Departments of Mathematics
Fall 2015
Montana State University
Prof. Kevin Wildrick
Measure Theory
Review Set 2
1. Define
• a σ-algebra
• a measure space
• a measurable set in a measure space
• a measurable function between two measure spaces
• a measurable function from a measure space to the extended real line
• a simple function
• the standard representation of a simple function
• the integral of a measurable simple function on a measure space
• the integral of a non-negative measurable function on a measure space
• the integral of a measurable function on a measure space
2. State and prove
• the fundamental approximation lemma
• the monotone convergence theorem
• Fatou’s lemma
• the “linearity” of the integral
3. Let (X, Σ, µ) be a measure space, and let {En }n∈N be a sequence of measurable subsets of
X. Show that
S
P
• µ( n∈N En ) ≤ n∈N µ(En ),
• if E1 ⊆ E2 , then µ(E1 ) ≤ µ(E2 ).
4. Let f : R → R be a continuous function, and let [a, b] be a compact interval in R. For each
integer N ≥ 1 and each i ∈ {1, . . . , N }, define
(i − 1)(b − a)
i(b − a)
N
Ii = a +
,a +
N
N
and
N
aN
.
i = min f (x) : x ∈ Ii
• Show that
φN =
N
X
i
i=1
is a measurable simple function.
aN
i χI N
• Show that
Z
Z
lim
N →∞ R
b
f (x) dx,
φN dm =
a
where the right hand side above denotes the Riemann integral of f from a to b.
• Show that {φN }N ≥1 is an increasing sequence and that
lim φN = f.
N →∞
• Show that
Z
b
Z
f dm.
f (x) dx =
[a,b]
a
5. Let (X, Σ, µ) be a measure space, and suppose that E ∈ Σ satisfies µ(E) = 0. Show that
for any measurable function f : (X, Σ, µ) → R,
Z
f dµ = 0.
E
(don’t forget to include a proof that the above integral is defined!)
6. Consider the measure space (N, P(N), µ), where every set is measurable and
µ(E) = card E
is the “counting measure”. Given a seqeuence {an }n∈N ⊆ [0, ∞], find a measurable function
f : N → [0, ∞] such that
Z
X
f dµ =
an .
N
What happens if we allow {an }n∈N ⊆ R?
n∈N