1. Find the Lebesgue measure µ0 of µ0 ([1/2, 2/3]), µ0 ({1/n : n = 1, 2, . . . }), µ0 (N), where N is set of natural numbers.

2. Let Ω = {1, 2, 3, 4, 5, 6} with uniform probability. Show that if A ⊂ Ω and B ⊂ Ω are independent and A has four elements, then B must have 0, 3, or 6 elements.

3. Let Ω = [0, 1] with Borel sets and Lebesgue measure. Find P(X ∈ [0, 1/2)) for X(ω) = ω 2 .

4. Show if X is a constant function, then it is a random variable with respect to any σ-algebra.

5. Let Ω = {1, 2, 3, 4} and F = {∅ , Ω , {1}, {2, 3, 4}}. Is X(ω) = 1 + ω a random variable with respect to the σ algebra F?  If not, give an example of a non-constant function which is.

6.  (Shreve, Exercise 2.4)  “Toss a coin repeatedly.  Assume the probability of head on each toss is , as is the probability of tail.  Let Xj   = 1 if the jth toss results in a head and Xj   = −1 if the jth toss results in a tail.  Consider the stochastic process M0 ,M1 ,M2 , . . . defined by M0  = 0 and

n

工

j=1

n ≥ 1.