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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:

integrals, wikipedia

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.

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    $\begingroup$ An integral is just a linear map from the vector space of real-valued functions to the reals. Using the axiom of choice, you can even "find" an integral that works for all functions, extends Riemann on compact support and even avoids $\pm\infty$ ... it just cannot be computed $\endgroup$ – Hagen von Eitzen Jun 16 '16 at 20:56
  • $\begingroup$ Fair - I could define the integral to be 0 any time Riemann is not defined. But what I'm asking is whether one of the known integrals is most general. $\endgroup$ – doublefelix Jun 16 '16 at 23:40
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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.

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    $\begingroup$ For sake of balance and interest, it seems worth pointing out the following quite old question: Why are gauge integrals not more popular? $\endgroup$ – Semiclassical Jun 17 '16 at 4:49
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    $\begingroup$ Sigh.No one can just make me look brilliant and leave it.........lol $\endgroup$ – Mathemagician1234 Jun 17 '16 at 6:10
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    $\begingroup$ I found two sources (en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral, encyclopediaofmath.org/index.php/Khinchin_integral) saying that the Khinchin integral is more general than the gauge integral. $\endgroup$ – Paul Jun 17 '16 at 13:08
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    $\begingroup$ @Paul While the Khinchin integral is somewhat more general then the gauge integral,it is also quite a bit more sophisticated in machinery and many of the more general aspects are lost on subsets of $\mathbb R^n$. For these purposes,the gauge integral is more general then the Lebesgue integral and therefore covers all cases in these spaces. $\endgroup$ – Mathemagician1234 Jun 17 '16 at 18:31

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