9.17. EXERCISES 233

8. Suppose un(t) is a differentiable function for t ∈ (a,b) and suppose that for t ∈ (a,b),|un(t)|, |u′n(t)|< Kn where ∑

∞n=1 Kn < ∞. Show(∑∞

n=1 un (t))′ = ∑

∞n=1 u′n(t).

Hint: This is an exercise in the use of the dominated convergence theorem and themean value theorem.

9. Suppose { fn} is a sequence of nonnegative measurable functions defined on a mea-sure space, (Ω,S ,µ). Show that

∫∑

∞k=1 fkdµ = ∑

∞k=1

∫fkdµ . Hint: Use the mon-

otone convergence theorem along with the fact the integral is linear.

10. Explain why for each t > 0,x→ e−tx is a function in L1 (R) and∫

∞

0 e−txdx = 1t . Thus∫ R

0

sin(t)t

dt =∫ R

0

∫∞

0sin(t)e−txdxdt

Now explain why you can change the order of integration in the above iterated in-tegral. Then compute what you get. Next pass to a limit as R → ∞ and show∫

∞

0sin(t)

t dt = 12 π. This is a very important integral. Note that the thing on the left

is an improper integral. sin(t)/t is not Lebesgue integrable because it is not ab-solutely integrable. That is

∫∞

0

∣∣ sin tt

∣∣dm = ∞. It is important to understand that theLebesgue theory of integration only applies to nonnegative functions and those whichare absolutely integrable.

11. Let the rational numbers in [0,1] be {rk}∞

k=1 and define

fn (t) ={

1 if t ∈ {r1, · · · ,rn}0 if t /∈ {r1, · · · ,rn}

Show that limn→∞ fn (t) = f (t) where f is one on the rational numbers and 0 on theirrational numbers. Explain why each fn is Riemann integrable but f is not. How-ever, each fn is actually a simple function and its Lebesgue and Riemann integral isequal to 0. Apply the monotone convergence theorem to conclude that f is Lebesgueintegrable and in fact,

∫f dm = 0.

12. Show limn→∞n2n ∑

nk=1

2k

k = 2. This problem was shown to me by Shane Tang, a for-mer student. It is a nice exercise in dominated convergence theorem if you massageit a little. Hint:

n2n

n

∑k=1

2k

k=

n

∑k=1

2k−n nk=

n−1

∑l=0

2−l nn− l

=n−1

∑l=0

2−l(

1+l

n− l

)≤

n−1

∑l

2−l (1+ l)

13. Give an example of a sequence of functions { fn} , fn ≥ 0 and a function f ≥ 0 suchthat f (x) = liminfn→∞ fn (x) but

∫f dm < liminfn→∞

∫fndm so you get strict in-

equality in Fatou’s lemma.

14. Let f be a nonnegative Riemann integrable function defined on [a,b] . Thus there isa unique number between all the upper sums and lower sums. First explain why,if ai ≥ 0,

∫∑

ni=1 aiX[ti,ti−1) (t)dm = ∑i ai (ti− ti−1) . Explain why there exists an in-

creasing sequence of Borel measurable functions {gn} converging to a Borel mea-surable function g, and a decreasing sequence of functions {hn} which are also Borelmeasurable converging to a Borel measurable function h such that gn ≤ f ≤ hn,∫

gndm equals a lower sum,∫

hndm equals an upper sum