John Franks
Published 2008-02-27, updated 2009-08-10Version 3
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.
Comments: This version corrects a few typos. An expanded version of this text has been published as "A (Terse) Introduction to Lebesgue Integration" as vol. 48 of the A.M.S. Student Mathematical Library
On the Wasserstein Distance between Classical Sequences and the Lebesgue Measure
A simplified construction of the Lebesgue integral
On equality of the $L^\infty$ norm of the gradient under the Hausdorff and Lebesgue measure
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