Resources for studying the Lebesgue integral at the undergraduate level

Question

I am a physics major in my second year, and I hope to learn about the Lebesgue integral. To give a sense of what I know so far: I went through 2 years worth of calculus courses and am currently doing a reading project based on the text "Understanding Analysis" by Stephen Abbott. What other texts/resources would you suggest for learning the about the Lebesgue integral?

Answer 1

You don't need a thorough grounding in Measure Theory to understand Lebesgue integration. The basic idea of Lebesgue integration over $\Bbb R^n$ is based on step functions, not Measure Theory. You do need to understand the idea of a null set, but that is as far as it goes.

One way of looking at the difference between Riemann integration and Lebesgue integration is that Riemann integrates by taking vertical slices of the area under the curve, whereas Lebesgue integrates by taking horizontal slices. The advantage of Lebesgue's approach is evident in that a Lebesgue integrable function is always Riemann integrable, but the converse is not true. When I was at university 40 years ago, they didn't even bother teaching us Riemann integration, because Lebesgue integration is so much better -- and just as easy to grasp, in my opinion.