Lebesgue Integration

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-inside 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;
8. Demonstrate knowledge of an advanced development/application of Lebesgue Integration:

This course is a rigorous 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 powerful theory due to Lebesgue is introduced. The emphasis is on examples and applications of the main theorems rather than proofs of the classical results. Students will be expected to study advanced applications via mini courses

Measure theory is one of the main areas of research in mathematical analysis; Lebesgue measure and integration is the most important example. This theory plays a central role in analysis, functional analysis and probability theory. There are applications to the modern theory of partial differential equations and financial modelling. Essential concepts are introduced for any student seeking a deeper understanding of mathematical analysis and its applications.

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'r Lemma. Dominated Convergence Theorem.
Lp spaces. Young's inequality. Holder's inequality. Minkowski's inequality. Completeness.
Construction of Lebesgue measure. Translational invariance. Completeness. Characterisation and approximation of Lebesgue measurable sets.
Product measures. Tonelli-Fubini Theorem.
Topics for mini courses include:
Fundamental Theorem of Calculus for absolutely continuous functions.
Rearrangements of functions. Hardy-Littlewood inequality, generalisations and applications.
Weak derivatives and Sobolev spaces in one dimension.
Lebesgue-Stieltjes integral.
Probability spaces, random variables, martingales.
Convolution integrals, smoothing properties.

Mae'r modiwl hwn yn cydymffurfio a FfCChC Lefel 7