mi lecture notes

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Lecture notes for Measure & Integration Theory, Copenhagen Uni 2011 Anders Munk-Nielsen 1 MI notes Disclaimer: These are my own, personal notes  they are subject to tons and tons of typos and misunderstandings. In particular, the lecturers and TAs are in no way responsible. However, please do circulate as you please if you feel they might help make the world a better place in any sense. I don t understand law stuff so Ill just say something like GPL 3 licensed or whatever also, throw a comment or a mail if you enjoyed these notes! Be awesome to hear from someone as geeky out there (and if these notes actually ended up being used for something unless youre building some weapon of mass destruction unless youd use that to enslave mankind and force everyone to chillax in which case by all means throw me an email!). Enjoy Measure Theory out there! Cheers, Anders Munk-Nielsen (superpronker at hot [albeit not directly sexy] mail d-to-the- o-to-the- tc o m) (( yeah, screw you spambots! )) Indhold 1 First lecture ............................................................................................................................................... 7 1.1 Properties of pavings ............................................................................................................................ 7  1.2 Comment on original example .............................................................................................................. 8  1.3 Sigma algebras ...................................................................................................................................... 8  1.3.1 Examples of σ-algebras ................................................................................................................ 9  1.4 Practical information ............................................................................................................................. 9  1.5 Appendix B: countability (tællelighed)................................................................................................. 9  1.5.1 Counting ....................................................................................................................................... 9  1.5.2 Lemma ........................................................................................................................................ 10  1.5.3 Subsets ........................................................................................................................................ 10  1.5.4 Product sets of countable are countable ..................................................................................... 10  1.5.5 Lemma is countable................................................................................................................ 11  1.5.6 0-1 sequences are not countable (Cantor’s diagonal argument) ................................................. 11  1.5.7 Uncountablly many subsets of ................................................................................................ 11  1.5.8 is uncountable ......................................................................................................................... 12  1.5.9 Theorem: Countable set operations ............................................................................................ 12  2 Goal today: measures .............................................................................................................................. 12  

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