Measure and Integration

Measure Theory:

•Lebesgue  Outer measure, Sigma-algebra of Lebesgue measurable sets. Completion of a measure, existence of  Non-measurable sets.

• Measurable functions and their properties. Some discussion on abstract measure.

Integration theory:

Lebesgue integral, Functions of bounded variation and absolutely continuous functions, Integration and Convergence theorems, Fundamental Theorem of Calculus for Lebesgue Integrals.

• Product measure spaces, Fubini-Tonelli theorem, Lebesgue-spaces, Reisz Representation Theorem for C([a, b]).

Evaluation: Quizzes will be during the lecture hours.

• Two 20-pointer quizzes,

• Four 10-pointer quizzes,

• Two 40-pointer exams.

Suggested Books:

1. Measure Theory and Integration by G. De Barra

2. Real Analysis By E. Stein and R. Shakarchi

3. Real and Complex Analysis By W. Rudin

4. Measure, Integration and Real Analysis by S. Axler