Transcription of The Lebesgue integral - MIT Mathematics
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The Lebesgue integral - MIT Mathematics
CHAPTER 2. The Lebesgue integral This part of the course, on Lebesgue integration, has evolved the most. Initially I followed the book of Debnaith and Mikusinski, completing the space of step functions on the line under the L1 norm. Since the Spring' semester of 2011, I. have decided to circumvent the discussion of step functions, proceeding directly by completing the Riemann integral . Some of the older material resurfaces in later sections on step functions, which are there in part to give students an opportunity to see something closer to a traditional development of measure and integration. The treatment of the Lebesgue integral here is intentionally compressed. In lectures everything is done for the real line but in such a way that the extension to higher dimensions carried out partly in the text but mostly in the problems is not much harder.
E;is sais to hold almost everywhere. In particular we write (2.19) f= ga:e:if f(x) = g(x) 8x2RnE; Eof measure zero. Of course as yet we are living dangerously because we have done nothing to show that sets of measure zero are ‘small’ let alone ‘ignorable’ as this de nition seems to imply. Beware of the trap of ‘proof by declaration’!
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