Intereting Posts

V = U⊕W then Prove that (V/W)* is isomorphic to W^0
Fourier series of function $f(x)=0$ if $-\pi<x<0$ and $f(x)=\sin(x)$ if $0<x<\pi$
Intuition on group homomorphisms
Proof that a sequence converges to a finite limit iff lim inf equals lim sup
Group of even order contains an element of order 2
Matrix with all 1's diagonalizable or not?
Theorems' names that don't credit the right people
Solution to differential equation $\left( 1-\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)=0$
Proving $\left(\sum_{n=-\infty}^\infty q^{n^2} \right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi \tau)}$
Show that this set is linearly independent
Maximal ideals in polynomial rings
Generalization of Dirichlet's theorem
Horizontal and vertical tangent space of Orthogonal group
Prove that formula is not tautology.
Conic by three points and two tangent lines

I would like to teach myself measure theory. Unfortunately most of the books that I’ve come across are very difficult and are quick to get into Lemmas and proofs. Can someone please recommend a layman’s guide to measure theory? Something that reads a bit like this blog post, starts out very gently and places much emphasis on the intuition behind the subject and the many lemmas.

- Hyperreal measure?
- Steinhaus theorem (sums version)
- Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…”
- Open source lecture notes and textbooks
- Prove Continuous functions are borel functions
- Understanding Fatou's lemma
- To define a measure, is it sufficient to define how to integrate continuous function?
- What are the most overpowered theorems in mathematics?
- Integrable function $f$ on $\mathbb R$ does not imply that limit $f(x)$ is zero
- Required reading on the Collatz Conjecture

Measures, Integrals and Martingales by René L. Schilling is a very gentle (mathematically rigorous, but that should be the case if you want to learn measure theory) introduction to measure theory. All the solutions to the exercises are available on the website of the author. Another advantage is that it is quite inexpensive.

However, I’d also suggest Measure and Integration Theory by Heinz Bauer. This is one of the best introductions to this subject I have ever seen (and my professor and some others seem to agree). One drawback is that it has a few typos but that keeps you sharp ;-). It is a translation of the author’s original book in German where only the relevant topics are kept.

Here (TU Delft) they first used the first book which I mentioned and this year they use Bauer.

Both books are an excellent basis if you want to go in the direction of analysis or probability theory. Both fields require at least what is in these books.

A companion to Bauer’s measure theory book if your goal is to learn probability theory is his probability theory book.

Another thing I would like to note is that you should have a reasonable knowledge of the foundations of real analysis before you embark on this. Measure theory is a “true” analytic topic and should not be treated like many calculus courses.

I would recommend “Lebesgue Integration on Euclidean space” by Frank Jones. The analysis texts by Stein and Shakarchi are also very accessible.

One of the very best books on analysis, which also contains so much more then just measure and integration theory,is also available very cheap from Dover Books: *General Theory Of Functions And Integration* by Angus Taylor. You can probably get a used copy for 2 bucks or less and it contains everything you ever wanted to know about not only measure and integration theory, but point set topology on Euclidean spaces. It also has some of the best exercises I’ve ever seen and all come with fantastic hints. This is my favorite book on analysis and I think you’ll find it immensely helpful for not only integration theory, but a whole lot more.

My favourite book on measure theory is Cohn’s. It has a manageable size and yet, it covers all the basics.

I suggest A Concise Introduction to the Theory of Integration by Daniel W. Stroock, which I found both a pleasure to read and straight to the point.

**Edit**: I was *almost* forgetting that it includes interesting exercises with hints/solutions, so it’s good for self study.

I very much enjoyed these lecture notes.

It’s written in a style that is suitable for self-study.

I found Robert Bartle’s book: Elements of Integration and Lebesgue Measure to be very helpful.

This book, **Problems in Mathematical Analysis III**, has plenty of exercises (with solutions!) on *The Lebesgue Integration.*

I really like Foundations of Modern Analysis by Avner Friedman. Excellent text on the essentials plus it is a “worker’s book on analysis” in the sense that it shows you how many of the tools you learn in a measure theory course are actually used to tackle problems in PDE, functional analysis, etc. Plus it is *cheap* ($12).

My other recommendation is a second nod to Lebesgue Integration on Euclidean Spaces by Frank Jones. *Very* accessible but astronomically expensive. Perhaps you can get a copy from the library (or interlibrary loan).

Thinking back very far, to when I was a student learning measure theory, I really liked “Introduction to measure and probability” by Kingman and Taylor. The measure theory part was also published as a separate book, “Introduction to measure and integration” by (only) Taylor.

- Is the degree of an infinite algebraic extensions always countable?
- $R$ is PID, so $R/I$ is PID, and application on $\mathbb{Z}$ and $\mathbb{N}$
- Find the eigenvalues of a projection operator
- Dividing a curve into chords of equal length
- Solve the integral $S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx$
- problem on intersecting circles
- Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?
- Finding primes so that $x^p+y^p=z^p$ is unsolvable in the p-adic units
- Smallest multiple whose digits are only ones and zeros
- Which of the following form an ideal in this ring?
- Find $C$ such that $x^2 – 47x – C = 0$ has integer roots, and further conditions
- Do we need Axiom of Choice to make infinite choices from a set?
- Cohomological Whitehead theorem
- Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$
- Proving that a sequence such that $|a_{n+1} – a_n| \le 2^{-n}$ is Cauchy