Prerequisite: Analysis in Rn

Extension theorems for measures and integrals. Basic convergence theorems. Measures with sign. Hahn-Jordan decomposition theorem. Absolutely continuous measures. Lebesgue decomposition theorem. Radon Nikodym theorem. Lp-spaces: Basic properties, duality. Product spaces. Fubini-Tonelli theorem. Riesz-Markov representation theorem. Convergence in measure. Relationship between differentiation and integration: Vitali’s theorem. Lebesgue’s differentiation theorem.

References
ARMANDO CASTRO JR, A. – Course in Measure Theory. Rio de Janeiro, IMPA, Projeto Euclides, 2nd edition, 2008.
BARTLE, R. – The Elements of Integration, New York, J. Wiley, 1966.
FERNANDEZ, P. – Measurement and Integration. Rio de Janeiro, IMPA, Projeto Euclides, 1976.
ISNARD, C. – Introduction to Measurement and Integration. Euclid Project, IMPA, 2007.
ROYDEN, M. – Real Analysis. New York, The MacMillan, 1963.
RUDIN, W. – Real and Complex Analysis. New York, Mc-Graw Hill, 1966.

* Basic syllabus. The teacher has the autonomy to make any changes.