238 CHAPTER 11. ABSTRACT MEASURE AND INTEGRATION

To verify the last claim, suppose M <∫

f dµ and let δ > 0 be given. Then there existsh > 0 such that

M <∞

∑i=1

hµ ([ih < f ])≤∫

f dµ.

By the first part of this lemma,

M <∞

∑i=1

h2

µ

([ih2< f])≤∫

f dµ

and continuing to apply the first part,

M <∞

∑i=1

h2n µ

([i

h2n < f

])≤∫

f dµ.

Choose n large enough that h/2n < δ . It follows

M < supδ>h>0

∞

∑i=1

hµ ([ih < f ])≤∫

f dµ.

Since M is arbitrary, this proves the last claim.

11.3.3 The Lebesgue Integral For Nonnegative Simple FunctionsDefinition 11.3.5 A function, s, is called simple if it is a measurable real valued functionand has only finitely many values. These values will never be ±∞. Thus a simple functionis one which may be written in the form

s(ω) =n

∑i=1

ciXEi (ω)

where the sets, Ei are disjoint and measurable. s takes the value ci at Ei.

Note that by taking the union of some of the Ei in the above definition, you can assumethat the numbers, ci are the distinct values of s. Simple functions are important because itwill turn out to be very easy to take their integrals as shown in the following lemma.

Lemma 11.3.6 Let s(ω) = ∑pi=1 aiXEi (ω) be a nonnegative simple function with the ai

the distinct non zero values of s. Then∫sdµ =

p

∑i=1

aiµ (Ei) . (11.3.11)

Also, for any nonnegative measurable function, f , if λ ≥ 0, then∫λ f dµ = λ

∫f dµ. (11.3.12)