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Measure Theory 
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#1
Aug2807, 10:31 PM

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How important is a measure theory course as an undergrad? I have to choose between taking an undergrad measure theory course and doing research. I'm already doing another research project, but I figure no grad school is going to penalize me for doing too much research. But how "bad" is it that I'm not taking a Measure Theory course? There is no way I can do measure theory and the research, it would put me at 6 courses which is just way too much for me given all the other nonacademic things I have to deal with.
I have taken an analysis course which did all the basics: sequences, series, derivative, all the way up to the Riemann integral. I have also taken a complex analysis course. 


#2
Aug2807, 11:12 PM

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Mmm. What are you interested doing research in? What are the other courses? If you've only gone up to Riemann integrals, measure theory is basically an extension of the notion of Lesbegue integration. Very important for some fields, not for others. Can you give us a context?



#3
Aug2907, 12:04 AM

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#4
Aug2907, 09:27 PM

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Measure Theory
this is a tough one, as both are valuable, but i will go with the research. measure theory is important for analysts and say people wanting to understand mathemaical foundations of quantum mechanics, but since you say you waNT differential geometry, i go with the research experience in geometry.



#5
Sep107, 02:19 PM

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Ditto mathwonk, FWI.
Measure theory is required for probablility theory and integration theory (in a graduate analysis course), which are required for ergodic theory, which is the most abstract part of dynamical systems theory. If you study Lie theory you'll run into "Haar measure", for example, and if you study anything involving operators you'll run into other measures. The most important theorem for 21st century mathematics is Szemeredi's lemma, which belongs to ergodic theory; see Terence Tao, "What is Good Mathematics?", http://www.arxiv.org/abs/math/0702396, which proposes Szemeredi as the canonical example of good mathematics. For philosophy as well as math students, there is no topic more important. Nonetheless, ditto mathwonk. See Folland, Real Analysis for all the measure theory you'll need presented compactly (no pun intended). ("Most important": taking a cue from mathwonk, I am being deliberately provocative in hope of impressing the impressionable, for the very best of pedagogical reasons!) ("Philosophy students": because the most important problem to which philosophers can contribute is this: "what if anything is the signifcance of statistics?" I've been saying that for years but finally decided I am going to start saying it every darn chance I get, because I am really serious about this claim.) 


#6
Sep107, 11:04 PM

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