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I am trying to learn about the Lebesgue integral and Lebesgue measurability. None of my text books really cover it from the basics, but I found this document online which seems to be pretty through in explaining the motivation behind developing the Lebesgue integral http://web.media.mit.edu/~lifton/snippets/measure_theory.pdf [Broken]

However, there is a statement I am having a hard time grasping, on the bottom of the first page:

"Third, all L_{p}spaces except for L_{∞}fail to be complete under the Riemann

integral"

Here is what I understand: when saying "L_{p}spaces" I'm assuming this means metric spaces, right? I know from functional analysis that a complete metric space is one where there are no "gaps", or formally, where every Cauchy sequence has a limit that's also in the space. (That's why, for example the rational numbers with the st. Euclidean metric (L_{p}with p=2) is not complete, because we have gaps at all the irrational places)

Here's what I don't understand: what does it mean to be completeunder the Riemann integral? I don't understand what this means. I thought a metric space would be a set of numbers, with a metric defined on it, and it would be complete or incomplete just based on that information alone. Where does the Reimann integral come in to play?

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# All Lp spaces, (except where p=∞) fail to be complete under the Reimann integral ?

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