The Mathematics of Financial Modelingand Investment Management

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).