The day I learned about the Lebesgue integral was very exciting. A more general integral than Riemann, which is equal to it for all Riemann integrable functions (on finite domains)? Very cool.
Unfortunately, my curiosity led me to google, and my search results showed:
It turns out I'm more naive than I ever knew.
The question: Is there a "most general integral" of real-valued functions on the real line? One which agrees with the others where they are defined, but is defined on a superset of their domains? ("defined", for me, includes infinite integrals). The Khinchin integral seems like a candidate.
Note: I saw another similar question but it didn't ask about $\mathbb{R}$ specifically, which is my interest.
Note2: I don't mean "trivial" integrals, like one which is defined to be 0 whenever the Riemann integral is not defined, or equal to it otherwise. The answer would presumably have its own wikipedia page.
Best Answer
The "gauge" integral,otherwise known as the Henstock–Kurzweil integral-is the most general integral known defined on subsets of $\mathbb R^n$, which of course includes the real line as a special case. Indeed, a number of mathematicians, including the late Robert Bartle, have suggested the gauge integral replace the Riemann integral in basic analysis/honors calculus courses because not only is it far more general then even the Lebesgue integral on these spaces, it's definition is much simpler. It results from a minor modification of the definition of a partition on a subset of $\mathbb R^n$.As a result, only a careful treatment of "$\epsilon-\delta$" calculus is needed to fully develop it.
A good brief introduction to the gauge integral-with references-can be found here.