Mathematics 2224: Lebesgue integral
Homework exercise sheet 5
Due 3:50pm, Wednesday 6th April 2011
1. For each j ∈ N, let fj : R → [0, ∞] be a non-negative Lebesgue measurable function. Use the
Monotone Convergence Theorem to show that
Z X
∞
∞ Z
X
fj dµ =
fj dµ.
R j=1
Here
∞
X
j=1
R
fj is the function f : R → [0, ∞] given by f (x) =
j=1
∞
X
fj (x) for x ∈ R.
j=1
2. Find a Lebesgue integrable function f so that f 2 is not Lebesgue integrable.
[This shows that although the Lebesgue integrable functions form a vector space over R by
Theorem 25, they don’t form a ring.]
3. Let f : R → [0, ∞] be a non-negative Lebesgue measurable function, and let E =
where E1 , E2 , E3 , · · · ∈ L.
Z
∞ Z
X
(a) If E1 , E2 , . . . are disjoint, show that
f dµ =
f dµ.
E
j=1
Z
S∞
j=1
Ej
Ej
Z
(b) If E1 ⊆ E2 ⊆ . . . , show that
f dµ = sup
f dµ.
j∈N
E
Ej
4. If f : R → [0, ∞] is any non-negative Lebesgue measurable function, show that the function
Z
λf : L → [0, ∞], λf (E) =
f dµ
E
is a measure. Which are the functions f so that λf is a probability measure?
5. Consider the equation
Z
Z
f dµ.
f dµ = lim
n→∞
R
(∗)
[−n,n]
(a) Show that (∗) holds if f : R → [0, ∞] is any non-negative Lebesgue measurable function.
(b) Show that (∗) holds if f : R → [−∞, ∞] is any Lebesgue integrable function.
(c) Describe both sides of (∗) for f = χ(−∞,0] − χ(0,∞) .
6. Let f (x) = max(x, 0).
Z
f dµ = ∞.
(a) Give a (very short) proof that
R
(b) Write down a monotone increasing sequence of simple non-negative Lebesgue measurable
functions which converges pointwise to χ[0,1] f . Then use this sequence with the Monotone
Convergence Theorem to prove that
Z
1
f dµ = .
2
[0,1]