221 Analysis 2, 2008–09
Exercise sheet 4
Due April 16th 2009
Let (X, M, m) be a measure space.
1. Fix a measurable set B ∈ M. Show that the function mB : M → [0, ∞]
given by mB (A) = m(A ∩ B) for A ∈ M, is a measure.
2. (a) Let f : X → [0, ∞] be a nonnegative function and suppose that An ⊆ X
with An ↑ X. Show that if fn = f χAn then fn ↑ f .
(b) Show that if f : R → R is integrable (with respect to Lebesgue measure)
then
Z n
Z
f (x) dx = lim
f (x) dx.
n→∞
−n
[First consider what happens if f (x) ≥ 0 for all x ∈ R and use the
Monotone Convergence Theorem].
(c) Give an example
Z nof a function f : R → R which is not integrable, but
for which lim
f (x) dx = 0.
n→∞
−n
For the next two questions, work with the measure space (R, M, m) where M
denotes the Lebesgue measurable subsets of R and m is Lebesgue measure.
3. (a) Let f : R → [0, ∞), f (x) = x χ[0,1) (x). Sketch the graph of f .
(b) Find a monotone increasing sequence (ϕn )n≥1 of simple nonnegative
measurable functions ϕn : R → [0, ∞) with ϕn ↑ f .
[Hint: look at the proof of Theorem 26].
R
(c) RFind ϕn dm and use the Monotone Convergence Theorem to find
f dm. [Don’t use the Fundamental Theorem of Calculus!]
(d) RLet I = R \ Q. Show that f = f χI almost everywhere, and compute
f dm.
I
4. (a) Let c ∈ R and let A be a Lebesgue
measurable
subset of R. Show that
R
R
χA (x − c) = χA+c (x) and χA+c (x) dx = χA (x) dx.
(b) Show that if ϕ : R → [0, ∞) is a simple nonnegative
measurable
function
R
R
and ϕc : R → [0, ∞), ϕc (x) = ϕ(x − c) then ϕc dm = ϕ dm.
(c) Use the Monotone Convergence Theorem to show that if f : R → [0, ∞]
is a nonnegative measurable
function
and fc : R → [0, ∞] is given by
R
R
fc (x) = f (x − c) then fc dm = f dm.
(d) Show that if f : R → R is an integrable functionR and fc : RR → R is given
by fc (x) = f (x − c) then fc is integrable, and fc dm = f dm.
[You may use the fact that fc is measurable without proof.]
The last two questions are optional.
Let (X, M, m) be a measure space. Recall that f : X → [−∞, ∞] is a
measurable function if
f −1 (α, ∞] = {x ∈ X : f (x) > α}
is a measurable set for every α ∈ R, and that this implies that f −1 (A) is
a measurable set for any Borel set A ⊆ R. It’s also not hard to see that
f −1 ({−∞} ∪ A) is a measurable set if A is a Borel set.
5. Let f, g : X → [−∞, ∞] be two measurable functions.
(a) If q ∈ Q, show that g −1 [−∞, q) and f −1 (q, ∞] ∩ g −1 [−∞, q) are both
measurable sets.
(b) Show that for any x ∈ X with f (x) > g(x), there is a rational number q
with g(x) < q < f (x). Deduce that
[
{x ∈ X : f (x) > g(x)} =
f −1 (q, ∞] ∩ g −1 [−∞, q).
q∈Q
Why does this show that this set is measurable?
(c) Deduce that the set {x ∈ X : f (x) 6= g(x)} is measurable.
6. Let f, g : X → [−∞, ∞] be two measurable functions.
(a) Show that if α ∈ R and q ∈ Q then f −1 (q, ∞] ∩ g −1(α − q, ∞] is a
measurable set.
(b) Suppose that g(x) 6= −∞ for all x ∈ X. The function f + g is then well
defined. Let α ∈ R.
Show that if f (x) + g(x) > α then there is a rational number q with
α − g(x) < q < f (x). Deduce that
[
(f + g)−1(α, ∞] =
f −1 (q, ∞] ∩ g −1(α − q, ∞].
q∈Q
Why does this show that this set is measurable?
(c) Deduce that f + g is a measurable function.
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