620 CHAPTER 20. REPRESENTATION THEOREMS

and so the same formula for the integral of a simple function is obtained in this case also.Now consider two simple functions

s =n

∑k=1

akXEk , t =m

∑j=1

b jXFj

where the ak and b j are the distinct values of the simple functions. Then from what wasjust shown,

∫(αs+β t)dν =

∫ ( n

∑k=1

m

∑j=1

αakXEk∩Fj +m

∑j=1

n

∑k=1

βb jXEk∩Fj

)dν

=∫ (

∑j,k

αakXEk∩Fj +βb jXEk∩Fj

)dν

= ∑j,k(αak +βb j)ν (Ek ∩Fj)

=n

∑k=1

m

∑j=1

αakν (Ek ∩Fj)+m

∑j=1

n

∑k=1

βb jν (Ek ∩Fj)

=n

∑k=1

αakν (Ek)+m

∑j=1

βb jν (Fj)

= α

∫sdν +β

∫tdν

Thus the integral is linear on simple functions so, in particular, the formula given in theabove definition is well defined regardless.

So what about the definition for f ∈ L∞ (Ω; µ)? Since f ∈ L∞, there is a set of µ mea-sure zero N such that on NC there exists a sequence of simple functions which convergesuniformly to f on NC. Consider sn and sm. As in the above, they can be written as

p

∑k=1

cnkXEk ,

p

∑k=1

cmk XEk

respectively, where the Ek are disjoint having union equal to Ω. Then by uniform conver-gence, if m,n are sufficiently large,

∣∣cnk− cm

k

∣∣< ε or else the corresponding Ek is containedin NC a set of ν measure 0 thanks to ν ≪ µ . Hence∣∣∣∣∫ sndν−

∫smdν

∣∣∣∣ =

∣∣∣∣∣ p

∑k=1

(cnk− cm

k )ν (Ek)

∣∣∣∣∣≤

p

∑k=1|cn

k− cmk | |ν (Ek)| ≤ ε ||ν ||

and so the integrals of these simple functions converge. Similar reasoning shows that thedefinition is not dependent on the choice of approximating sequence.

620 CHAPTER 20. REPRESENTATION THEOREMSand so the same formula for the integral of a simple function is obtained in this case also.Now consider two simple functionswhere the a; and b; are the distinct values of the simple functions. Then from what wasjust shown,| (as+Brav = / (z Ys aa, Zinn + y y; been dvB= / [Ean tin + Bin dvDkY (cay + Bbj) V (Ex OF)Lky Aagv (Ex Fj) +imye BbjV (Ex N Fj)db.Y eaves (Ex) Lave= x fats +p [uvThus the integral is linear on simple functions so, in particular, the formula given in theabove definition is well defined regardless.So what about the definition for f € L® (Q;u)? Since f € L®, there is a set of fp mea-sure zero N such that on N© there exists a sequence of simple functions which convergesuniformly to f on NC. Consider s, and s,,. As in the above, they can be written asP PLen, Vet 2,k=1 k=lrespectively, where the F;, are disjoint having union equal to Q. Then by uniform conver-gence, if m,n are sufficiently large, ct| < € or else the corresponding EF; is containedin N© a set of v measure 0 thanks to v < pt. Hence[sav- [nav(ck — cf’) V (Ex)MsIl1Ms»< Ice — cK || (Ex)| S € |||kIlmnand so the integrals of these simple functions converge. Similar reasoning shows that thedefinition is not dependent on the choice of approximating sequence. J