Exercise Sheet 5, ST213
1) Let (Ω, F, P) be a probability space and G ⊆ F a sub-σ-algebra generated by a countable partition of
Ω.
(a) If X1 , X2 : Ω → R are integrable and b1 , b2 ∈ R then a.s.
E[b1 X1 + b2 X2 | G] = b1 E[X1 | G] + b2 E[X2 | G].
(b) If X : Ω → R is integrable then
E(E[X | G]) = E(X).
2) Let Ω = [0, 1], F = {B ∩ [0, 1] : B ∈ B(R)} = P([0, 1]) ∩ B(R) and let P be the restriction of the
Lebesgue measure ` on (R, B(R)) to F. Realise that P is a probability measure. Define Xn : Ω → R
by letting
2
n : 0 ≤ ω ≤ 1/n,
Xn (ω) =
0 : ω > 1/n,
and show that, as n → ∞, Xn converges to zero in probability and P-a.s. but does not converge in
mean to any random variable at all.
3) Let Ω = [0, 1), F = P([0, 1)) ∩ B(R) and let P be the restriction of the Lebesgue measure ` on
(R, B(R)) to F. Again realise that P is a probability measure. For n ∈ N, let k ∈ N ∪ {0} and then
m ∈ {0, . . . , 2k − 1} be the unique natural numbers such that n = 2k + m, and define Xn = 1[ m , m+1 ) .
2k
2k
Show that, as n → ∞, Xn converges to zero in probability as well as in mean but it does not converge
P-a.s. to any random variable at all (in fact, show that for every ω ∈ Ω, Xn (ω) does not converge, as
n → ∞).
4) Show that a differentiable function F : R → R has a Borel measurable derivative. Next, let F : R → R
be a nondecreasing and continuously differentiable function. Remark that F 0 is then Borel measurable
and nonnegative. Let furthermore µ be the measure on B(R) for which µ((a, b]) = F (b) − F (a), for all
−∞ < a ≤ b < +∞, and let l be the Lebesgue measure on B(R). Show that µ l and dµ/dl = F 0
a.e.-l.
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