Ch1

Question 1

Sigma Algebra

Answer 1

1) empty set in F; 2) if A\in F, A^c\in F; 3) countable union of sets is in F

Question 2

Measure Space

Answer 2

Pair (\Omega, F) where F is the sigma algebra.

Question 3

sigma algebra generated by A

Answer 3

Let A\subseteq \Omega, the smallest sigma algebra containing A

Question 4

Borel sigma-algebra

Answer 4

Omega needs to be endowed with a topology. Then the borel sigma algebra is the sigma algebra generated by open sub sets of Omega.

Question 5

Borel Sigma Algebra on R

Answer 5

Also generated by all open, half-open, closed, and (-\infty,x) intervals.

Question 6

Restrictions on Algebras

Answer 6

Let (\Omega, F) be a measurable space. Let \Omega’\in F, then the restriction F’={A\cap \Omega’:A\in F} is a sigma algebra, and generated by A’={A\cap\Omega’:A\in scriptA} if scriptA generated F.

Question 7

Measures

Answer 7

Measures are defined on measurable spaces, and have the following two properties: 1) \mu(\emptyset)=0; 2) countable additivity (sum of disjoint equals measure of union)

Question 8

Pi system

Answer 8

Let \Omega be a set. P is a pi-system if it closed under finite intersection.

Question 9

Lambda system

Answer 9

Let \Omega be a set. L is a lambda system if L is a collection of \Omega subsets and satisfies: 1) \emptyset \in L; 2) complements in L; 3) union of countable disjoint is in set.

Question 10

Dynkin’s Pi-Lambda theorem

Answer 10

Let \Omega be a set. If there is a P pi-system contained within a L lambda-system, then the sigma algebra generated by P is within L.

Question 11

When are measures unique?

Answer 11

if \mu_1 and \mu_2 are both measures that agree on P, and there is an increasing sequence of sets in P that converge to \Omega, then they also agree on \sigma(P).

Question 12

Outer Measure vs. Measure

Answer 12

1) Defined on power set; 2) \phi(union) \leq sum \phi(A-i)

Question 13

B is phi measurable if

Answer 13

phi(A)=phi(A\cap B)+phi(A\cap B^c)

Question 14

Sigma Algebra and Phi Measurability

Answer 14

Let Omega be a set, and F be the set of phi-measurable sets in Omega. Then F is a sigma algebra, and phi is the measure on (Omega, F).

Question 15

Caratheodory’s Extension Theorem

Answer 15

Let A be an algebra on Omega, and let there be a measure on A. Then mu has an extension on the sigma algebra generated by the algebra, and moreover, if mu is sigma finite, the measure is unique.
In particular, we define the extension as the infimum of the cover over A, where A_i is taken from the algebra which approximates A.

Question 16

Lebesgue Measure

Answer 16

For any half open interval (a,b], define the Lebesgue measure in the usual sense. It has a unique extension on the Borel Sigma algebra on R, which is generated by the collection of scriptA={(a,b]:a<b></b>