6-ConceptsProbability Page 173 Wednesday, February 4, 2004 3:00 PM
Concepts of Probability 173
For each measure M, the integral is a number that is associated to
every integrable function f. It is defined in the following two steps:
■ Step 1. Suppose that f is a measurable, non-negative function and con-
sider a finite decomposition of the space Ω, that is to say a finite collection
of disjoint subsets Ai ⊂ Ω whose union is Ω:
Ai ⊂ Ω such that Ai ∩ Ai = ∅ for i ≠ j and ∪ Ai = Ω
Consider the sum
∑inf(f ()ω: ω ∈ Ai )MA() i
i
The integral
∫ fMd
Ω
is defined as the supremum, if it exists, of all these sums over all possible
decompositions of Ω. Suppose that f is bounded and non-negative and
M(Ω) < ∞. Let’s call
S– = sup∑(inf f ()ωMA()i)
ω ∈ Ai
i
the lower integral and
()MA
S i
+
= inf∑(sup f ω ())
i ω ∈ Ai
the upper integral. It can be demonstrated that if the integral exists then
S+= S–. It is possible to define the integral as the common value S = S+=
S–. This approach is the Darboux-Young approach to integration.^7
■ Step 2. Given a measurable function f not necessarily non-negative,
consider its decomposition in its positive and negative parts f = f +– f –.
The integral of f is defined as the difference, if a difference exists,
between the integrals of its positive and negative parts.
(^7) See Patrick Billingsley, Probability and Measure, Second edition (New York: Wiley,
1985).