Lebesgue integration

1. Aug 13, 2004

rick1138

I am looking for some good materials on Lebesgue integrals, especially anything with a geometric / visual flavor. Any suggestions would be greatly appreciated.

2. Aug 17, 2004

sal

It's not clear to me (or, I suspect, others) what you're aiming at here.

As I recall, roughly speaking, the Lebesgue integral is evaluated by dividing the interval over which it's being integrated into ranges of equal value instead of just putting the interval through a "bread slicer" as is done in the Riemann integral. The sum is then taken over these (typically disconnected) sets on which the function takes particular values, and the limit is taken as the range of values in each section is reduced to zero. (This was a very rough description...)

The reason it's interesting is that many more functions can be Lebesgue-integrated than Riemann-integrated. But -- and it's a big But -- when the Riemann integral exists for a function, the Lebesgue integral is equal to it, and if a function is so badly behaved that you can't Riemann-integrate it, then it's a pretty strange function, or the set over which it's being integrated is perverse.

So, the "meaning" of the Lebesgue integral is identical to the "meaning" of the Riemann integral; only the procedure is different.

With that in mind, what sort of thing are you looking for? Are you looking for illustrations of how a set would be divided up when taking the "Lebesgue measure" of it? Or are you looking for examples of Lebesgue-measurable sets which are not Riemann-measurable, or Lebesgue-integrable functions which are not Riemann-integrable? Or are you looking for general material on Lebesgue theory? The latter can be found in just about any analysis text (e.g., Rudin, "Principles of Mathematical Analysis", or Rudin, "Real and complex analysis", or browse any college or online bookstore looking for analysis texts).