Qualifying Examination in Real Variables, August 2010
(1) (a) Give an example of a sequence (fn) in L1[0,1] such that
limn→∞ kfnkL1= 0, but (fn) does not converge to 0 almost
everywhere.
(b) Show that if a sequence (fk) in L1[0,1] satisfies kfkkL1≤
2−kfor k≥1, then fk→0 almost everywhere.
(2) Let Ebe a subset of [0,1] with positive outer Lebesgue measure,
i.e. m∗(E)>0. Show that for each α∈(0,1) there is an
interval I⊂[0,1] so that
m∗(E∩I)≥αlength(I).
(3) Let Xbe a Banach space and let (xn) be a sequence from X
that converges weakly to 0. Prove that the sequence (kxnk) is
bounded.
(4) (a) Let (fn) be a bounded sequence in C[0,1]. Prove that
(fn) converges weakly to 0 ⇐⇒ (fn) converges pointwise to 0.
(b) Assume that (fn)⊂C[0,1] converges in the weak topology.
Show that fnis norm convergent in L1[0,1].
[For part (b) you may use problem (3).]
(5) Let f:R→Rbe a measurable function such that for some
C > 0
m{x:|f(x)| ≥ λ} ≤ Cλ−2,for all λ > 0.
Prove that there is some C0>0 so that
ZE
|f(x)|dx ≤C0pm(E),for all measurable E⊂R.
(6) Let f(x) be a continuous function on [0,1] with a continuous
derivative f0(x). Given ε > 0, prove that there is a polynomial
p(x) so that
kf(x)−p(x)k∞+kf0(x)−p0(x)k∞< ε.
(7) Let Xbe a non-empty complete metric space and let
{fn:X→R}∞
n=1
be a sequence of continuous functions with the following prop-
erty: for each x∈X, there exists an integer Nxso that {fn(x)}n≥Nx
is either a monotone increasing or decreasing sequence. Prove
that there is a non-empty open subset U⊆Xand an integer
Nso that the sequence {fn(x)}n≥Nis monotone for all x∈U.
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