***Warning: Youâ€™re not allowed to use Riemannâ€“Lebesgue theorem in this
problem set.***
(1) Define f : [0, 1] â†’ R by
f(x) =
ï£±
ï£´ï£´ï£´ï£²
ï£´ï£´ï£´ï£³
1 if 1 âˆ’ 2
âˆ’2k â‰¤ x â‰¤ 1 âˆ’ 2
âˆ’(2k+1) for k = 0, 1, 2, . . .
0 if 1 âˆ’ 2
âˆ’(2k+1) < x < 1 âˆ’ 2
âˆ’(2k+2) for k = 0, 1, 2, . . .
0 if x = 1
R 1
0
Prove that f is integrable on [0, 1] , and compute f(x)dx .
(2) For a bounded function f : [0, 1] â†’ R , define
Rn :=
1
n
k=1
f
k
n
Xn .
(a) Prove that if f is integrable, then limnâ†’âˆž Rn =
R 1
0
f(x)dx .
(b) Find an example of f that is not integrable, but limnâ†’âˆž Rn exists.
(7) Let f be a continuous function on [a, b] . Suppose that
Z b
a
x
n
f(x)dx = 0
for n = 0, 1, 2, . . . . Prove that f(x) = 0 for any x âˆˆ [a, b] .
(8) Let f, g be integrable functions on [a, b] . Prove that
Z b
a
f(x)g(x)
2
â‰¤
Z b
a
f(x)
2
dx Z b
a
g(x)
2
dx
.
When does the equality hold?
Hint: Consider R b
a
(
R b
a
(f(x)g(y) âˆ’ f(y)g(x))2dx)dy .

***Warning: Youâ€™re not allowed to use Riemannâ€“Lebesgue theorem...