with other variants of this notation being similar. Also, the convention, 0 ⋅∞ = 0 will be used to simplify
the presentation whenever it is convenient to do so. The notation a ∧ b means the minimum of a and
b.
Definition 6.1.1Let f :
[a,b]
→
[0,∞ ]
be decreasing. Define
∫ b ∫ b ∫ b
f (λ) dλ ≡ lim M ∧f (λ )dλ = sup M ∧ f (λ)dλ
a M →∞ a M a
where a ∧ b means the minimum of a and b. Note that for f bounded,
∫ b ∫ b
sup M ∧ f (λ)dλ = f (λ)dλ
M a a
where the integral on the right is the usual Riemann integral because eventually M > f. For f anonnegative decreasing function defined on [0,∞),
∫ ∫ ∫ ∫
∞ R R R
0 fdλ ≡ Rli→m∞ 0 fdλ = suRp>1 0 f dλ = suRpMsup>0 0 f ∧ M dλ
Since decreasing bounded functions are Riemann integrable, the above definition is well defined. This
claim should have been seen in calculus, but if not, see Problem 10 on Page 391. Now here is an obvious
property.
Lemma 6.1.2Let f be a decreasing nonnegative function defined on an interval
[a,b]
. Then if
[a,b]
= ∪k=1mIkwhere Ik≡
[ak,bk]
and the intervals Ikare non overlapping, it follows
∫ b ∑m ∫ bk
f dλ = fdλ.
a k=1 ak
Proof: This follows from the computation,
∫ ∫
b b
a fdλ ≡ Mli→m∞ a f ∧ M dλ
∑m ∫ bk m∑ ∫ bk
= lim f ∧ M dλ = fdλ
M→ ∞ k=1 ak k=1 ak