Homework - I (Probability tutorial)
Algorithmical and Statistical Modeling, Fall 2012.
Due: Oct. 2, Tuesday (to be submitted in the Lecture)
1. Exhibit two real valued sequences {xn } and {yn } such that
lim sup xn + lim sup yn > lim sup(xn + yn ).
n→∞
n→∞
(5 points)
n→∞
2. If A1 , A2 , . . . are elements belonging to a sigma-algebra F then show that the set
∞ ∪
∞
∩
Ak
n=1 k=n
belongs to F. (10 points)
3. Let Ω = {a, b, c}. Exhibit two sigma algebras on Ω such that their union is not a sigmaalgebra. (15 points)
4. Let µ be a measure on a sigma-algebra F.
(a) Let A1 ⊂ A2 ⊂ A3 ⊂ · · ·.
If A =
(b) Let A1 ⊃ A2 ⊃ A3 ⊃ · · ·.
(30 points)
If A =
∪∞
i=1 Ai
show that limn→∞ µ(An ) = µ(A).
i=1 Ai
show that limn→∞ µ(An ) = µ(A).
∩∞
5. Let Ω = {a, b, c, d}. Let S = {1, 2, 3}. Construct (nontrivial) sigma algebras F and B
on Ω and S respectively such that f : Ω → S is measurable. Also construct a g : Ω → S
which is not measurable (here, a sigma-algebra is trivial if it has just the null set and
the whole space). (15 points)
6. Let (Ω1 , F1 ), (Ω2 , F2 ) and (Ω3 , F3 ) be measurable spaces. If f : Ω1 → Ω2 and f : Ω2 →
Ω3 are respectively (F1 , F2 ) and (F2 , F3 ) measurable functions, prove that f2 ◦ f1 :
Ω1 → Ω3 , where f2 ◦ f1 (x) := f2 (f1 (x)) is (F1 , F3 ) measurable. (10 points)
7. Let (Ωi , Fi ), i = 1, 2 be measurable spaces and let f : Ω1 → Ω2 be a measurable
function. Then for any measure µ on (Ωi , Fi ), verify the function on Ω2 given by
λ(A) := µ(f −1 (A)),
A ∈ F2
is a measure on F2 . The measure λ is called the induced measure (or the measure
induced by f ). Show that if µ is a probability measure then so is λ. (15 points)