1
exercise 1: R
R
(i). Compute lim inf fn and lim inf fn for the sequence of functions from R to R
fn = 1[n,n+1] .
(ii). Find a sequence of measurable functions fn from [0, 1] to [0, ∞] such that
Z
Z
lim inf fn 6= lim inf fn
(iii). Find a sequence of measurable functions fn from R to [0, ∞] such that fn converges
uniformly to zero and
Z
Z
lim inf fn 6= lim inf fn
exercise 2:
Find (with proof)
Z
∞
lim
n→∞
−∞
(sin x)n
dx
x2
.
exercise 3:
(a). Let fn be a sequence of measurable functions on [a, b] valued in R and converging uniZ b
formly to zero. Use the D.C.T. to show that
fn converges to 0.
a
1
(b). Let
Z fn be a sequence of measurable functions in L (R) converging uniformly to zero.
Does
fn converge to 0?
R
exercise 4:
Problem 20, chapter 4: only the first question
exercise 5:
Problem 21, chapter 4.
exercise 6:
Z
For t in [0, ∞), define F (t) =
0
∞
e−x
2 − t2
x2
dx.
a. Show that F is continuous on [0, ∞).
b. Show that F is differentiable
on (0, ∞) and form a differential equation for F .
√
π
c. Using that F (0) = 2 , express F (t) using known functions.
2
exercise 7
Let
Z
F (u) =
0
∞
e−xu
sin x
dx
x
a. Show that F is differentiable on (0, ∞) and that F 0 (u) = −1/(1 + u2 ).
b. Show that lim F (u) = 0.
u→∞
c. Show that the function x → sinx x is not in L1 ([0, ∞)). Show that the limit of
Z A
sin x
dx
as A tends to infinity exists (Hint: integrate by parts ). That limit is
x Z
0
∞
sin x
dx.
denoted
x
0
Z ∞
sin x
d. Infer the value of
dx. Hint:
x
0
Z
e−xu cos (x) ue−xu sin (x)
1 − e−xu sin (x) dx =
+
− cos (x)
u2 + 1
u2 + 1