Measure Theory Lecture 10 01/11/2019

An introductory lecture on measure theory given at Cardiff Univeristy in autumn 2019. Syllabus content:

- Sigma-algebras and measures
- Lebesgue outer measure, Lebesgue measurable sets and Lebesgue measure.
- Non Lebesgue measurable sets and the axiom of choice.
- Hausdorff measure and Hausdorff dimension of fractal objects
- Measurable functions and image measure
- Integrability (in the sense of Lebesgue) and almost everywhere prevailing properties
- Integral convergence theorems
- Radon-Nikodym theorem
- Spaces of integrable functions
- Dynkin (d-) systems and the monotone class theorem with applications
- Product measures, product Sigma-algebras and Fubini's theorem.