I'm not sure why you consider the gauge integral to be easier to understand the Lebesgue integral. It may be due to learning Lebesgue first, but I find it much more intuitive.

Also:

Royden is slightly better in this respect. The first four chapters are excellent, but still probably too theoretical. Further, eventually one will encounter measure spaces that aren't based on the real numbers and the Lebesgue measure,

Yes, this is a good thing. One doesn't understand a structure until one understands which parts of a structure are forcing which properties. Moreover, this supplies useful counterexamples that helps one understand what sort of things one necessarily will need to invoke if one wants certain results.

Hmm. Upvoted for contributing to a good topic but I'm not sure I agree.

I just looked up the gauge integral because I wasn't familiar with it. For those curious about the debate, here's the introduction to the gauge integral I found, which has a lot of relevant information. My beef with this is precisely that it doesn't use the general background of measure theory (sigma-algebras, measurable functions, etc.) and you're going to need that background to do useful things. The gauge integral approach doesn't give you the tools to generalize to scenarios like Brownian motion where you need to construct different measures; also, the gauge integral doesn't come with a lot of nice convergence theorems the way the Lebesgue measure does.

I don't find the standard treatment of measure theory especially hard; it takes about a month to understand everything up to the Lebesgue integral, which isn't an obscene time commitment.

Also, there's some virtue to just being familiar with the definitions and concepts that everybody else is. (It's not just mathematicians "refusing to update." I know for sure that economists, and potentially people in other fields, speak the language of standard measure theory. But maybe it's not everyone. What are you using measure theory for?)

If you're looking for an easier, more straightforward treatment than Rudin, I'd recommend Cohn's Measure Theory. I'm not sure why, but it feels friendlier and less digressive.

I haven't studied real analysis, could you explain what advantages the guage integral is better than the lebesgue integral? Edit: maybe just respond to SarahC.