Notes on Measure and Integration

Details Notes on Measure and Integration Notes on Measure and Integration

2008-02-27

arxiv.org/abs/0802.4076v3

Mathematics

Classical Analysis and ODEs

This version corrects a few typos. An expanded version of this text has been published as "A (Terse) Introduction to Lebesgue

Scientific paper

This text grew out of notes I have used in teaching a one quarter course on integration at the advanced undergraduate level. My intent is to introduce the Lebesgue integral in a quick, and hopefully painless, way and then go on to investigate the standard convergence theorems and a brief introduction to the Hilbert space of $L^2$ functions on the interval. The actual construction of Lebesgue measure and proofs of its key properties are relegated to an appendix. Instead the text introduces Lebesgue measure as a generalization of the concept of length and motivates its key properties: monotonicity, countable additivity, and translation invariance.

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