Measure theory and topology books that have solution manuals I am trying to find a book to learn measure theory that contains complete solutions manual. Does someone know of any?
Also, I would like to know if there is a book with solutions manuals about topology.
thanks.
 A: There are two parts of this answer:


*

*Textbooks (or problem books) that have solution manual sold along with them;

*Good and classic textbook most of whose problems have solution online.



For 1, here are are some recommendations:

1a) "Principles of Real Analysis 3rd Edition" by Charalambos D. Aliprantis and Owen Burkinshw.

This is the original book, and it has a companion solution manual of all the exercises in this book:

1a') "Problems in Real Analysis, A Workbook with Solutions 2nd Edition" by Charalambos D. Aliprantis and Owen Burkinshw.

I have to say that I never had any professors or courses that used this book. This book is good if you are not the first time learner, since it introduces measure, integrable functions, etc, in a way that first-time learner may find hard to understand and to relate to other classic books like Royden, Rudin, or Stein. It is more like a Folland-approach. But if you just want to do some extra exercises and you have learnt measure theory more than once, this is good.
By the way, this book also contains some basic analysis problem, so you can review the undergraduate stuff.
Another book is a problem book:

1b) "Problems in Mathematical Analysis III --- Integration" by W.J. Kaczor and M.T. Nowak.

It has detailed solution inside the book. This is pretty user friendly if you are the first-time learner, but not really useful if learn it in a second time and in a deeper level. The exercise simply does not cover the deeper stuff. But if you learn the measure theory first time in the first semester PhD course or so, this can be pretty good book to supplement your course.
This is pretty much of it.

For 2), I will refer the books and list some links of solution I found.

2a) "Real Analysis: Modern Techniques and Their Applications" by Gerald B. Folland.

Most of Chapter 1 to Chapter 6
Most of Chapter 7
Most of Chapter 8
Q1 and Q10 of Chapter 9
Another solution set from Chapter 1 to Chapter 9
Some problems in Folland are pretty hard and technical so they don't have solution online (and they are likely in your homework if you are with a harsh professor), so go here and ask.

2b) "Real Analysis" by H.L. Royden and P.M. Fitzpartrick

A really good solution set that contains most of the exercises in the whole book
Despite being commonly used, I personally hate this book. Perhaps my first time learning experience was with this book, and the proof in this book was not readable for the first time learning. However, it does cover a pretty broad of topics, if you are not the first time, companying Folland with Royden will be a perfect choice.

2c) "Probability: Theory and Examples" by Rick Durrett.

Some solutions as solutions of homework
Never imaged a customized solution manual can be sold...
This is a really commonly used book to introduce measure theoretic probability, and MSE (and internet) contains the solutions of pretty much all the exercises. However, be careful with the question numbering. The book now is of 5th edition, despite no change in the exercise, the ordering has a drastic change, and solutions were created during the 2nd of 3rd edition of this book.
I must say that, if you are for a serious treatment (and I think this is vital) of measure theoretic probability, do NOT read this book. This book is somehow not deep enough and not standard enough. Measure theoretic probability contains an inevitable and hard experience of getting your technicality perfect. This is a "must-experience" experience. This book tries to skip them (or cannot cover them in details) or tries to taught them in a not standard way. The book becomes funny when it treats conditional probability distribution, like it goes crazy.

For a serious treatment of measure theoretic probability, here are some really good choices:

3a) Amir Dembo's note:  This book is really technical;
3b) "Probability Second Edition" by A.N. Shiryaev;
3c) "Probability - 2" by A.N. Shiryaev.

Shiryaev's book is really good and really detailed. It also covers a really wide range of topics, from basic measure theory to orthogonal random measure (so discrete spectrum theory) ect. You can not only learn measure theoretic probability, but can learn stochastic process a lot from the discrete view (from discrete to continuous, it is pretty simple).
Dembo's note didnt do a great job when it introduces Brownian motion, but it is more of a topic of stochastic process. If you need, I also have some recommendation.
But for now I think above is enough for you :)
A: http://jianfeishen.weebly.com/uploads/4/7/2/6/4726705/solution-measure-integration.pdf .This contains some exercises and solutions.
The following has the solutions of the selected exercises of the book 'A  First Look at Rigorous Probability Theory'.
http://www.worldscientific.com/doi/suppl/10.1142/6300/suppl_file/6300-solutionsmanual_free.pdf.
Goodluck
A: Perano, most textbooks on measure theory and topology are considered too high level to have solutions manuals in the usual sense-students at that level who need solutions manuals to get through their courses are considered doomed to failure. I don't agree with this thinking,I think all textbooks,regardless of level,should have complete solutions manuals. But most books at that level don't. But I do know a few exceptions and they're mainly problem course texts. 
Ian Adamson's A General Topology Workbook covers all the main topics of point set topology-open and closed sets,subspaces, general convergence,etc.-through a series of beautiful exercises,all with complete solutions in the second half of the book.The only really "standard" textbook I know on measure theory that has a conventional solutions manual is Robert Bartle's A Modern Theory of Integration-which isn't really a conventional graduate course on measure and integration, but rather a development based on the Henstock-Kurtzwell integral. While I think this is a subject that's underused in teaching analysis and it's quite well presented in this book, it isn't really what you're looking for. 
Lastly, there's a terrific problem course in measure and integration that comes with complete solutions-Problems in Mathematical Analysis III:Integration by W.J. Kaczor and M.T. Nowak. The exercises are immense, clear and not too difficult and come with complete solutions in the back. Since the book is so comprehensive and the courses in the subject have become so standardized-you may find all the solutions you need in the second half of this book. I'd also recommend getting the earlier 2 volumes in the same series-they provide great practice and additional training in real variables for the serious student.
Good luck!  
A: René Schilling: Measures, Integrals and Martingales 
The solution manual is not contained in the book, but available on the web page.
A: This is neither topology nor measure theory but rather functional analysis (a subject which uses both).... Paul Halmos's "A Hilbert Space Problem Book" is wonderfully constructed, it  has a brief discussion at the start of every chapter then follows this formula (definition, problem, definition, problem,...). But this is only the first third of the book; the middle part of the book gives a complete set of hints to every problem from all chapters, and the final third section gives complete solutions. 
A: Hmm, the most excellent Stein and Shakarchi book, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, has a solutions manual that is pretty good. I think if you google around for it you can find it. The solutions seem pretty complete too. 
A: You can try Terence Tao's An Introduction to Measure Theory. The book is one of the best in the fields and costs less than 70$ on Amazon. The exposition is brilliant and the exercises are doable.
The solutions are available under http://math.solverer.com/library/terence_tao/an_introduction_to_measure_theory
