Elements of Measure Theory and Functional Analysis

Programme

Alexander Meskhi

a) Measurable space; measure space; elementary properties of general measure; complete measure; example of non-complete measure; outer measure; Borel sets; Lebesgue measure; an example of non-measurable set; signed measure; Hahn decomposition theorem; extension of measure; Caratheodory extension theorem; product measures.

b) measurable functions; elementary properties of measurable functions; convergence almost everywhere; convergence in measure; Egorov’s theorem; Lusin’s theorem; simple functions.

c) general integral; elementary properties of general integral; Chebishev’s inequality;; absolutely continuity of general integral; Fatou’s lemma; Lebesgue monotone convergence theorem; Lebesgue dominated convergence theorem; Fubini’s theorem; Tonelli’s theorem; comparison of Lebesgue and Riemann integrals; comparison of Lebesgue and improper Riemann integrals; absolutely continuous and singular measures; Radon-Nykodim theorem; Lebesgue decomposition theorem.

d) Functions of bounded variation; Jordan’s theorem on representation of functions of bounded variation; absolutely continuous functions; relationship between the class of absolutely continuous functions and the class of functions of bounded variation; Cantor set; Cantor-Lebesgue function; differentiation of monotonic functions; rising sun (Riesz) lemma; Lebesgue theorem on differentiation of a monotonic function; Lebesgue differentiation on differentiation of indefinite integral; reconstruction of function from its derivative; Lebesgue theorem on derivative of absolutely continuous functions; a full characterization of functions which can be represented as indefinite integral.

e) Metric spaces; examples of metric spaces; space of continuous functions; Lebesgue spaces; Schwartz ‘s inequality; Holder’s inequality; Minkowski’s inequality; complete metric spaces; continuous functions on metric spaces; principle of nested balls; completion of a metric space; compact sets in metric spaces; Arzela’s theorem; Kolmogorov’s theorem; totally bounded sets; principle of contraction mappings and its applications.

f) Normed linear spaces; Banach spaces; linear functionals; conjugate spaces; extension of linear functionals; Hahn-Banach theorem; weak convergence; linear operators; conjugate linear operators; inverse operators; Banach theorem on invertible bounded linear operators; compact linear operators; Banach- Steinhaus theorem; Hilbert spaces; complete orthonormal system; Hilbert-Schmidt theorem; Riesz representation theorem.

References:
1) H. L. Royden, Real Analysis, Macmillan Publishing Company, USA, 1988;
2) A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover Publications, INC, Mineola, New-York, 1961;
3) A. N. Kolmogorov and S. V. Fomin, Introduction to Real Analysis Dover Publications, INC, New-York, 1970;
4) W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill International Editions, Mathematical Series, 1976;
5) W. Rudin, Real and Complex Analysis; McGraw-Hill Book Company, New York.
6) R. L. Wheeden and A. Zygmund, Measure and Integral. Introduction to Real Analysis; Pure and Applied Mathematics, A Series of Monographs and textbooks, Marcel Dekker, INC, New York, 1977.