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 I’ll 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 you’re building some weapon of mass destruction … unless you’d 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- “t” c o m) (( yeah, screw you spambots! )) Indhold 1 First lecture ............................................................................................................................................... 7 1.1Properties of pavings ............................................................................................................................ 7 1.2Comment on original example .............................................................................................................. 8 1.3Sigma algebras ...................................................................................................................................... 8 1.3.1Examples of σ-algebras ................................................................................................................ 9 1.4Practical information ............................................................................................................................. 9 1.5Appendix B: countability (tællelighed)................................................................................................. 9 1.5.1Counting ....................................................................................................................................... 9 1.5.2Lemma ........................................................................................................................................ 10 1.5.3Subsets ........................................................................................................................................ 10 1.5.4Product sets of countable are countable ..................................................................................... 10 1.5.5Lemma ℚ is countable................................................................................................................ 11 1.5.60-1 sequences are not countable (Cantor’s diagonal argument) ................................................. 11 1.5.7Uncountablly many subsets ofℕ ................................................................................................ 11 1.5.8ℝ is uncountable ......................................................................................................................... 12 1.5.9Theorem: Countable set operations ............................................................................................ 12 2Goal today: measures .............................................................................................................................. 12