The course has two components : Lectures and tutorials. (Credit: 3.5)
Lebesgue Measure: Introduction, Outer measure, Lebesgue measure measurable sets, Properties of measurable sets, Borel sets and their measurability, Non-measurable sets.
Measurable Functions: Definition and properties of measurable functions, Step functions, Characteristic functions, Simple functions Littlewoods' three principles,
Lebesgue Integral: Lebesgue integral of bounded function, Integration of non-negative functions, General Lebesgue integrals, Integration of series, Comparison of Riemann and Lebesgue integrals. Convergence in measure.
Differentiation and Integration: Differentiation of monotone functions, Functions of bounded variation, Lebesgue differentiation theorem, Differentiation of an integral, Absolute Continuity,
Course learning outcomes: Upon completion of this course, the student will be able to:
1) define Lebesgue measure on R
2) describe measurable functions and its properties.
3) apply measures to construct integrals.
4) explain the basic convergence theorems for the Lebesgue integral,
5) analyze the relation between differentiation and Lebesgue integration.
Recommended Books:
1) Royden, H.L. & P. M. Fitzpatrick, Real Analysis, Pearson Education (2011) 4th Edition.
2) Barra, G.de, Measure Theory and Integration, Wiley Eastern Ltd. (2012).
3) Jain, P.K., and Gupta, V.P., Lebesgue Measure and Integration, New Age International Ltd. (2010) 2nd Edition.
4) Rana, I.K., An Introduction to Measure and Integration, Narosa Publication House (2010) 2nd Edition.