2) Probability Spaces

Question 1

What is a Probability Space

Answer 1

(Ω, F, P) satisfying the following properties:
1. Ω is a nonempty set
2. F is a σ-algebra of subsets of Ω
3. P : F → R is a probability measure

Question 2

What is σ-algebra

Answer 2

Question 3

What is a Probability Measure

Answer 3

If property 9 is not met it is called a measure

Question 4

What are the generators of σ-Algebra

Answer 4

If C ⊆ 2^Ω then the smallest σ-algebra on Ω containing C is denoted by σ(C)

Question 5

What is the Borel σ-algebra on R

Answer 5

The Borel σ-algebra, B(R), is the smallest σ-algebra on R containing all open sets. It is generated by the collection of open sets and is denoted as B(R)=σ(C) where C is the set of open subsets of R

Question 6

What are the properties of a Probability Space related to Set Operations

Answer 6

Question 7

What are the properties of a Probability Space related to Limits and Continuity

Answer 7

Question 8

What is lim sup and lim inf

Answer 8

Question 9

What are the inequalities related to lim sup and lim inf

Answer 9

Question 10

What is the First Borel-Contelli lemma

Answer 10

Question 11

What is the Lebesgue measure

Answer 11

Question 12

What is a π-system of sets

Answer 12

Question 13

What is the uniquness of measures

Answer 13