Exercise Sheet 4, ST213
1) Let (Ω, F, P) be a measure space and X a non-negative Borel function. Show that
Z
Q(A) = 1A X dP,
A∈F
defines a measure on (Ω, F).
2) Let (Ω, F, P) be a measure space and Q be absolutely continuous with respect to P (we write Q P)
with density D = dQ\dP. Let further X be a Borel real-valued function on Ω. Show that the integral of
X with respect to Q is defined if and only if the integral of X D with respect to P is defined, in which
case they are equal, i.e.:
Z
Z
X dQ =
X D dP.
Show this first for X a simple Borel function.
3) Let (Ω, F, P) be a measure space, X and Y two Borel real-valued functions.
R
R
(a) Show that if X is integrable and X = Y a.e. then Y is integrable and XdP = Y dP.
(b) Let Q be a measure on (Ω, F, P) such that Q P, and A ∈ F. Show that if P(A) = 0 then Q(A) = 0,
but (in general) not vice versa.
4) Let Pn , n = 1, 2, . . ., be a sequence of measures on a measurable space (Ω, F).
P∞
a) Show that Q = n=1 Pn is a measure on (Ω, F).
b) Let X be a non-negative Borel function. Show that
Z
X dQ =
∞ Z
X
X dPn .
n=1
Remark that Q(A) might be infinite for some A ∈ F. In such a case the definition of the integral implies
that
Z
(a1A ) dQ = a · Q(A) = a · ∞ for a ≥ 0
and the only meaningful definition of this a · ∞ in integration theory is a · ∞ = ∞ if a > 0 but a · ∞ = 0
if a = 0. Furthermore a + ∞ = ∞ for a ≥ 0 including a = ∞. NEVER calculate anything like +∞ − ∞
without extra consideration!
5) Let (Ω, F) be a measurable space.
Pn
(a) Let X = i=1 ai 1Ai , Ai ∈ F, ai ∈ [0, +∞), i = 1, . . . , n, n ∈ N. Show that X is a (simple, of course)
Borel function.
(b) Let X : Ω → R be nonnegative, Borel measurable, i ∈ N. Define:
 i

i2
X
k
−
1
i
i  + i1{X≥i} .
Xi = 
1
2i {(k−1)/2 ≤X<k/2 }
k=1
Show that Xi is a simple Borel function and that Xi ↑ X as i → ∞, uniformly on every set on which
X is bounded.
6) Let Ω be any set and ω ∈ Ω. Define δω : P(Ω) → [0, 1] by letting δωP
(A) = 1A (ω). Realise that δω is a
∞
probability measure on P(Ω), the Dirac measure at ω. Conclude that n=1 δn is a measure, the so-called
counting measure, on P(N).
1