User:Caliburn/s/mt/CohnRoad

Misc

 * Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum - Proof 2 based on atoms

1.1 Algebras and $\sigma$-algebras
Incomplete


 * Definition:Measurable Set
 * Examples 1.1
 * (a) Power Set is Sigma-Algebra
 * (b) Definition:Trivial Sigma-Algebra
 * (b) Trivial Sigma-Algebra is Sigma-Algebra
 * (c) Collection of Infinite Subsets of Finite Set is not Algebra of Sets
 * (d) Collection of Finite and Cofinite Subsets of Infinite Set is Algebra of Sets but not Sigma-Algebra - may need choice, see "Amorphous set"
 * (e) Collection of Countable Subsets of Uncountable Set is not Algebra of Sets
 * (f) Collection of Co-Countable Subsets of Set is Sigma-Algebra
 * (g) no idea how to name this
 * Proposition 1.2 - Intersection of Sigma-Algebras
 * Union of Sigma-Algebras may not be Sigma-Algebra
 * Existence and Uniqueness of Generated Sigma-Algebra

1.2 Measures
Incomplete


 * Definition:Countably Additive Function
 * Definition:Measure (Measure Theory)
 * Definition:Finitely Additive Function
 * Definition:Finitely Additive Measure
 * Measure is Finitely Additive Function
 * Definition:Measure Space
 * Definition:Measurable Space
 * clarification of vocabulary for Definition:Measure (Measure Theory)
 * Examples 1.2.1
 * Definition:Counting Measure
 * Definition:Dirac Measure
 * mention of Lebesgue measure
 * Definition:Infinite Measure
 * define $\mu : \Sigma \to [0, \infty]$ by $\map \mu A = 1$ if $A \ne \emptyset$ and $\map \mu A = 0$ if $A = \emptyset$. Then $\mu$ is not a finitely additive measure

4.1 Signed and Complex Measures
done up to end of 4.1.7