Measure Theory: Course Outline (MAT-610)
Motivation and introduction to Measure theory on abstract sets, Algebras, sigma algebras, Borel sets, measure space, measurable sets, complete measure. Outer Measure, on abstract set. Construction of measure on abstract sets using elementary family and pre measure. Borel Measure on Real Line, Measure induced by an increasing function. Lebesgue measure and Lebesgue Steiltjes measure. Inner and outer regular measure. General Measurable Functions, Approximation of measurable function by continuous functions, modes of convergence, Egorov’s theorem on abstract measure space. Integration with respect to general measure, Levi’s theorem on abstract measure space, Fatou’s Lemma on abstract measure space, Dominated convegence theorem on abstract measure space, Chebyshev’s Inequality, comparison of Riemann and Lebesgue Intgeral. Lp- Spaces defined with respect to general measure, Holder’s Inequality and Minkowski’s inequality and convergence in norm.Inclusions of Lp.