I am working on some notes on Hilbert spaces for physics students, and I am exposing the need for an integration method without some of the limitations of the Riemann integral (the Lebesgue integral).
I know the very basics of measure theory and understand the construction of the Lebesgue integral, and that it is better for building metric spaces thanks to the dominated convergence theorem.
However, I wonder if it would make sense to speak of a "Borel integral" cunstructed in the same way as the Lebesgue integral, but using the Borel measure instead (and thus more limited), just to present a "step between" the Riemann and Lebesgue integrals. if it worked the way I imagine, the Dirichlet function would be Borel-integrable, since the Dirichlet set is a Borel set, but it would still fail to integrate the indicator functions of sets that are Lebesgue-measurable, but not Borel.
I know it wouldn't really be practical, since it would still carry the flaws of the Lebesgue integrals, while not being as powerful. It's merely a question out of curiosity.
Thank you!