# Reference for measure theory book with 'many' examples of different Measures

I post this question with some personal specifications. I hope it does not overlap with old posted questions

I am looking for a clear way to learn measure theoretic probability theory.I have already done a course in measure theory so i already know all basic concepts of Measure theory which are usually covered in $1$ semester course like Caratheodory's extension theorem,integration,Fubini's theorem,$L^p$-spaces,Radon Nikodyn Theorem,Riesz Representation theorem for compact metric spaces etc but i realize that although i have done a course in measure theory but i don't still know many examples of measures other than some usual Lebesgue measure,counting measure,Jordan Measure and Haar Measure etc.In think any theory without lot of examples is mostly useless.So i am looking for a book of measure theory which discuss probability theory and also discusses many examples of spaces like Coin tossing space and discuss measures on different spaces.Most of the books which i have seen yet does not really give many examples of measures.

Is there any book of Measure theory which develops measure theoretic probability theory as well as discuss many examples of 'different' measure?

• It could be useful to take a look into books written by mathematicians who are into probability theory. These present naturally examples like the coin tossing space. Especially I am thinking about "Measure and Integration Theory" by H. Bauer. – Fritz Sep 28 '16 at 7:04
• Long time ago, but I seem to remember Billingsley has such examples. – BruceET Sep 28 '16 at 7:08

See, the point is, when a probabilist writes a measure theory book, the focus is on probability measures on Euclidean spaces, and distributions, expectations, etc.. Rarely does a probabilist even consider unbounded measures, but one needs to be very careful with unbounded measures. So that's the most fundamental, and sometimes, only example one gets.

When a pure mathematician writes a measure theory book, the focus is on the analysis rather than the examples themselves. Analysts, in my opinion, do not generally spend a lot of time on examples: they give a theory and when the question of examples arises, they scornfully say, "Talk to the Calculus-ians" or "Look at exercises".

The last advice is not too bad. After all, I believe that measures are very natural mathematical functions. You want to measure something, and different functions give probably different labels of the same thing. Mostly while working through exercises, you'll get to learn something is, or is not, a measure. For example, to show something is NOT a measure, you need to find disjoint sets $A_n$ such that $\mu(\cup_n A_n)\neq \sum_n\mu(A_n)$.

So here's a suggestion: take whatever measure theory book you have in your library. You know the theory, so directly go to the exercises. Almost all books I have seen contain lots of examples of measures in exercises.

Finally, I would recommend George Roussas' "An Introduction to Measure Theoretic Probability". It's very readable and not at all terse. And, it has lots of examples AND exercises.

• Sorry..i don't quite agree with the statement that 'all books contains lots of examples of measures?Suppose i ask you to give me lets say $10$ examples of 'different'(not the ones which are absolutely continuous to either of one) measures what would you give me?Thanks. – Dontknowanything Sep 28 '16 at 16:39
• "...in the exercises". Incidentally, there are some interesting measures than can be constructed on the set of natural numbers. And when I said "lots of examples" I was talking about checking whether some functions are measures or not. Not really discriminating among whether they are abs. continuous with respect to the same measure. – Landon Carter Sep 28 '16 at 17:00
• Dear @Landon: Sorry i don't know those 'interesting' measures on $\mathbb N$ other than the obvious ones like couting measure etc.I'd be happy you would add those measures either in comment or in your answer.Thanks! – Dontknowanything Sep 28 '16 at 17:27
• – Landon Carter Sep 29 '16 at 4:31
• Thank you,i'll have a look. – Dontknowanything Oct 3 '16 at 17:16