Lebesgue Integral and Measure Theory for an undergraduate?

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KennethF

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Hi all.

I am a freshman studying in Asia with a Bachelor's degree concentrating in Finance and Mathematics, so here's the problem - our maths department doesn't offer courses on Lebesgue Integral and Measure Theory (except in the graduate level) - and they seem to be very important in this field(?). So are there any books that you guys could recommend me to take a look at, and at which time should I do so (like after which courses to take)?

Thanks so much!
 
Spiegel "Real Variables Lebesgue Measure and Integration" (Schaum Outlines)

Very practical

Later : Rudin

~ 2nd year university pure maths

To be honest, not offering Lebesgque at undergrad maths says a lot about program and its quality.
 
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Thanks a lot for all of your suggestions. I am not really sure at the moment whether it will be exactly taught as I cannot find it in any course outlines yet we would be using Rudin's book in our second and third analysis course. (There is a sequence of 3 analysis courses for undergrad) But I would take a look at those books recommended.
 
Daniel Duffy said:
To be honest, not offering Lebesgque at undergrad maths says a lot about program and its quality.
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Not sure I agree. We have analysis at undergraduate, of course, but the idea is that if you are interested in taking a course on Measure theory then you are quite ready to take classes within the graduate department anyway, and there's no issue.
 
euroazn said:
Not sure I agree. We have analysis at undergraduate, of course, but the idea is that if you are interested in taking a course on Measure theory then you are quite ready to take classes within the graduate department anyway, and there's no issue.
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Fair enough.

But MT and Lebesgue at graduate level is leaving it a bit late IMO. Probably too late.
 
At my university it is mandatory for all math students to take 2 measure theory courses. In the first course we used Measures, Integrals and Martingales by Schilling. The book might be good for self study because there are solutions to all the exercises on the books homepage. I didn't like the book because it lacked reasons for e.g. why it is smart that the sigma algebra is defined like it is. However, some people like it that way.

In the second course we used a book that isnt available online, however, it is comparable to Probability and Measure by Billingsley. I bought the book and I really liked it.
 
I didn't like the book because it lacked reasons for e.g. why it is smart that the sigma algebra is defined like it is.


Many treatments tend to lack motivation and tend to be a bunch of axioms. And lack of good concrete examples.

A good way to get some feeling for measurable sets is to take a monotonic function and compute by hand the upper and lower Lebesgue sums numerically.


http://demonstrations.wolfram.com/LebesgueIntegration/
 
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Daniel Duffy said:
Fair enough.

But MT and Lebesgue at graduate level is leaving it a bit late IMO. Probably too late.
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I agree with this point - most capable undergrads take it their junior or senior year here - but they are officially "graduate" courses.
 
euroazn said:
I agree with this point - most capable undergrads take it their junior or senior year here - but they are officially "graduate" courses.
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In finance I suppose you mean?
In pure maths it was (is?) 2nd-year stuff.
 
euroazn said:
No, in math. Do Europeans typically take Real Analysis their first year of undergraduate studies? If so, what age do you enter university?
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In England, 18. But the undergrads are coming with "A" levels in pure math and one extra math (applied math or further math). They're already competent at differential and integral calculus. If they're not taking analysis in their first year, definitely in their second. Universities like Oxford start with (or used to) with the Lebesgue integral in the first course on analysis, and not bother with the Riemann integral at all. Which makes sense, as there's nothing inherently abstruse about the Lebesgue integral.
 
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That's interesting. Undergrads come in with calculus classes from high school as well, but are exposed to a bit more breadth in their first two years of undergrad before taking analysis (e.g linear algebra, discrete mathematics, etc.)
 
euroazn said:
That's interesting. Undergrads come in with calculus classes from high school as well, but are exposed to a bit more breadth in their first two years of undergrad before taking analysis (e.g linear algebra, discrete mathematics, etc.)
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Maths undergrads do analysis and algebra simultaneously. I would say it's a must.

Algebra is easy; analysis should be done at an early age. It counteracts linear algebra that teaches you that everything you come in contact with is a finite-dimensional vector space.
 
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euroazn said:
That's interesting. Undergrads come in with calculus classes from high school as well, but are exposed to a bit more breadth in their first two years of undergrad before taking analysis (e.g linear algebra, discrete mathematics, etc.)
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In the US system they keep tediously repeating the same material again and again. The best math undergrad students have typically taken AP calc, and at the BC level rather than the AB level. So they're spared the first two semesters of calc. Vector calc is what they'll start with followed by a course in diff eqs. Those who took high school calc without taking or clearing the AP exam have to go through the same dreary rigmarole again. It makes a joke of university education. This is why the 4-year US degree typically covers not much more than the first two years of an English undergrad program. The first US analysis course (usually taken in the junior or senior year) will be pitched at the level of baby Rudin -- covered in the first or second year of an English undergrad program. The first complex analysis course will be pitched at the level of Churchill, again at the level of a second year English undergrad program.
 
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bigbadwolf said:
In the US system they keep tediously repeating the same material again and again. The best math undergrad students have typically taken AP calc, and at the BC level rather than the AB level. So they're spared the first two semesters of calc. Vector calc is what they'll start with followed by a course in diff eqs. Those who took high school calc without taking or clearing the AP exam have to go through the same dreary rigmarole again. It makes a joke of university education. This is why the 4-year US degree typically covers not much more than the first two years of an English undergrad program. The first US analysis course (usually taken in the junior or senior year) will be pitched at the level of baby Rudin -- covered in the first or second year of an English undergrad program. The first complex analysis course will be pitched at the level of Churchill, again at the level of a second year English undergrad program.
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Didn't know. Why do you think US Universities are typically held in so much regard, then?
 
euroazn said:
Didn't know. Why do you think US Universities are typically held in so much regard, then?
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The grad programs, where there's a big disconnect with the undergrad programs and many of the students come from abroad. It's an uneasy welding together of a 19th century English undergrad humanistic education (hence the stupid course requirements of lit, history and other baloney for science students) and a German grad education directed towards research students. The welding isn't good so many students fall down the abyss between the undergrad program and the grad one.

Not to be cynical but most US math departments are service departments whose bread and butter is calc 1 and 2 and various remedial courses. This is what puts food on the table for profs and teaching assistants. And they also love repeat customers -- those unfortunates who need, say, college algebra or calc 1 to graduate or for graduate programs but can't get through at the first pass. Ultimately what it means is inefficient money-gobbling and time-consuming programs run by a self-serving bureaucracy, with all sort of useless hoops to be jumped through.

American students are getting shafted at every level -- middle school, high school and undergrad.
 

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