Probability Spaces 2

Question 1

If A <= P(omega), then what is sigma(A)

Answer 1

If A <= P(omega), then sigma(A) is sigma field generated by A, smallest sigma field on omega that contains A

Question 2

What is an example of sigma(A)

Answer 2

An example of sigma(A) is:
Sigma(E) is (empty set, omega, E,E^C)
Take (E1,E2) = A, then sigma(A) = (empty set, omega, E1,E2,E1^C,E2^C, E1 U E2, E1^C N E2^C, E1^C U E2^C, E1 U E2^C, E1^C U E2…) no need to list

Question 3

What is sigma(A) (intersection)

Answer 3

Sigma(A) (intersection) is intersection of sigma fields containing A

Question 4

What is Borel sigma-algebra

Answer 4

Borel sigma-algebra is:
Let omega = R, consider G = (-inf,x] ; x E R) <= P(omega) and generate sigma(G)
This sigma-field is Borel sigma-algebra B(R)/B
Doesn’t contain everything, contains finite intervals