334 CHAPTER 14. THE LEBESGUE INTEGRAL
14.6 Riemann Integrals for Decreasing FunctionsThis continues the abstract development but here it is tied in to the ordinary theory ofRiemann integration for real valued functions. A decreasing function is always Riemannintegrable with respect to the integrator function F (t) = t. This is because the function isof bounded variation and the integrator function is continuous. You can also show directlythat there is a unique number between the upper and lower sums. I will define the Lebesgueintegral for a nonnegative function in terms of an improper Riemann integral which involvesa decreasing function.
Definition 14.6.1 Let f : [a,b]→ [0,∞] be decreasing. Define∫ b
af (λ )dλ ≡ lim
M→∞
∫ b
aM∧ f (λ )dλ = sup
M
∫ b
aM∧ f (λ )dλ
where A∧B means the minimum of A and B. Note that for f bounded,
supM
∫ b
aM∧ f (λ )dλ =
∫ b
af (λ )dλ
where the integral on the right is the usual Riemann integral because eventually M > f .For f a nonnegative decreasing function defined on [0,∞),∫
∞
0f dλ ≡ lim
R→∞
∫ R
0f dλ = sup
R>1
∫ R
0f dλ = sup
RsupM>0
∫ R
0f ∧Mdλ
Now here is an obvious property.
Lemma 14.6.2 Let f be a decreasing nonnegative function defined on an interval [a,b] .Then if [a,b] =∪m
k=1Ik where Ik ≡ [ak,bk] and the intervals Ik are non overlapping, it follows
∫ b
af dλ =
m
∑k=1
∫ bk
ak
f dλ .
Proof: This follows from the computation,∫ b
af dλ ≡ lim
M→∞
∫ b
af ∧Mdλ = lim
M→∞
m
∑k=1
∫ bk
ak
f ∧Mdλ =m
∑k=1
∫ bk
ak
f dλ
Note both sides could equal +∞.
14.7 Lebesgue Integrals of Nonnegative FunctionsHere is the definition of the Lebesgue integral of a function which is measurable and hasvalues in [0,∞]. The idea is motivated by the following picture in which f−1 (λ i,∞) isA∪B∪C and we take the measure of this set, multiply by λ i−λ i−1 and do this for each λ iin an increasing sequence of points, λ 0 ≡ 0. Then we add the “areas” of the little horizontal“rectangles” in order to approximate the “area” under the curve. The difference here isthat the “rectangles” in the sum are horizontal whereas with the Riemann integral, theyare vertical. Note how it is important to be able to measure f−1 (λ ,∞)≡ {x : f (x)> λ} ≡