I am expanding the notes last year (still working in progress, and should be finished at the beginning of July):
http://ihome.ust.hk/~cclee/document/WorkshopinAnalysis.pdf
Last year I stopped at littlewood's 3 principles (last section in measurable functions). Having the preparation last year, I plan to quickly go through abstract outer measure and measure (skip some of the proofs which can be copied word by word) and go directly into some important extension theorem on "pre-measure" and "pre-sigma algebra" (in my case, semi-ring), they will be used in the construction of product measure.
My schedule is the following sequence:
Basic Measure Theory
1. Outer measure and Measure
2. Integration
3. Construction of particular measure, go back to R^n
4. ???
5. ???
6. Duality of L^p space on sigma-finite measure space
(possibly need Radon–Nikodym, so 4, 5 may be signed measure and relevant theorems, that's will be big task).
Measure and Topology
7. Brief review of topology material (if necessary)
8. Facts and measure on locally compact Hausdorff space
9. Try to prove the Riesz-Markov theorem
http://en.wikipedia.org/wiki/Riesz_representation_theorem
Hopefully (???) if I have time I plan to apply knowledge in functional analysis and the Riesz-Markov theorem to show Haar measure always exists on compact group, which is unique up to a +ve-constant.
The workshop will begin at the middle of July, and as before, I would ask Kin Li to book me a room. If you are to-be year 2 students and also want to join us, it would be better to digest the material in chapter 2 and 3. We will apply some of the technique implicitly.
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