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This question has been edited in view of comments received.Hope it is better now.

Consider the set of all points on the circumference of a circle and the set of all points inside a circle. what is difference in the measure of these two sets.

Is Bertrand paradox related to measure theory ? Can bertrand paradox be resolved with the notion of a measure of a set. If so please suggest some good references/books for measure theory.

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-1; why haven't you consulted the Wikipedia articles first? They are long, well-written, and have many references. –  Qiaochu Yuan Nov 29 '10 at 20:10
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@ Rajesh D: I think you should try to ask one or just a few questions at the time. This will generate better answers for you. –  Max Muller Nov 29 '10 at 20:35
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@Rajesh D: I hope one doesn't have to be a genius to read the Wikipedia articles on these topics, but I admit that some of them can be difficult to read. (For some of these topics I don't think there's any truly easy way to learn about them.) I am pretty sure that one does not have to be a genius to scroll to the bottom and consult the list of references. –  Pete L. Clark Nov 29 '10 at 21:29
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@Rajesh: my point is that if you found the Wikipedia articles confusing or otherwise uninformative, you could ask about the specific things you found confusing and the specific things you wanted more information on. If you found the list of references in the Wikipedia article somehow lacking, you could be more specific about what you're looking for in a reference. There is no point in us duplicating a function Wikipedia already serves (answering general questions); this site exists precisely because it can answer specific, detailed questions. –  Qiaochu Yuan Nov 30 '10 at 15:53
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@Rajesh: I do not see how this constitutes "showing off." It is site policy that people should justify their downvotes whenever possible. Anyway, we are getting off-topic: I have started a meta thread at meta.math.stackexchange.com/questions/1241/… if you wish to discuss this further. –  Qiaochu Yuan Nov 30 '10 at 16:56
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up vote 4 down vote accepted

Lang's book is a fairly comprehensive treatise on Differential Geometry which will introduce you to Riemannian manifold and other notions you mention in first paragraph

http://books.google.com/books?id=VfxGB5nYv1MC

Measure theory can be picked up from Paul Halmos' book

http://www.amazon.com/dp/0387900888

Edit: Following Rajesh D's comment below.

I liked Munkres book on point set Topology. It also gives a flavor of Algebraic Topology towards the end. Though I have heard mixed opinions from others about this. But it can be read with almost zero background in any sort of math. It builds up from basic set theory (this was incidentally the first mathematics text I ever read).

http://www.amazon.com/dp/0131816292

Janich book on basic topology is wonderful and reads like a novel.

http://www.amazon.com/dp/0387908927

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@Timothy: Do you mind if I make the links into inline links? –  Arturo Magidin Nov 29 '10 at 20:10
    
@Arturo: That's fine with me. It's just that I find sometimes people tend to miss inline links. Alternately, I can tinyurl them. –  Timothy Wagner Nov 29 '10 at 20:12
    
@Timothy: Nevermind then; if it is on purpose, then let it stand. –  Arturo Magidin Nov 29 '10 at 20:14
    
@Timothy Wagner: I forgot to mention that i don't know Topology either. that was the first topic in contents of Lang's book on Differential geometry. –  Rajesh D Nov 29 '10 at 20:22
    
@Tim: I hope you don't mind if I stripped the cruft off your URLs; for reasons I don't want to get into, the metadata are tricksy ones. –  J. M. Nov 30 '10 at 1:38
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Consider the set of all points on the circumference of a circle and the set of all points inside a circle. what is difference in the measure of these two sets.

The circumference of a circle has a measure of length (e.g. inches) and the set of all points inside the circle has a measure of area (e.g. square inches).

Cantor proved that there are not "more" (as measured by one-to-one correspondence) points in the area of a circle than there are on the circumference! So it is wrong to thing of measures simply a count of the number of points.

Is Bertrand paradox related to measure theory?

Absolutely. That is the reason that the paradox is so popular. The paradox twists three measures together in what is confusing until you apply measure theory.

Can Bertrand paradox be resolved with the notion of a measure of a set.

Essentially, but it is more in understanding how the different ways that different measures and different sets can interact.

If so please suggest some good references/books for measure theory.

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