1. Here we consider convolution and Lp-functions with respect to the Lebesgue measure m on Rd .

Let ϕ 1  : Rd  → [0, ∞) be a smooth non-negative function which is supported on the unit ball B1 (0) and satisfies lRd  ϕ 1  = 1. For σ > 0 define ϕσ  : Rd  → [0, ∞) by ϕσ (x) = σ −n ϕ 1 (x/σ) so that ϕσ  is supported on the ball Bσ (0) and satisfies lRd  ϕσ  = 1.

Let 1 ≤ p < ∞ and consider any function f ∈ Lp (Rd ). For each σ > 0, let

\

for each x ∈ Rd .

(a) Use H¨older’s inequality to show that ∥fσ ∥Lp   ≤ ∥f∥Lp   for all f ∈ Lp (Rd ).

(b) Show that for each f ∈ Lp (Rd )

lim  ∥fσ  − f∥Lp   = 0.

σ →0+

2. Let X be a set and µ∗ be an outer measure on X . Let µ be the measure defined by restricting µ∗ to the sets which are Carath´eodory measurable with respect to µ∗ .

Arguing as we did in the proof of the Extension Theorem, we can define an outer measure µ+  on X by

µ+ (E) = inf { µ(Ej ) : E ⊆  Ej  and Ej  are Carath´eodory measurable w.r.t. µ∗ }

for each set E ⊆ X .

(a) Show that for each set E ⊆ X we have µ∗ (E) ≤ µ+ (E).

(b) For a given set E, show that µ∗ (E) = µ+ (E) if and only if there is a µ∗-measurable set A such that E ⊆ A and µ(A) = µ∗ (E).

3. Consider the measurable space ([0, 1], B), where B is the collection of all Borel subsets of [0, 1]. m : B → [0, ∞] be the Lebesgue measure and let c : B → [0, ∞] be the counting measure, i.e. c(E) equals the number of elements of E .

Let m × c be the product measure on [0, 1] × [0, 1] as constructed in lecture.  In particular, let (m × c)∗ be the outer measure on [0, 1] × [0, 1] defined by

(m × c)∗ (E) = inf { m(Aj )c(Bj ) : E ⊆ Aj  × Bj  and Aj ,Bj  ∈ B} ,