MX4083: MEASURE THEORY (2019-2020)

1. Lebesgue extension of a measure;Algebras and σ-algebras of sets;Measure spaces;Properties of the Lebesgue measure;
2. Lebesgue extension of a measure defined on a ring of sets;
3. σ-additive and σ-semi-additive measures;
4. Extension a measure from a semi-ring of sets to the corresponding ring of sets;
5. Measurable functions;Convergence almost everywhere;
6. Measurable functions and uniform convergence; the Egorov theorem;
7. Definition and properties of measurable functions;
8. Lebesgue integral;The definition and the properties of the Lebesgue integral;The Lebesgue, the Levi and the Fatou convergence theorems for the Lebesgue integral;
9. Comparison of the Lebesgue integral and of the Riemann integral.
10. Absolute continuity and σ-additivity of the Lebesgue integral; the Chebyshev inequality;
11. Lebesgue integral for simple functions;

Syllabus

• Lebesgue extension of a measure;

          Extension a measure from a semi-ring of sets to the corresponding ring of sets;

          Algebras and σ-algebras of sets;

          σ-additive and σ-semi-additive measures;

          Measure spaces;

          Lebesgue extension of a measure defined on a ring of sets;

          Properties of the Lebesgue measure;

• Measurable functions;

          Definition and properties of measurable functions;

          Convergence almost everywhere;

          Measurable functions and uniform convergence; the Egorov theorem;

• Lebesgue integral;

           Lebesgue integral for simple functions;

          The definition and the properties of the Lebesgue integral;

          Absolute continuity and σ-additivity of the Lebesgue integral; the Chebyshev inequality;

          The Lebesgue, the Levi and the Fatou convergence theorems for the Lebesgue integral;

          Comparison of the Lebesgue integral and of the Riemann integral.

Information on contact teaching time is available from the course guide.

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There are no assessments for this course.