221 Analysis 2, 2008–09
Exercise sheet 3
Due March 9th 2009 March 30th 2009
1. For each of the following collections S of subsets of R, determine whether
or not σ(S) is equal to the collection of Borel subsets of R. Justify your
answers.
(a) S = P(R)
(b) S = {the closed subsets of R}
(c) S = {(a, b) : a, b ∈ R}
(d) S = {(−a, a) : a ≥ 0}
2. Let (X, M, m) be a measure space. Show that if f : X → [−∞, ∞] is a
measurable function then A = {A ⊆ R : f −1 (A) ∈ M} is a σ-algebra which
contains every Borel set.
[As usual, f −1 (A) = {x ∈ X : f (x) ∈ A}.]
3. Let (X, M, m) be a measure space and for A ⊆ X, let χA : X → [0, ∞) be
the characteristic function of the set A.
(a) Show that χA is measurable if and only if A ∈ M.
(b) Show that χA∁ = 1 −χA . [This means: χA∁ (x) = 1 −χA (x) for all x ∈ X.]
(c) Show that if A, B ⊆ X then χA χB = χA∩B .
(d) Find an expression for χA∪B in terms of χA and χB .
4. Let X be a non-empty set and let ϕ : X → R be a simple function; that is,
ϕ(X) is a finite set.
(a) Show that if ϕ(X) = {a1 , . . . , an } where ai 6= aj for i 6= j and if we define
Ai = ϕ−1 (ai ) then A1 ∪ · · · ∪ An is a partition of X into (non-empty) subsets,
and
(∗)
ϕ = a1 χA1 + · · · + an χAn .
(b) Show that if m ≥ 1 and b1 , . . . , bm are any distinct real numbers and
B1 ∪ · · · ∪ Bm is a partition of X into (non-empty) subsets with
ϕ = b1 χB1 + · · · + bm χBm ,
then n = m and there is a permutation π ∈ Sn with bi = aπ(i) and Bi = Aπ(i)
for each i = 1, 2, . . . , n.
(c) If (X, M, m) is a measure space, show that ϕ is a measurable function if
and only if ϕ can be written in the form (∗) for some A1 , A2 , . . . , An in M.
5. Let (X, M, m) be a measure space and let A ∈ M. Show that for every
measurable function f : X → [−∞, ∞], the function f χA : X → [−∞, ∞],
x 7→ f (x)χA (x) is also measurable.