# Real analysis : Preliminary topics for - Measure Theory, Integration Theory, Differentiation and Integration [closed]

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 -
**1
. 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.

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

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• Please assume me beginner. – Omkara Jun 25 '15 at 10:16
• Rudin's book on real and complex analysis? – user190080 Jun 25 '15 at 10:17
• I am refering "principle of mathematical analysis". – Omkara Jun 25 '15 at 10:55
• In what kind of context are you attending this real analysis course? – user190080 Jun 25 '15 at 11:17
• 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. – 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.

bests

• 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. – Omkara Jun 26 '15 at 5:36