# 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

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• 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