Probability and Stochastic Processes

(1) Let Ω = R, F = {A ⊂ Ω : A is countable or Ac
is countable}. Define a function
P : F → [0, 1] as P(A) = 0 if A is countable and P(A) = 1 if A is uncountable. Show that
(Ω, F, P) is a probability space.
(2) Let (Fn)n∈N be a sequence of σ-algebras such that F1 ⊂ F2 ⊂ F3 ⊂ · · · . Is ∪n∈NFn a
σ-algebra in general? If yes, prove it. If no, give a counter example.
(3) Show that the Borel σ-algebra on R
d
(the σ-algebra generated by all open sets in R
d
)
is also generated by the following collection of subsets
S =
n
(a1, b1] × (a2, b2] × · · · × (ad, bd] ∈ R
d
: ai < bi
for i = 1, 2, …, do
(4) Suppose X and Y are random variables on (Ω, F, P) and let A ∈ F. Let Z : Ω → R
be
Z(ω) = (
X(w) if ω ∈ A
Y (ω) if ω /∈ A
.
Show that Z is also a random variable on (Ω, F, P).
(5) Read the proof of Theorem 1.1.4 in the textbook.

Sample Solution

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