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I want to get deeper into the real analysis and measure theory(in order to build good mathematical basis for learning stochastic proceses,probability theory etc.).

I want to ask you an advice about good real analyse's book: I have background in Single Variable Calculus and Multivariable calculus(Stewart's books). Linear Algebra(G.Strang book), Probability Theory and Inferential Statistics.

Which book will be the most appropriate variant for such background(I mean, it should me hard/informative enough, but not as hard, that it would destroy me).

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    $\begingroup$ Do you know what $\limsup$ means? Do you know what uniform convergence means? How about metric spaces, countable vs uncountable sets, compactness, connectness, the Weierstrass approximation theorem, nowhere differentiable functions that are continuous? If not, then you will be well-advised to take a basic real analysis course first - the kind of course most students take before getting into measure theory. $\endgroup$ – zhw. Oct 5 '16 at 17:11
  • $\begingroup$ @zhw. Thanks you for the advice! Can you please provide some good books for real analysis? $\endgroup$ – Daniel Yefimov Oct 5 '16 at 17:38
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Well, according to your question, I think you are not familiar with higher level mathematics(so called measure theory).

First of all, you must study undergraduate real analysis because the theory of probability and stochastic process are based on analysis. If you want to know this subject deeply, you have to take a class on analysis or self-study.

Before to study measure theory, I recommend two books.

Two books does not require measure theory, but it will help you to understand basic idea of probability theory.

One possible choice is Probability and Measure Theory written by Robert B. Ash and Catherine A. Doléans-Dade but it requires some maturity on mathematics https://www.amazon.com/Probability-Measure-Theory-Second-Robert/dp/0120652021

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  • $\begingroup$ Thank you for the answer! I have read and did examples of the second book(Dimitri P. Bertsekas and John N. Tsitsiklis), so i am familiar about probability theory. Do you know some good books about real analysis for undergrades? $\endgroup$ – Daniel Yefimov Oct 5 '16 at 13:26
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    $\begingroup$ @DanielYefimov This link would be helpful I guess. See my answer. math.stackexchange.com/questions/1895625/… $\endgroup$ – Will Kwon Oct 5 '16 at 14:03

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