HOMEWORK ASSIGNMENT #5, MATH 5225 (DUE BY
MON. OCT. 20)
(1) Show that there are f∈L1(R) and a sequence {fn}with fn∈L1(R)
such that
kf−fnk1→0,
but fn(x)→f(x) for no x.
Hint: Let fn=χInwith Inappropriately chosen sequence of in-
tervals with l(In)→0.
(2) Suppose Fis a closed set in R, whose complement ha s finite measure,
and let δ(x) denote the distance from xto F, that is
δ(x) = d(x, F ) = inf{|x−y|:y∈F}.
Consider
I(x) = ZR
δ(y)
|x−y|2dy .
(a) Prove that δis continuous, by showing that it satisfies the
Lipshitz condition
|δ(x)−δ(y)|≤|x−y|.
(b) Show that I(x) = ∞for each x /∈F.
(c) Show that I(x)<∞for a.e. x∈F.
Hint: For the later part investigate RFI(x)dx, assuming that it is
allowed to switch the order of the integrals in the corresponding dou-
ble integral (Tonelli’s theorem that we will prove later).
(3) Prove that if fis integrable on R, and fis not identically zero, then
f∗(x)≥c
|x|,for some c > 0 and all |x| ≥ 1.
Conclude that f∗/∈L1(R). Then show that the weak type estimate
m({f∗> α})≤c
α
1
