PROBLEM SET 3, 18.155
DUE MIDNIGHT FRIDAY 30 SEPTEMBER, 2011
(1) Consider the space L1loc (Rn ) of locally Lebesgue integrable functions, defined by
(
f (x) if |x| < R ∀ R > 0
1
n
f ∈ Lloc (R ) ⇐⇒ fR (x) =
∈ L1 (Rn ),
0
if |x| ≥ R
L1loc (Rn ) = L1loc (Rn )/a.e.
Show that the kfN kL1 , N ∈ N, form a countable set of seminorms with respect to which L1loc (Rn ) is complete, i.e. it is complete with respect to the metric
X
kfN − gN kL1
d(f, g) =
2−N
.
1
+
kf
N − gN kL1
N
(2) Prove the Monotonicity Lemma. If fj ∈ L1 (Rn )Rare real-valued,
fj (x) is monotonic increasing for each x and fj is bounded
1
n
then there
R exists
R f ∈ L (R ) such that fj (x) → f (x) a.e. and
limj→∞ fj → f.
(3) We say that U ⊂ Rn is Lebesgue measurable if the characteristic
function χU (x) = 1 if x ∈ U, χU (x) = 0 otherwise is an element
of L1loc (Rn ). It is of finite measure if χU ∈ L1 (Rn ) and the
measure (volume) of U is the integral. Show that the Lebesgue
measurable sets form a σ-algebra – the collection of measurable
sets is closed under complements (in Rn ) and countable unions
(and hence countable intersections).
(4) Give an example of a continuous function u ∈ C 0 (Rn ) which is
not polynomially bounded yet is such that
Z
n
∞
(1)
Cc (R ) 3 φ 7−→ u(x)φ(x)
extends by continuity to an element of S 0 (Rn ). Hint: The simplest example I know is the derivative of a function like exp(iex ).
(5) Show that the Fourier transform defines a continuous linear map
F : L1 (Rn ) −→ C00 (Rn )
= {u : Rn −→ C; u is continuous and lim sup |u(x)| = 0}.
R→∞ |x|≥R
1
2
PROBLEM SET 3, 18.155 DUE MIDNIGHT FRIDAY 30 SEPTEMBER, 2011
Hint: Check it for a dense subspace, such as S(Rn ), and then
get an estimate which proves continuity.