1. Measures: Classes of sets. The Lebesgue-Caratheodory extension. Borel measures in metric spaces. Borel measures on the real line. The Lebesgue measure in R^d.

2. Measurable functions: Approximation by simple functions. Almost everywhere convergence and convergence in measure. Borel-Cantelli lemma. Egorov's theorem.

3. Integration: The Lebesgue integral. The monotone convergence theorem. Fatou's lemma. The L^p-spaces, completeness. Basic integral inequalities. The dominated convergence theorem. Vitali's convergence theorem. Product of measures. Fubini's theorem and its applications (convolution of measures, layer itnegration).

4. Differentiation of measures: The Radon-Nikodym theorem. Differentiation in R^d: Hardy-Littlewood maximal function, Vitali's covering lemma, the Lebesgue differentiation theorem and its applications (convolution with good kernels), differentiation of singular measures. 

5. Charges: Signed measures. The Hahn decomposition. Complex-valued measures. Total variation. 

6. Functions of bounded variation on R: Basic properties. The Banach indicatrix. Lebesgue-Stieltjes integrals (if time permits).

G. B. Folland, Real Analysis. Modern Techniques and Their Applications, Chapters 1, 2, 3.

B. Makarov, A. Podkorytov, Real Analysis: Measures, Integrals and Aplications, Chapters 1-5, 9, 11.

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Additional literature:

R. M. Dudley, Real Analysis and Probability

E. H. Lieb, M. Loss, Analysis

W. Rudin, Real and Complex Analysis

E. M. Stein, R. Shakarchi, Real Analysis. Measure theory, Integration and Hilbert Spaces.

Measurable functions. Integration. Differentiation of measures. Charges and complex-valued measures. Functions of bounded variation.

G.B.Folland, Real