2.9 Exercises 41
proofs and the book is comprehensive. The factorization lemma (Lemma 2.5) is
stated in the book by Kallenberg [2002] (Lemma 1.13 there). Kallenberg calls this
lemma the “functional representation” lemma and attributes it to Joseph Doob.
Theorem 2.4 is a Corollary of Caratheodory’s extension theorem, which says
that probability measures defined on semirings of sets have a unique extension
to the generatedσ-algebra. The remaining results can be found in either of the
three books mentioned above. Finally, for something older and less technical we
recommend the philosophical essays on probability by Pierre Laplace, which was
recently reprinted [Laplace, 2012].
2.9 Exercises
2.1(Composing random elements) Show that iffisF/G-measurable andg
isG/H-measurable for sigma algebrasF,GandHover appropriate spaces then
their composition,g◦f(defined the usual way: (g◦f)(ω) =g(f(ω)),ω∈Ω), is
F/H-measurable.
2.2 LetX 1 ,...,Xnbe random variables on (Ω,F). Prove that (X 1 ,...,Xn) is
a random vector.
2.3(Random-variable inducedσ-algebra) LetUbe an arbitrary set and
(V,Σ) a measurable space andX:U → Van arbitrary function. Show that
ΣX={X−^1 (A) :A∈Σ}is aσ-algebra overU.
2.4 Let (Ω,F) be a measurable space andA⊆Ω andF|A={A∩B:B∈F}.
(a) Show that (A,F|A) is a measurable space.
(b) Show that ifA∈F, thenF|A={B:B∈F,B⊆A}.
2.5 LetG ⊆ 2 Ωbe a nonempty collection of sets and defineσ(G) as the smallest
σ-algebra that containsG. By ‘smallest’ we mean thatF ∈ 2 Ωis smaller than
F′∈ 2 ΩifF ⊂F′.
(a)Show thatσ(G) exists and contains exactly those setsAthat are in every
σ-algebra that containsG.
(b)Suppose (Ω′,F) is a measurable space andX: Ω′→Ω beF/G-measurable.
Show thatXis alsoF/σ(G)-measurable. (We often use this result to simplify
the job of checking whether a random variable satisfies some measurability
property).
(c) Prove that ifA∈FwhereFis aσ-algebra thenI{A}isF-measurable.
2.6(Knowledge andσ-algebras: A pathological example) In the
context of Lemma 2.5, show an example whereY=Xand yetYis notσ(X)
measurable.
