Measure Theory

InbredDummy

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.

Dick

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?

InbredDummy

My ultimate goal is to research differential geometry; I have been reading very broad overviews of Perelman's work, Richard Hamilton's founding work on Ricci flows and I am enamored with this field of geometry. I would also like to delve into more mathematical General Relativity, maybe the manifolds used in quantum gravity. I want to delve into a heavy dose of geometry, geometric analysis, and some topology. But I do see the value of taking measure theory as an undergrad, getting exposed to the material before taking in the same time a year from now in grad school. Some of my peers have told me if I plan on going into geometry, go for the geometry research project. I'm torn on the decision because up till now I have been able to take every course and fit it anything else into my schedule.

mathwonk

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.

Chris Hillman

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.)

InbredDummy

I want to take it because he does a chapter on the space of measurable functions, a little sneak peek to functional analysis. arg, I might just take both...screw it. this is my last semester to take real math courses since i'm graduating in May and in Spring I need to take 4 general ed courses to graduate. i'm going to do both because i want to do both.