ADVANCED REAL ANALYSIS-II (Optional Paper)

Derivative: Banach-Zarecki theorem, Derivative and integrability of absolutely continuous functions, Lebesgue point of a function, Determining a function by its derivative.

General Measure and Integration : Additive set functions, Measure and signed measure, Limit theorems, Jordan and Hahn decomposition theorems, Complete measures, Integrals of non- negative functions, Integrable functions, Absolute continuous and singular measures, Radon-Nikodym theorem, Radon- Nikodyn derivative in a measure space.

Fourier Series: Fourier series of functions of class L, Fejer-Lebesgue theorem, Integration of Fourier series, Cantor- Lebesgue theorem on trigonometric series, Riemann’s theorem on trigonometric series, Uniqueness of trigonometric series.

Distribution Theory: Test functions, Compact support functions, Distributions, Operation on distributions, Local properties of distributions, Convergence of distributions.

Differentiation of distributions and some examples, Derivative of locally integrable functions, Distribution of compact support, Direct product of distributions and its properties, Convolution and properties of convolutions.