I am currently taking an introductory course to real analysis and my professor has decided to leave Rudin's "Principles of Mathematical Analysis" when teaching us the concepts of Lebesgue integration. He is using a "measure-free" approach to Lebesgue integration, in that he is avoiding teaching us general measure theory first. He is basing his lectures on Konigsberger's "Analysis 2", which unfortunately for me, is written in German. I personally find supplementing lectures with reading the textbook very helpful, but I can't do that here. Does anyone know of other analysis textbooks that are comparable to this book? (I don't think an english translation exists)
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I don't know Konigsberger's "Analysis 2" book, but I think Apostol's "Analisys" could help you (it doesn't use much measure theory in order to explain Lebesgue's integral, only zero measure and some other things).
Again, I don't know Konigsberger's book. 'Introduction to Integration' by Hilary Priestley does Lebesgue integration without using measures.