221 Analysis 2, 2008–09
Exercise sheet 5
Due April 30th 2009
In these questions, the words “measurable” and “integrable” should be taken
to mean “Lebesgue measurable” and “Lebesgue integrable”.
1. Either give an example, or explain why no such example exists, of:
(a) a (Lebesgue) measurable function f : R → R which is not integrable;
(b) an integrable function f : R → R such that −f is not integrable;
(c) an integrable function f : R → R such that f 2 is not integrable;
(d) a convergent sequence of integrable functions fn : R → R such that
lim fn is not integrable.
n→∞
2. Evaluate the following expressions, carefully justifying your working.
Z ∞
Z n
Z 1
1
sin(n2 x2 )
1 + nx2
√
(a)
dx
(c)
lim
dx
dx
(b) lim
n→∞ −n n2 |x|3/2
n→∞ 0 (1 + x)n
1 + x2
−∞
3. (a) Let a, b ∈ R with a < b and let f, g : [a, b] → R be continuously differentiable. Use the Fundamental Theorem of Calculus to show that
Z b
Z b
′
f (x)g (x) dx = f (b)g(b) − f (a)g(a) −
f ′ (x)g(x) dx.
a
[Hint: what is
Rb
a
a
(f g)′(x) dx?]
(b) Show that if f, g : R → R are continuously differentiable functions and
f (x)g(x) → L ∈ R as x → ∞, and f g ′ χ[0,∞) and f ′ gχ[0,∞) are both
integrable, then
Z ∞
Z ∞
′
f (x)g (x) dx = L − f (0)g(0) −
f ′ (x)g(x) dx.
0
0
R∞
(c) Fix a number t ∈ R. Show that 0 e−x cos(xt) dx = (1+t2)−1 , justifying
your working.
[Hint: call the value of this integral J. Integrate by parts twice (justify
using (b)) to find an equation that J satisfies, and then solve it.]
R∞
(d) For t ∈ R, let I(t) = 0 x−1 e−x sin(xt) dx. Show that I ′ (t) = (1 + t2)−1 ,
and solve this differential equation to find I(t).
4. Let (an )n≥1 be a sequence in [−∞, ∞]. Prove the following assertions.
(a) lim sup(−an ) = − lim inf an
n→∞
n→∞
(b) an converges as n → ∞ if and only if lim inf an = lim sup an .
n→∞
n→∞
(c) If an → a as n → ∞ then a = lim inf an = lim sup an .
n→∞
n→∞