Measure on the Real Line: Lebesgue Outer Measure – Measurable Sets – Regularity – Measurable Functions – Borel and Lebesgue Measurability 

Integration of Functions of a Real Variable: Integration of Non-Negative Functions – The General Integral – Integration of Series – Riemann and Lebesgue Integrals. 

Abstract Measure Spaces: Measures and Outer Measures – Extension of a Measure -Uniqueness ofthe Extension – Completion of a Measure – Measure Spaces – Integration with Respect to a Measure. 

L p Inequalities L p and the Spaces: The Spaces – Convex Functions – Jensen’s Inequality – The Inequalities of Holder and Minkowski – Completeness of L p(u). 

Signed Measures and their Derivatives: Signed Measures and the Decomposition – The Jordan Decomposition – The Radon-Nikodym Theorem – Some Applications of the Radon-Nikodym Theorem. 

1. Measure Theory and Integration by G.de Barra. First Edition. New Age International Publishers, Reprint 2000. Reference Book: 
2. Real Analysis by H.L. Royden and P.M.Fitzpatrick. Fourth Edition. Pearson Education Asia Limited, 2010. 
3. Elias M. Stein & Rami Shakarchi, Real Analysis Measure Theory, Integration, and Hilbert Spaces (Princeton Lectures in Analysis), Princeton university press, 2007.