Summary of contents: sigma-algebras; De Morgan's Laws; measurable spaces and measurable subsets; extended real line; measures and measure spaces; example: N and the counting measure; proof of basic properties of measures; proof that measures are continuous from above and from below; proof that measures are countably sub-additive; finite measures; sigma-algebras generated by a collection of subsets; Borel sigma-algebra on a topological space; example: the Borel sigma-algebra of R; null sets and almost everywhere; complete and translation-invariant measures; Lebesgue measure on R^d; proof that the Lebesgue measure is finite; measurable maps; composition of measurable maps; pointwise converging sequences of measurable maps; push-forward of a measure.
Full lecture notes (work in progress): click here
Lecture Notes for this lecture:
No comments :
Post a Comment