Mathematics 2224: Lebesgue integral
Homework exercise sheet 4
Due 3:50pm, Wednesday 23rd March 2011
1. Let f : R → R and g : R → R be two Lebesgue measurable functions, and let E
be a Lebesgue measurable subset of R. Let h = f χE + gχE c . Give a short proof
that h is Lebesgue measurable, and explain why
(
f (x) if x ∈ E,
h(x) =
g(x) if x 6∈ E.
2. Show that if f, g : R → R are two Lebesgue measurable functions, then min(f, g)
is also Lebesgue measurable. What happens if f, g : R → [−∞, ∞]?
3. (a) Give an example of a set E ⊆ R so that χE is not a Lebesgue measurable
function.
(b) Give an example of a function f : R → R which is not a simple function, and
is not a Lebesgue measurable function.
4. Let f, g : R → R.
(a) Show that if E ⊆ R then (f ◦ g)−1 (E) = g −1 (f −1 (E)).
[Note: this question is about preimages, not about inverse functions.]
(b) Show that if f is continuous and g is Lebesgue measurable, then f ◦ g is
Lebesgue measurable.
5. The only functions f : R → R which are both simple and continuous are the
constant functions. Why?
6. Let f = χ[−1,1] + 3χ[−2,2] + χ[0,∞) − χ(3,∞) . Show that f is a simple, non-negative
Lebesgue measurable function, and compute its standard representation and its
Lebesgue integral.
7. Show that if f, g : R → [−∞, ∞] are two Lebesgue measurable functions, then
{x ∈ R : f (x) < g(x)} ∈ L and {x ∈ R : f (x) ≤ g(x)} ∈ L.
8. Let f : R → R be a simple function. Prove that, up to reordering the sum, the
standard representation of f is unique.
P
P
[That is: if f = ni=1 ai χEi = m
j=1 bj χFj are two standard representations of f ,
show that n = m and {a1 , . . . , an } = {b1 , . . . , bn }, and that ai = bj =⇒ Ei = Fj
for 1 ≤ i, j ≤ n.]