11.8. EXERCISES 267

11. Let (Ω,F ,µ) be a measure space and suppose fn converges uniformly to f and thatfn is in L1(Ω). When is

limn→∞

∫fndµ =

∫f dµ?

12. 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.

13. Show that {∑∞i=1 2−nµ ([i2−n < f ])} for f a nonnegative measurable function is an

increasing sequence. Could you define∫f dµ ≡ lim

n→∞

∞

∑i=1

2−nµ([

i2−n < f])

and would it be equivalent to the above definitions of the Lebesgue integral?

14. 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 monotone convergence theorem along with the fact the integral islinear.