I have following syllabus to study in Real Analysis Subject. I want to know, What are necessary topics that I have to cover as a prerequisite for below syllabus.
Actually I am unable to get direction as this subject is very big. Can someone help me in providing point wise preliminaries topics for below syllabus?

Below is syllabus -
. Measure Theory: Preliminaries, Exterior Measure, Measurable Sets and Lebesgue Measure, Measurable Functions.
2. Integration Theory: The Lebesgue Integral, basic properties and convergence theorems. The space $\mathcal{L}^1$ of integrable functions, Fubini’s theorem.
3. Differentiation and Integration: Differentiation of the integral, Good kernels and approximation to the identity, differentiation of functions.**

Already I am studying Rudin's book. Still I don't know how much it will help me in scoring.

Thanks in advance.


closed as primarily opinion-based by Adam Hughes, Grigory M, user147263, anomaly, TravisJ Jun 25 '15 at 23:21

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  • $\begingroup$ Please assume me beginner. $\endgroup$ – Omkara Jun 25 '15 at 10:16
  • $\begingroup$ Rudin's book on real and complex analysis? $\endgroup$ – user190080 Jun 25 '15 at 10:17
  • $\begingroup$ I am refering "principle of mathematical analysis". $\endgroup$ – Omkara Jun 25 '15 at 10:55
  • $\begingroup$ In what kind of context are you attending this real analysis course? $\endgroup$ – user190080 Jun 25 '15 at 11:17
  • $\begingroup$ This is outline of course that I have started on my own. =>Real number system and its order completeness, sequences and series of real numbers. Metric spaces: Basic concepts, continuous functions, completeness, contraction mapping theorem, connectedness, Intermediate Value Theorem, Compactness, Heine-Borel Theorem. Differentiation, Taylor's theorem, Riemann Integral, Improper integrals Sequences and series of functions, Uniform convergence, power series, Weierstrass approximation theorem, equicontinuity, Arzela-Ascoli theorem. But I have to reach to => Measure Theory. $\endgroup$ – Omkara Jun 25 '15 at 12:42

I did a swift check on Rudin's book on Principles of Mathematical Analysis:
The book covers in my opinion (and Rudin's as well:) a 1-year undergraduate course in analysis (at least that is what is taught in German universities) and a bit beyond (for example chapter 10).

All your topics (measure theory, integration, differentiation) are subject of one or more chapter in this very book. Although I haven't read this book, I think it would be a good choice to stick to the outline of this book by Rudin if you want to work with it - I am sure it is not a bad choice. I also could find some solution manuals on the internet, which might become in handy if you're doing a self-study course.

I would say the outcome of working through the book is a very solid knowledge on analysis, a good companion for advanced studies in mathematics.


  • $\begingroup$ Thanks for answer. Will continue reading "Principle of mathematical analysis" book as this covers basic concepts. Let me check on internet again what are pre topics. $\endgroup$ – Omkara Jun 26 '15 at 5:36

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