the Lebesgue integral. For instance, the Henstock-Kurzweil (aka Perron or Luzin integral) is a notion of integral that extends the Riemann integral and has different properties than the Lebesgue integral.

Different theories of integration should be compared on the basis of the properties you mention as well as the resulting convergence theorems (i.e., exchangeability of limits and integration), properties of absolute convergence, and the the versions of the fundamental theorem of calculus they permit. Some integrals are designed to have particularly nice properties with respect to, e.g., convergence theorems (i.e., Lebesgue integration), while others are designed to have particularly strong fundamental theorems of calculus (i.e., Henstock-Kurzweil integration). A particularly nice book that studies several integration theories and compares them is this book.

Solution 2:

To answer my own question, the answer seems to be "yes," with details given in the following MathOverflow thread, particularly in the answer of G. Rodrigues:

https://mathoverflow.net/questions/38439/integrals-from-a-non-analytic-point-of-view