Acknowledgments xv Remark 0.0.6. The question of how much of real analysis still survives when one is not permitted to use the axiom of choice is a delicate one, involving a fair amount of logic and descriptive set theory to answer. We will not discuss these matters in this text. We will, however, note a theorem of odel [Go1938] that states that any statement that can be phrased in the first-order language of Peano arithmetic, and which is proven with the axiom of choice, can also be proven without the axiom of choice. So, roughly speaking, odel’s theorem tells us that for any “finitary” application of real analysis (which includes most of the “practical” applications of the subject), it is safe to use the axiom of choice it is only when asking questions about “infinitary” objects that are beyond the scope of Peano arithmetic that one can encounter statements that are provable using the axiom of choice, but are not provable without it. Acknowledgments This text was strongly influenced by the real analysis text of Stein and Shakarchi [StSk2005], which was used as a secondary text when teaching the course on which these notes were based. In particular, the strategy of focusing first on Lebesgue measure and Lebesgue integration, before moving onwards to abstract measure and integration theory, was directly inspired by the treatment in [StSk2005], and the material on differentiation theorems also closely follows that in [StSk2005]. On the other hand, our discussion here differs from that in [StSk2005] in other respects for instance, a far greater emphasis is placed on Jordan measure and the Riemann integral as being an elementary precursor to Lebesgue measure and the Lebesgue integral. I am greatly indebted to my students of the course on which this text was based, as well as many further commenters on my blog, including Maria Alfonseca, Marco Angulo, J. Balachandran, Farzin Barekat, Marek Bern´at, Lewis Bowen, Chris Breeden, Danny Calegari, Yu Cao, Chandrasekhar, David Chang, Nick Cook, Damek Davis, Eric Davis, Marton Eekes, Wenying Gan, Nick Gill, Ulrich Groh, Tim Gowers, Laurens Gunnarsen, Tobias Hagge, Xueping Huang, Bo Jacoby, Apoorva Khare, Shiping Liu, Colin McQuillan, David Milovich, Hossein Naderi, Brent Nelson, Constantin Niculescu, Mircea Petrache, Walt Pohl, Jim Ralston, David Roberts, Mark Schwarzmann, Vladimir Slepnev, David Speyer, Blake Stacey, Tim Sulli- van, Jonathan Weinstein, Duke Zhang, Lei Zhang, Pavel Zorin, and several anonymous commenters, for providing corrections and useful commentary on the material here. These comments can be viewed online at: terrytao.wordpress.com/category/teaching/245a-real-analysis
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