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Is there a book with simplest examples one can ever imagine? For example: "Lets say we have "tree" "apple" "1" . . ." What is sigma algebra of this set, what is sigma algebra generated by something in this set, what is borel sigma algebra etc. It would be awesome if book covered main topics in measure theory that are important to probability and stochastic processes. Do such books exist? For Aduh: Books i have read on measure theory use notation like this below. I want author instead to explain intuition in simple way.

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  • $\begingroup$ Measure theory for third graders ? $\endgroup$ – Rene Schipperus Oct 15 '16 at 18:12
  • $\begingroup$ If you restrict attention to positive measures on finite sets, then up to some fairly simple modifications you would be studying finite probability spaces. This does help give a basic intuition and might be taught in high school/gymnasium or in college as "discrete mathematics". $\endgroup$ – hardmath Oct 15 '16 at 19:24
  • $\begingroup$ No matter how good the author, measure theory can only be simplified so much before it ceases to be measure theory. Countable additivity $$\mu\left(\bigcup_{k=1}^\infty E_k\right) = \sum_{k=1}^\infty \mu(E_k)$$ is part of the definition of a measure, so if you aren't willing to learn some set notation, then you can't go very far unless you only consider very simple examples. $\endgroup$ – Alexis Olson Oct 18 '16 at 4:09
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Every textbook on measure theory that I've looked at has plenty of simple examples of the kind you mention (not with trees and apples, but simple nonetheless). What do you find lacking in the texts you've read?

Regarding your example $A:= \{\text{tree,apple,1} \}$, it's a mistake to ask what is the sigma algebra of this set (I hope I understand your question correctly here). There exist multiple sigma algebras of members of $A$: $\{ \emptyset, A \}$ is one, the power set of $A$ is another. Again, any of the standard references should make this very clear.

When I started learning about Lebesgue measure and integration, I found Taylor's General Theory of Functions and Integration very helpful (and still do). It moves slowly and gives lots of examples. It also has a Dover edition and so is very affordable.

If you're interested in an introductory text on measure-theoretic probability, I can recommend Rosenthal's A First Look at Rigorous Probability Theory. I would not consider this a textbook in measure theory proper, but it explains and makes use of the basic measure-theoretic concepts needed for probability and the exercises are not too difficult.

Addendum. I will stick with my original recommendations in light of your edit. You should keep in mind that, in learning mathematics, part of your job as a reader is to think of intuitive explanations and simple examples of the new concepts that are introduced. That's how one learns. Reading math is an active affair; you have to struggle with examples and exercises until the concepts become familiar. No matter how clear and simple your author may be, you'll never learn math by just passively absorbing a textbook.

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  • $\begingroup$ I have edited my post $\endgroup$ – Markoff Chainz Oct 15 '16 at 18:48
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Personally, I do not know of a book that simple. With that being said, Terrence Tao's An Introduction to Measure Theory is quite approachable and readable as an introduction to Measure Theory, assuming you have the prerequisite background.

More particularly, if you want simple examples, focus first on the Lebesgue Theory. It is more geometric and a bit less abstract, but it provides a firm base for the pursuit of abstract Measure Theory later.

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