Measure Theory (08/15) - The sigma-algebra generated by a collection of subsets (2/2)

Properties of the sigma-algebra on a set X generated by a collection of subsets.
Methods for showing inclusion or equality of sigma-algebras generated by two different collections of subsets.
The Borel sets in the real line: definition and examples.

Notes:
- Just after time 10:00 I said "contained in C_2" when I meant "generated by C_2"
- Sometimes I said "sigma algebra generated by" without saying "on X" (or "on R"). This is OK as long as there is no ambiguity about which set X you are working on.
- Between times 20:18 and 21:20 I wrote sigma_X instead of sigma_R a few times. (I was thinking of R as the whole space here.)

For more information and links to further resources, see my blog page
https://explainingmaths.wordpress.com/measure-theory/