A.4. WHICH FUNCTIONS ARE INTEGRABLE? 855

Theorem A.4.10 If a bounded set E, has Jordan content 0, then E is a Jordan (contented)set and if f is any bounded function defined on E, then f XE ∈R (Rn) and∫

Ef dV = 0.

Proof: Let m be a lower bound for f and let M be an upper bound. Let G be a grid with

∑Q∩E ̸= /0

v(Q)<ε

1+(M−m).

ThenUG ( f XE)≤ ∑

Q∩E ̸= /0Mv(Q)≤ εM

1+(M−m)

andLG ( f XE)≥ ∑

Q∩E ̸= /0mv(Q)≥ εm

1+(M−m)

and so

UG ( f XE)−LG ( f XE) ≤ ∑Q∩E ̸= /0

Mv(Q)− ∑Q∩E ̸= /0

mv(Q)

= (M−m) ∑Q∩E ̸= /0

v(Q)<ε (M−m)

1+(M−m)< ε.

This shows f XE ∈R (Rn). Now also,

mε ≤∫

f XE dV ≤Mε

and since ε is arbitrary, this shows∫E

f dV ≡∫

f XE dV = 0

Why is E contented? Let G be a grid for which

∑Q∩E ̸= /0

v(Q)< ε

Then for this grid,UG (XE)−LG (XE)≤ ∑

Q∩E ̸= /0v(Q)< ε

and this proves the theorem. ■

Corollary A.4.11 If fXEi ∈R (Rn) for i = 1,2, · · · ,r and for all i ̸= j,Ei∩E j is either theempty set or a set of Jordan content 0, then letting F ≡ ∪r

i=1Ei, it follows f XF ∈R (Rn)and ∫

f XF dV ≡∫

Ff dV =

r

∑i=1

∫Ei

f dV.