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# Resources for Measure theory and Lebesgue integration?

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1. Does anyone know of any good online resources to learn measure theory and Lebesgue integration?

I found some good pdf's but they were a bit too advanced (I have no prior knowledge) and too long.

Thanks!
2. First of all, what knowledge do you have already? You'll need set theory; so that means point-set topology as well probably, since it's really just set theory on steroids. It'd also be nice to know some real analysis, so you can draw parallels from Riemann integration and the real line. Also, if you don't know Riemann integration and start learning it after Lebesgue integration, you might start using properties erroneously from Lebesgue integration.

As for resources, you'll find plenty just be searching the internet. As long as you have sufficient knowledge, you should be able to dive right in to measure theory.
3. (Original post by cliveb2016)
Does anyone know of any good online resources to learn measure theory and Lebesgue integration?

I found some good pdf's but they were a bit too advanced (I have no prior knowledge) and too long.

Thanks!
As already suggested, it would be helpful to know the stage that you're at, so we can adapt our answers to your needs. In the UK, measure theory is usually introduced in the third year of the undergraduate maths degree; therefore students starting to learn it already have a grounding in point set topology, basic real analysis, set theory and so forth.

Having said that, the beginnings of measure theory and integration are, from a pedagogical point of view, pretty standard. You introduce sigma algebras of sets, think about pi-systems and d-systems of sets and go about proving existence and uniqueness of measures. So, a pretty good introduction to this sort of stuff can be found in the first chapters of Bass's online book "Real analysis for graduate students". (This is designed for American grad students facing their qualofying exams for the PhD degree - about the level of UK 3rd years/1st year grad.)

Beyond online materials, I usually recommend Schilling's "Measures, Integrals and Martingales" as the best first book on measure theory.
4. (Original post by Gregorius)
As already suggested, it would be helpful to know the stage that you're at, so we can adapt our answers to your needs. In the UK, measure theory is usually introduced in the third year of the undergraduate maths degree; therefore students starting to learn it already have a grounding in point set topology, basic real analysis, set theory and so forth.

Having said that, the beginnings of measure theory and integration are, from a pedagogical point of view, pretty standard. You introduce sigma algebras of sets, think about pi-systems and d-systems of sets and go about proving existence and uniqueness of measures. So, a pretty good introduction to this sort of stuff can be found in the first chapters of Bass's online book "Real analysis for graduate students". (This is designed for American grad students facing their qualofying exams for the PhD degree - about the level of UK 3rd years/1st year grad.)

Beyond online materials, I usually recommend Schilling's "Measures, Integrals and Martingales" as the best first book on measure theory.
I have taken courses in Real analysis and set theory so I should be all good on that front. I have studied Riemann integration as well. I am going to start yr3 in October but I am not sure which modules I will be taking yet so I want to get a taste for a few to see which I prefer. Thanks for the advice so far I will check those books out.

The Bass one looks very nice as it has a good amount of topics in it and should come in handy for topology as well!
5. (Original post by Alex:)
First of all, what knowledge do you have already? You'll need set theory; so that means point-set topology as well probably, since it's really just set theory on steroids. It'd also be nice to know some real analysis, so you can draw parallels from Riemann integration and the real line. Also, if you don't know Riemann integration and start learning it after Lebesgue integration, you might start using properties erroneously from Lebesgue integration.

As for resources, you'll find plenty just be searching the internet. As long as you have sufficient knowledge, you should be able to dive right in to measure theory.
I have a good basis in each of those topics so I am ready to start the topic but I wanted to have something that was around 100 pages or less to get me into it without being too excessive i.e 500+ pages on graduate measure theory yikes.

Thanks!

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