User:Caliburn/s/mt/CohnRoad

Misc

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

Chapter 1

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

Chapter 4

4.1 Signed and Complex Measures

done up to end of 4.1.7