Math 524
Homework due 01/10/01
Problem 1. (Prelim) Let f: (0,∞)→Rbe in L1(m). For x > 0 define
g(x) = Z∞
√x/2
f(t)
t2dt.
Show that g∈L1(m), where mdenotes the Lebesque measure, and that
Z∞
0
g(x)dx = 2 Z∞
0
f(x)dx.
Problem 2. (Prelim) Let f: [0,1] →R+be Lebesgue measurable. Let
A={g: [0,1] →R:g∈L1(m),|g| ≤ fa.e. on [0,1]}.
2.1 Show that Ais a closed subset of L1.
2.2 Show that if Ais compact in L1(m) then fmust be in L1(m).
Problem 3. (Prelim) Show that for every continuous real valued function fdefined on
[-1,1]
lim
n→∞ R1
−1f(x)x2ndx
R1
−1x2ndx
exists. Give the value of the limit in terms of f.
Problems from Folland
Chapter 3, Section 5: problem 41, 42.
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