328 CHAPTER 14. THE LEBESGUE INTEGRAL

where xk+1− xk ≤ 2ε and xm+1− xm ≤ 2ε . In writing down a sum corresponding to thisδ ε division, it reduced to F (xm+1)−F (xk). Letting ε → 0 yields the integral and it equalsF (d+)−F (c−). this shows 1.).

For 2.) (c,d) = ∪k[c+ 1

k ,d−1k

]for all j suitably large. Thus from Lemma 14.1.4,

µF ((c,d)) = limk→∞

µF

([c+

1k,d− 1

k

])= lim

k→∞

(F((

d− 1k

)+

)−F

((c+

1k

)−))

= limk→∞

(F(

d− 1k

)−F

(c+

1k

))= F (d−)−F (c+)

For 3.) similar reasoning to the above using (c,d] = ∪k[c+ 1

k ,d],

µF ((c,d]) = limk→∞

F (d+)−F(

c+1k

)= F (d+)−F (c+) .

Part 4.) is entirely similar.

14.4 RegularityThis has to do with approximating with certain special sets. This is of utmost importance ifyou want to use the Lebesgue integral in any significant way, especially for various functionspaces. I will show that under reasonable conditions this needed regularity is automatic.

Definition 14.4.1 A set is called Fσ if it is the countable union of closed sets. Aset is called Gδ if it is the countable intersection of open sets.

Lemma 14.4.2 If A is an Fσ set, then if I is any interval, finite or infinite, A∩ I is alsoan Fσ set. If A is a Gδ set, then if I is any interval, then I∩A is also a Gδ set.

Proof: Consider the following example in which I = [a,b). Say A = ∪∞k=1Hk where Hk

is closed. I = ∪∞j=1

[a,b− 1

j

]and so

A∩ I = (∪∞k=1Hk)∩∪∞

j=1

[a,b− 1

j

]= ∪∞

j=1

[a,b− 1

j

]∩∪∞

k=1Hk

= ∪∞j=1∪∞

k=1

(Hk ∩

[a,b− 1

j

])which is still an Fσ set. It is still a countable union of closed sets. Other cases are similar.

Now consider the case where A is Gδ . Say A = ∩∞k=1Vk for Vk open. Consider the same

half open interval. In this case, [a,b) = ∩∞j=1

(a− 1

j ,b)

. Then

A∩ I = ∩∞k=1Vk ∩∩∞

j=1

(a− 1

j,b)= ∩∞

j=1∩∞k=1

(Vk ∩

(a− 1

j,b))

which is a Gδ set, still being a countable intersection of open sets. Other cases are similar.