Good book on measure theory

[Mathias]

I was wondering if you know any books on measure theory that takes the student from a relatively low level to a relatively high (and perhaps abstract) level?

I can see that there is a ton of literature out there and some free notes as well... I would really appreciate if you have any suggestions!

 

[bigbadwolf]

I was wondering if you know any books on measure theory that takes the student from a relatively low level to a relatively high (and perhaps abstract) level?

I can see that there is a ton of literature out there and some free notes as well... I would really appreciate if you have any suggestions!

What is your own level and what books have you found so far? I presume your interest is in measure theory directed towards probability?

 

[Mathias]

I presume your interest is in measure theory directed towards probability?

Correct. I'm about to take a course in measure theory which is notorious difficult and I want to have a book that builds up to that.

The book we use is 'Measure theory 4th edt.' by Ernst Hansen which is relatively unknown but a good book anyways. I want something that starts at a lower level than this book but reaches this level (if possible). The course content is:

1) Probability measures: properties, uniqueness, measures with density.​

2) Random variables and their distributions.​

3) Image measures, transformation of probability measures.​

4) Moments, distribution functions and quantiles.​

5) Conditional expectations.​

6) Product measures: definitions, Tonelli & Fubini, independence of random variables.​

7) Density transformation in one and several dimensions.​

8) The central limit theorem.​

and they expect us to be able to:​

At the end of the course the students should be able to​

• Apply measure theory to propose models in probability theory.
• Translate between probabilistic statements, using the language of random variables, and measure theoretic statements.
• Transform densities in an abstract set-up as well as the set-up of densities on R with respect to Lebesgue measure.
• Identify the most common probability measures and recall their basic properties.
• Apply theorems on successive integration.
• Master computations with univariate and multivariate moments and conditional expectations.
• Apply the central limit theorem to approximate the distribution of an average.

 

[bigbadwolf]

The book we use is 'Measure theory 4th edt.' by Ernst Hansen which is relatively unknown but a good book anyways.

I didn't know the book prior to your mentioning it. I think it has been published only in Danish.

Some good books I've found in English are

1) Probability-Through-Problems
2) Measure-Integral-Probability
3) Probability Theory in Finance. Dineen. AMS.
4) Introduction Measure theoretic Probability
5) Probability: A Graduate Course. Gut. Springer.
6) A Probability Path. Resnick. Birkhauser.

Maybe 4) is closest to your text.

 

[Mathias]

Tanks a lot the book has been published in english, but it used to be in danish.

In my experience, books from the springer verlag is usually quite thorough... thanks a lot - I'll try to look at them and especially 4).

 

[Jose T]

I like this book:

Measure Theory and Probability Theory, Athreya and Lahiri

 

[FedericoM]

This book covers ALL the topics you listed. I've been studying it for one semester, and I can definitely state that it was worth it
It is also the prerequisite book required to attend the ETH Zurich MSc in Quantitative Finance, maybe the most prestigious quant school in Europe.

Probability Essentials

 

[Mathias]

This book covers ALL the topics you listed. I've been studying it for one semester, and I can definitely state that it was worth it
It is also the prerequisite book required to attend the ETH Zurich MSc in Quantitative Finance, maybe the most prestigious quant school in Europe.

Probability Essentials

It looks good - thanks for the suggestion. On amazon.co.uk, it says the language is german (which is wrong if you look inside the book, but a bit funny though).

 

[jackcoke]

For a quick introduction to measure and probability theory over 4 lectures, you can check out the following video links (under probability theory section).

http://finmath.uchicago.edu/students/sept_review.shtml