[SOLVED] Math154 homework 2 -probability spaces p0

MATH154
Homework 2
Probability spaces
Problem 2.1: Verify the following properties from the axioms. a) P[∅] = 0.
b) A ⊂ B ⇒ P[A] ≤ P[B].
c) P[Sn An] ≤ Pn P[An].
d) P[Ac] = 1 − P[A].
e) 0 ≤ P[A] ≤ 1.
f) A1 ⊂ A2,⊂ ··· with An ∈ A then P[
Problem 2.2: Let Ω be a set. Let A be the set of countable or cocountable subsets of Ω.
a) Verify that A satisfies all the ring axioms of Boolean algebra.
b) Verify that A is a π-system.
c) Verify that A is a λ-system.
e) Verify that A is the smallest σ algebra containing the cofinite topology.
Problem 2.3: Let Ω = [0,1]2. Let I = {[a,b) × [c,d)} denote the set of all left-bottom closed right-top open rectangles. a) Verify that this is a π-system.
b) Verify that P[a,b) × [c,d)] = (d − c)(b − a) is a probability measure on this π system.
c) Why can the measure P be extended to the smallest σ-algebra containing I?
d) Under which conditions are two elements in I independent? Probability Theory
Problem 2.4: Verify the following properties. The first four are known as Keynes postulates, the fifth is called Bayes Theorem.
1) P[A|B] ≥ 0.
2) P[A|A] = 1.
3) P[A|B] + P[Ac|B] = 1.
4)
Problem 2.5: Prove the ΠΣΛ sorority theorem in the text. It states ”The smallest λsystem A containing a π-system I is the smallest σ algebra containing I.”

Figure 1. To the left an example of a ΠΣΛ chapter (in this case Oxford MS). To the right, a brooch from BU in the shape of a Marguerite daisy (or A∩B when intersecting two sets in a Venn Diagram) also in the order of the mathematical order ΠΛΣ: to check that we have a σ-algebra, we have to check it is a π-system and a λsystem.