Lebesgue Integration

Cod y Modiwl
MA37010
Teitl y Modiwl
Lebesgue Integration
Blwyddyn Academaidd
2027/2028
Semester
Semester 2
Cyd-gysylltydd y Modiwl
Dr Robert Douglas
Rhestr Ddarllen
Gweld ar Aspire
Rhagofynion
MA30210 neu MT30210
Anghymharus (Unrhyw Flwyddyn Acad)
MAM7020
Staff Eraill sy'n Cyfrannu

Dulliau Asesu

Math o Asesiad

Manylion Asesiad

Cyfran
Arholiad Semester Written Examination: 2 Awr 100%
Arholiad Ailsefyll Written Examination: 2 Awr 100%

Canlyniadau Dysgu

Wedi cwblhau'r modiwl dylai'r myfyrwyr fedru:

  1. Demonstrate knowledge of examples of sigma-algebras and measures
  2. Determine the Lebesgue measure of certain Borel sets
  3. Determine whether a function is measurable
  4. Determine whether a function is integrable
  5. Demonstrate that the Riemann and Lebesgue integrals co-incide for a class of functions
  6. Apply convergence theorems to justify the exchange of limiting processes for integrals
  7. Demonstrate knowledge of Lebesgue measure and its properties

Disgrifiad cryno

This course is a rigourous practical guide to the basic technical foundations and main principles which underpin the classical notions of area, volume, and the related idea of an integral. After reviewing the Riemann integral, its properties and limitations, the beautiful and poweful theory due to Lebesgue is introduced. The emphasis is on examples and applications of the main theorems rather than proofs of the classical results.

Cynnwys

Riemann integrability. Fundamental Theorem of Calculus. Interchange of limiting processes.

Set theory. Sigma-algebras, generated sigma-algebras. Borel sets. Measures. Examples. Lebesgue measure.

Measurable functions. Simple functions. Fundamental Approximation Lemma. Lebesgue integrable functions.

Monotone Convergence Theorem. Fatou's Lemma. Dominated Convergence Theorem.

Lp spaces. Young's inequality. Holder's inequality. Mikowski's inequality.
Completeness.

Construction of Lebesgue measure. Translational invariance. Completeness.
Characterisation and approximation of Lebesgue measurable sets.

Product measures. Tonelli-Fubini theorem.

Sgiliau Modiwl

Math o Sgiliau

Manylion Sgiliau
Cyfathrebu Written answers to exercises must be clear and well structured.
Datblygu personol a chynllunio gyrfa Completion of tasks (assignments) to set deadlines will aid personal development. The course will give indications of whether a student wants to further pursue mathematical analysis and its applications.
Datrys Problemau The Assignments will give the students opportunities to show creativity in finding solutions and develop their problem solving skills.
Gwaith Tim Students will be encouraged to work together on questions during problem classes.
Gwella dysgu a pherfformiad ei hun Students are expected to develop their own approach to time management regarding completion of assignments on time and preparation between lectures.
Rhifedd Required throughout the course
Sgiliau pwnc penodol Broadens exposure of student to topics in mathematics.

Nodau

Mae'r modiwl hwn yn cydymffurfio a FfCChC Lefel 6